arXiv:1201.1324v1 [math.AG] 5 Jan 2012 The geometry of blueprints Part II: Tits-Weyl models of algebraic groups Oliver Lorscheid ∗ Abstract This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called F 1 , the field with one element. Based on Part I of The geometry of blueprints, we introduce the class of Tits morphisms between blue schemes. The resulting Tits category Sch T comes together with a base extension to (semiring) schemes and the so-called Weyl extension to sets. We prove for G in a wide class of Chevalley groups—which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups—that a linear representation of G defines a model G in Sch T whose Weyl exten- sion is the Weyl group W of G . We call such models Tits-Weyl models. The potential of Tits-Weyl models lies in (a) their intrinsic definition that is given by a linear representa- tion; (b) the (yet to be formulated) unified approach towards thick and thin geometries; and (c) the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry. ∗ Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42097 Wuppertal, Germany, [email protected]. 1
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12 The geometry of blueprints
Part II: Tits-Weyl models of algebraic groups
Oliver Lorscheid∗
Abstract
This paper is dedicated to a problem raised by Jacquet Tits in1956: the Weyl group ofa Chevalley group should find an interpretation as a group over what is nowadays calledF1,the field with one element. Based on Part I of The geometry of blueprints, we introduce theclass ofTits morphismsbetween blue schemes. The resultingTits categorySchT comestogether with a base extension to (semiring) schemes and theso-calledWeyl extensiontosets.
We prove forG in a wide class of Chevalley groups—which includes the special andgeneral linear groups, symplectic and special orthogonal groups, and all types of adjointgroups—that a linear representation ofG defines a modelG in SchT whose Weyl exten-sion is the Weyl groupW of G . We call such modelsTits-Weyl models. The potential ofTits-Weyl models lies in(a) their intrinsic definition that is given by a linear representa-tion; (b) the (yet to be formulated) unified approach towards thick andthin geometries;and (c) the extension of a Chevalley group to a functor on blueprints, which makes it, inparticular, possible to consider Chevalley groups over semirings. This opens applicationsto idempotent analysis and tropical geometry.
∗Department of Mathematics, University of Wuppertal, Gaußstr. 20, 42097 Wuppertal, Germany,[email protected].
One of the main themes ofF1-geometry was and is to give meaning to an idea of Jacques Titsthat dates back to 1956 (see Section 13 in [33]). Namely, Titsproposed that there should be atheory of algebraic groups over a field of “caracteristiqueune”, which explains certain analogiesbetween geometries over finite fields and combinatorics.
There are good expositions of Tits’ ideas from a modern viewpoint (for instance, [31], [10]or [24]). We restrict ourselves to the following example that falls into this line of thought. Thenumber ofFq-rational points GLn(Fq) of the general linear group is counted by a polynomialN(q) in q with integral coefficients. The limit limq→1N(q)/(q−1)n counts the elements of theWeyl groupW = Sn of GLn. The same holds for any standard parabolic subgroupP of GLn
whose Weyl groupWP is a parabolic subgroup of the Weyl groupW. While the group GLn(Fq)acts on the coset space GLn/P(Fq), which are theFq-rational points of a flag variety, the WeylgroupW = Sn acts on the quotientW/WP, which is the set of decompositions of{1, . . . ,n} intosubsets of cardinalities that correspond to the flag type of GLn/P.
The analogy of Chevalley groups over finite fields and their Weyl groups enteredF1-geometryas the slogan:F1-geometry should provide anF1-modelG of every Chevalley groupG whosegroupG(F1) = Hom(SpecF1,G) of F1-rational points equals the Weyl groupW of G . Manyauthors contributed to this problem: see [21], [26], [31], [15], [19], [35], [10], [23], [6], [24],[16] (this list is roughly in the order of appearance, without claiming to be complete).
However, there is a drawback to this philosophy. Recall thatthe Weyl groupW of a Cheval-ley groupG is defined as the quotientW = N(Z)/T(Z) whereT is a split maximal torus ofG and N is its normalizer inG . Under certain natural assumptions, a group isomorphismG(F1)
∼→W yields an embeddingW → N(Z) of groups that is a section of the quotient mapN(Z)→W. However, such a section does exist in general as the exampleG = SL2 witnesses(see Problem B in the introduction of [24] for more detail).
This problem was circumvented in different ways. While someapproaches restrict them-selves to treat only a subclass of Chevalley groups overF1 (in the case of GLn, for instance, onecan embed the Weyl group as the group of permutation matrices), other papers describe Cheval-ley groups merely as schemes without mentioning a group law.The more rigorous attempts toestablish Chevalley groups overF1 are the following two approaches. In the spirit of Tits’ laterpaper [34], which describes the extended Weyl group, Connesand Consani tackled the problemby considering schemes overF12 (see [10]), which stay in connection with the extended Weylgroup in the case of Chevalley groups. In the author’s earlier paper [24], two different classes ofmorphisms were considered: while rational points are so-called strong morphisms, group lawsare so-calledweak morphisms.
In this paper, we choose a different approach: we break with the convention thatG(F1)should be the Weyl group ofG . Instead, we consider a certain category SchT of F1-schemes
3
that comes together with “base extension” functors(−)Z : SchT →SchZ1 to usual schemes andW : SchT →Setsto sets. Roughly speaking, aTits-Weyl modelof a Chevalley groupG is anobjectG in SchT together with a morphismµ : G×G→ G such thatGZ together withµZ isisomorphic toG as a group scheme and such thatW (G) together withW (µ) is isomorphic tothe Weyl group ofG . We call the category SchT theTits categoryand the functorW the Weylextension.
A first heuristic
Before we proceed with a more detailed description of the Tits category, we explain the funda-mental idea of Tits-Weyl models in the case of the Chevalley group SL2. The standard definitionof the scheme SL2,Z is as the spectrum ofZ[SL2] = Z[T1,T2,T3,T4]/(T1T4−T2T3−1), which isa closed subscheme ofA4
Z = SpecZ[T1,T2,T3,T4]. The affine spaceA4Z has anF1-model in the
language of Deitmar’sF1-geometry (see [15]). Namely,A4F1
= SpecF1[T1,T2,T3,T4] where
F1[T1,T2,T3,T4] = {Tn11 Tn2
2 Tn33 Tn4
4 }n1,n2,n3,n4≥0
is the monoid2 of all monomials inT1, T2, T3 andT4. Its prime ideals are the subsets
(Ti)i∈I = {Tn11 Tn2
2 Tn33 Tn4
4 }ni>0 for onei∈I
of F1[T1,T2,T3,T4] whereI ranges to all subsets of{1,2,3,4}. Note that this means(Ti)i∈I = /0for I = /0. ThusA4
F1= {(Ti)i∈I}I⊂{1,2,3,4}.
If one applies the naive intuition that prime ideals are closed under addition and subtractionto the equation
T1T4 − T2T3 = 1,
then the points ofSL2,F1 should be the prime ideals(Ti)i∈I that do not contain both termsT1T4andT2T3. This yields the set SL2,F1 = {( /0),(T1),(T2),(T3),(T4),(T1,T4),(T2,T3)}, which canbe illustrated as
(T3) (T4)
(T1,T4)(T2,T3)
(T2) (T1)
/0
1Note a slight incoherence with the notation of the main text of this paper where the functor(−)Z is denotedby (−)+Z . We will omit the superscript “+” also at other places of the introduction to be closer to the standardnotation of algebraic geometry. An explanation for the needof the additional superscript is given in Section 1.1.
2For the sake of simplification, we do not requireF1[T1,T2,T3,T4] to have a zero. This differs from the con-ventions that are used in the main text, but this incoherencedoes not have any consequences for the followingconsiderations.
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where the vertical lines express the inclusion relation(Ti)i∈J ⊂ (Ti)i∈I . The crucial observationis that the two maximal ideals(T2,T3) and (T1,T4) of this set correspond to the subscheme{( ∗ 0
0 ∗)}
of diagonal matrices and the subscheme{(
0 ∗∗ 0
)}of anti-diagonal matrices of SL2,Z,
respectively, which, in turn, correspond to the elements ofthe Weyl groupW = N(Z)/T(Z)whereT =
{( ∗ 00 ∗)}
is the diagonal torus andN its normalizer.This example was the starting point for the development of the geometry of blueprints. A
formalism that puts the above ideas on a solid base is explained in the preceeding Part I of thispaper (see [25]). Please note that we give brief definitions of blueprints and blue schemes in theintroduction of Part I. In the proceeding, we will assume that the reader is familiar with this.
The Tits category
It is the topic of this paper to generalize the above heuristics to other Chevalley groups andto introduce a class of morphisms that allows us to descend group laws to morphisms of theF1-model of Chevalley groups. Note that the approach of [24] isof a certain formal similarity:the tori of minimal rank in a torification of SL2,Z are the diagonal torus and the anti-diagonaltorus. Indeed the ideas of [24] carry over to our situation.
The rank space Xrk of a blue schemeX is the set of the so-called “points of minimal rank”(which would be the points(T2,T3) and (T1,T4) in the above example) together with certainalgebraic data, which makes it a discrete blue scheme. ATits morphismϕ : X→Y between twoblue schemesX andY will be a pairϕ= (ϕrk,ϕ+) of a morphismϕrk : X→Y between the rankspaces and a morphismϕ+ : X+→Y+ between the associated semiring schemes3 X+ =XN andY+ =YN that satisfy a certain compatibility condition.
The Tits category SchT is defined as the category of blue schemes together with Tits mor-phisms. The Weyl extensionW : SchT → Sets is the functor that sends a blue schemeXto the underlying setW (X) of its rank spaceXrk and a Tits morphismϕ : X → Y to the un-derlying mapW (ϕ) : W (X)→ W (Y) of the morphismϕrk : Xrk→ Yrk. The base extension(−)Z : SchT → SchZ sends a blue schemeX to the schemeX+
Zand a Tits morphismϕ : X→Y
to the morphismϕ+Z
: X+Z→ Y+
Z. Note that we can replaceZ by a semiringk, which yields a
base extension(−)k : SchT → Schk for every semiringk. We obtain the diagram
Sets
SchT
W22❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢
(−)+,,❳❳❳❳❳
❳❳❳❳❳❳
❳❳❳❳
SchN(−)k // Schk
.
3Please note that we avoid the notation “XN” from the preceeding Part I of this paper for reasons that areexplained in Section 1.1.
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Results and applications
The main result of this paper is that a wide class of Chevalleygroups has a Tits-Weyl model.This includes the special and the general linear groups, symplectic groups, special orthogonalgroups (of both typesBn andDn) and all Chevalley groups of adjoint type. Next to this, weobtain Tits-Weyl models for split tori, parabolic subgroups of Chevalley groups and their Levi-subgroups.
The strength of the theory of Tits-Weyl models can be seen in the following reasons. Thisputs it, in particular, in contrast to earlier approaches towardsF1-models of algebraic groups,
Intrinsic definition through explicit formulas
Tits-Weyl models are determined by explicit formulas (asT1T4−T2T3 = 1 in the case of SL2),which shows that Tits-Weyl models are geometric objects that are intrinsically associated torepresentations in terms of generators and relations of theunderlying scheme. The examplesin Appendix A show that they are indeed accessible via explicit calculations. In other words,we can say that every linear representation of a group schemeG yields anF1-modelG. Thegroup law ofG descends uniquely (if at all) to a Tits morphismµ : G×G→ G that makesG aTits-Weyl model ofG .
Unified approach towards thick and thin geometries
Tits-Weyl models combine the geometry of algebraic groups (over fields) and the associatedgeometry of their Weyl groups in a functorial way. This has applications to a unified approachtowards thick and thin geometries as alluded by Jacques Titsin [33]. A treatment of this will bethe matter of subsequent work.
Functorial extension to blueprints and semirings
A Chevalley groupG can be seen as a functorhG from rings to groups. A Tits-Weyl modelG of G can be seen as an extension ofhG to a functorhG from blueprints to monoids whosevalueshG(F1) andhG(F12) stay in close connection to the Weyl group and the extended Weylgroup (see Theorem 3.14). In particular,hG is a functor on the subclass of semirings. Thisopens applications to geometry that is build on semirings; by name, to idempotent analysis asconsidered by Kolokoltsov and Maslov, et al. (see, for instance, [27]), tropical geometry as con-sidered Itenberg, Mikhalkin, et al. (see, for instance, [18], [28] and, in particular, [29, Chapter2]), idempotent geometry that mimicsF1-geometry (see [11], [22] and [32]) and analytic ge-ometry from the perspective of Paugam (see [30]), which generalizes Berkovich’s and Huber’sviewpoints on (non-archimedean) analytic geometry (see [4], [5] and [20]).
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Remarks and open problems
The guiding idea in the formulation of the theory of Tits-Weyl models is to descend algebraicgroups “as much as possible”. This requires us to relinquishmany properties that are knownfrom the theory of group schemes, and to substitute these losses by a formalism that has all thedesired properties, which are, roughly speaking, that the category and functors of interest areCartesian and that Chevalley groups have a model such that its Weyl group is given functorially.As a consequence, we yield only monoids instead of group objects and there are no directgeneralizations to relative theories—with one exception:there is a good relative theory overF12.Tits monoids overF12 are actually much easier to treat: the rank space has a simpler definitionthat does not require inverse closures, the universal semiring scheme is a scheme, Tits-Weylmodels overF12 are groups in SchT and many subtleties in the proofs about the existence of−1 in certain blueprints vanish. Note that the Tits-Weyl models that are established in thispaper, immediately yield Tits-Weyl models overF12 by the base extension−⊗F1 F12 from F1
to F12.The strategy of this paper is to establish Tits-Weyl models by a case-by-case study. There are
many (less prominent) Chevalley groups that are left out. Only for adjoint Chevalley groups, weconstruct Tits-Weyl models in a systematic way by considering their root systems. This raisesthe problem of the classification of Tits-Weyl models of Chevalley groups. In particular, thefollowing questions suggest themselves.
• Does every Chevalley group have a Tits-Weyl model? Is there asystematic way to estab-lish such Tits-Weyl models?
• As explained before, a linear representation of a Chevalleygroup defines a unique Tits-Weyl model if at all. When do different linear representations of Chevalley groups leadto isomorphic Tits-Weyl models? Can one classify all Tits-Weyl models in a reasonableway?
• Every Tits-Weyl model of a Chevalley group in this text comesfrom a “standard” rep-resentation of the Chevalley group. Can one find a “canonical” Tits-Weyl model? Whatproperties would such a canonical Tits-Weyl model have among all Tits-Weyl models ofthe Chevalley group?
See Appendix A.2 for the explicit description of some Tits-Weyl models of typeA1.
Content overview
The paper is organized as follows. In Section 1, we provide the necessary background onblue schemes to define the rank space of a blue scheme and the Tits category. This section
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contains a series of results that are of interest of its own while other parts are straightforwardgeneralizations of facts that hold in usual scheme theory (as the results on sober spaces, closedimmersions, reduced blueprints and fibres of morphisms). Wetry to keep these parts short andomit some proofs that are in complete analogy with usual scheme theory. Instead, we remarkoccasionally on differences between the theory for blue schemes and classical results.
The more innovative parts of Section 1 are the following. In Section 1.5, we investigate thefact that a blue field can admit embeddings into semifields of different characteristics, whichleads to the distinction of thearithmetic characteristicand thepotential characteristicsof a bluefield and of a pointx of a blue scheme. Section 1.6 shows that the base extension morphismαX : XN → X is surjective; in caseX is cancellative, also the base extension morphismβX :XZ→ X is surjective. From the characterization of prime semifields in Section 1.5, it followsthat the points of a blue scheme are dominated by algebraic geometry over algebraically closedfields and idempotent geometry over the semifieldB1 = {0,1}�〈1+1≡ 1〉. In Section 1.7,we investigate the underlying topological space of the fibreproduct of two blue schemes. Incontrast to usual scheme theory, these fibre products are always a subset of the Cartesian productof the underlying sets. In Section 1.8, we definerelative additive closures, a natural procedure,which will be of importance for the definition of rank spaces in the form ofinverse closures.As a last piece of preliminarily theory, we introduceunit fieldsandunit schemesin Section 1.9.Namely, the unit field of a blueprintB is the subblueprintB⋆ = {0}∪B× of B, which is a bluefield.
In Section 2, we introduce the Tits category. In particular,we definepseudo-Hopf pointsand therank spacein Section 2.1 and investigate the subcategory Schrk
F1of blue scheme that
consists of rank spaces. Such blue schemes are calledblue schemes of pure rank. In Section2.2, we defineTits morphismsand investigate its connections with usual morphisms betweenblue schemes. In particular, we will see that the notions of usual morphisms and Tits morphismscoincide on the common subcategories of semiring schemes and blue schemes of pure rank.
In Section 3, we introduce the notions of aTits monoidand of Tits-Weyl models. Afterrecalling basic definitions and facts on groups and monoids in Cartesian categoriesin Section3.1, we show in Section 3.2 that the Tits category as well as some other categories and functorsbetween them are Cartesian. In Section 3.3, we are finally prepared to define a Tits monoid asa monoid in SchT and a Tits-Weyl model of a group schemeG as a Tits monoid with certainadditional properties as described before. As first applications, we establish constant groupschemes and tori as Tits monoids in Schrk
F1in Section 3.4. Tori and certain semi-direct products
of tori by constant group schemes, as they occur as normalizers of maximal tori in Chevalleygroups, have Tits-Weyl models in Schrk
F1.
In Section 4, we establish Tits-Weyl models for a wide range of Chevalley groups. As a firststep, we introduce the Tits-Weyl model SLn of the special linear group in Section 4.1. All otherTits-Weyl models of Chevalley groups will be realized by an embedding of the Chevalley group
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into a special linear group. In order to do so, we will frequently use an argument, which wecall thecube lemma, to descend morphisms. In Section 4.3, we prove the core result Theorem4.7, which provides a Tits-Weyl model for subgroups of a group scheme with a Tits-Weyl modelunder a certain hypothesis on the position of a maximal torusand its normalizer in the subgroup.We apply this to describe the Tits-Weyl model of general linear groups, symplectic groups andspecial orthogonal groups and some of their isogenies like adjoint Chevalley groups of typeAn
and orthogonal groups of typeDn. In Section 4.4, we describe Tits-Weyl models of Chevalleygroups of adjoint type that come from the adjoint representation of the Chevalley group on itsLie algebra. This requires a different strategy from the cases before and is based on formulasfor the adjoint action over algebraically closed fields.
In Section 5, we draw further conclusions from Theorem 4.7. If G is a Chevalley groupwith a Tits Weyl model, then certain parabolic subgroups ofG and their Levi subgroups haveTits-Weyl models. We comment on unipotent radicals, but theproblem of Tits-Weyl models oftheir unipotent radicals stays open.
We conclude the paper with Appendix A, which contains examples of non-standard Tits-Weyl models of tori and explicit calculations for three Tits-Weyl models of typeA1.
1 Background on blue schemes
In this first part of the paper, we establish several general results on blue schemes that we willneed to introduce the Tits category and Tits-Weyl models.
1.1 Notations and conventions
To start with, we will establish certain notations and conventions used throughout the paper. Weassume in general that the reader is familiar with the first part [25] of this work. Occasionally,we will repeat facts if it eases the understanding, or if a presentation in a different shape isuseful. For the purposes of this paper, we will, however, slightly alter notations from [25] asexplained in the following.
All blueprints are proper and with a zero
The most important convention—which might lead to confusion if not noticed—is that wechange a definition of the preceding paper [25], in which we introduced blueprints and blueschemes:
Whenever we refer to a blueprint or a blue scheme in this paper, we understandthat it is proper and with0.
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When we make occasional use of the more general definition of ablueprint as in [25], then wewill refer to it as ageneral blueprint. In [25], we denoted the category of proper blueprints with0 byBl pr0. There is a functor(−)0 from the categoryBl pr of general blueprints toBl pr0.
While for a monoidA and a pre-additionR on A, we denoted byB = A�R the generalblueprint with underlying monoidA, we mean in this paper byA�R the proper blueprintBprop
with 0, whose underlying monoidA′ differs in general fromA. Namely,A′ is a quotient ofA∪{0}.
To acknowledge this behaviour, we will callA�R a representation of Bif B= A�R. If Ais the underlying monoid ofB, then we callA�R theproper representation of B (with0).
We say that a morphism between blueprint issurjectiveif it is a surjective map betweenthe underlying monoids. In other words,f : B→ C is surjective if for allb ∈ C, there is ana∈ B such thatb= f (a). If B= A�R andC = Spec�A′R ′ are representations, which do notnecessarily have to be proper, andf : A→ A′ is a surjective map, thenf : B→C is a surjectivemorphism of blueprints.
Note that the canonical morphismB→ B0 for a general blueprintB induces a homeomor-phism between their spectra. To see this, remember that the proper quotient is formed by iden-tifying a,b ∈ B if they satisfya≡ b. If a≡ b, then a prime ideal ofB contains either bothelements or none. Since every ideal contains 0 ifB has a zero, it follows that the spectra ofBandBprop are homeomorphic.
Accordingly, we refer to proper blue schemes with 0 simply byblue schemes, and call blueschemes in the sense of [25]generalblue schemes. IfX is a general blue scheme, thenX0→ Xis a homeomorphism, thus we might make occasional use of general blue schemes if we areonly concerned with topological questions. We denote the category of blue schemes (in thesense of this paper) by SchF1.
Note that we do not require that blueprints are global. We will not mention this anymore, butremark here that all explicit examples of blueprints in thistext are global. For general argumentsthat need the fact that morphisms between blue schemes are locally algebraic (Theorem 3.23 of[25]), we take care to work with the coordinate blueprintsΓX = Γ(X,OX) andΓY = Γ(Y,OY),which are by definition global blueprints.
Blue schemes versus semiring schemes
By a (semiring) scheme, we mean a blue scheme whose coordinate blueprints are (semi-) rings.We denote the category of semiring schemes by Sch+
Nand the category of schemes by Sch+
Z.
Though the categories Sch+N
and Sch+Z
embed as full subcategories into SchF1, and these embed-dings have left-adjoints, one has to be careful with certaincategorical constructions like fibreproducts or affine spaces, whose outcome depends on the chosen category. Roughly speaking,we will apply the usual notation from algebraic geometry if we carry out a construction in the
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larger category SchF1, and we will use a superscript+ if we refer to the classical construction inthe category of schemes. Usually, constructions in Sch+
Ncoincide with constructions in Sch+
Z,
so that we can use the superscript+ also for constructions in Sch+N
.We explain in the following, which constructions are concerned, and how the superscript+
is used.
Tensor products and fibre products
We denote the functor that associates to a blueprint the generated semiring by(−)+. Thuswe write B+ for the associated semiring (which isBN in the notation of [25]), andX+ forthe semiring scheme associated to a blue schemeX. These come with canonical morphismsB→ B+ andβ : X+→ X.
We have seen in [25] that the category of blueprints containstensor productsB⊗D C. Todistinguish these from the tensor product of semirings in caseB, C andD are semirings, wewrite for the latter constructionB⊗+D C.
Since(−)+ : Bl pr0→ SRings is the right-adjoint of the forgetful functorSRings→Bl pr0, we have that(B⊗D C)+ = B+⊗+D+ C+. Since we are considering onlyBl pr0, thefunctor(−)inv from [25], which adjoins additive inverses to a blueprintB is isomorphic to thefunctor (−)⊗F1 F12 (recall from [25, Lemma 1.4] that a blueprint is with inverses if and onlyif it is with −1). This implies thatB+⊗+D+ C+ if and only if one ofB, C or D is with a−1. Inparticular,B⊗+D C is a ring if B, C andD are rings; and(B⊗F1 F12)+ is the ring generated by ablueprintB.
The corresponding properties of the tensor product hold forfibre products of blue schemes.We denote byX×Z Y the fibre product in SchF1, while X×+Z Y stays for the fibre product inSch+
N. Then we have(X×Z Y)+ = X+×+Z+ Y+. For a blue schemeX we denoteXB or X×F1 B
the base extensionX×SpecF1 SpecB. If B is a semiring, thenX+B stays for(XB)
+. Note that ingeneral(X+)B is not a semiring scheme. In particularX+
N= X+ andX+
Z= (XF12)
+, which isthe scheme associated toX.
Free algebras and affine space
Another construction that needs a specification of the category is the functor of free algebras.We denote thefree object in a set{Ti}i∈I over a blueprint Bin the categoryBl pr0 by B[Ti ]. If Bis a semiring, we denote the free object inSRingsby B[Ti]
+. If B is a ring, thenB[Ti]+ is a ring.
The spectrum of the free object onn generators isn-dimensional affine space:AnB = SpecB[Ti]
if B is a blueprint, and+AnB = SpecB[Ti ]
+ is B is a semiring.Note that localizations coincide for blueprints and semirings, i.e. ifS is a multiplicative
subset of a blueprintB andσ : B→ B+ is the canonical map, then(S−1B)+ = σ(S)−1B+. We
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denote the localization of the free blueprint inT over B by B[T±1] or B[T±1]+, dependingwhether we formed the free algebra inBl pr0 or SRingsThe corresponding geometric objectsare themultiplicative group schemesGm,B and+Gm,B, respectively. Thehigher-dimensionaltori Gn
m,B and+Gnm,B are defined in the obvious way.
There are other schemes that can be defined either category SchF1 and Sch+N
. For example,the definition of projectiven-space (as a scheme) by gluingn-dimensional affine spaces alongtheir intersections generalizes to semiring schemes and blue schemes. We definePn
B as theprojective n-spaceobtained by gluing affine planesAn
B if B is a blueprint,+PnB as the projective
n-space obtained by gluing+AnB if B is a semiring and+Pn
B as the projectiven-space obtainedby gluing+An
B if B is a ring. A more conceptual viewpoint on this is given in a subsequent paperwhere we introduce the functor Proj for graded blueprints and graded semirings.
1.2 Sober and locally finite spaces
While the underlying topological space of a scheme of finite type over an (algebraically closed)field consists typically of infinitely many points, a scheme of finite type overF1 has only finitelymany points. This allows a more combinatorial view for the latter spaces, which is the objectiveof this section.
To begin with, recall that a topological space issoberif every irreducible closed subset hasa unique generic point.
Proposition 1.1. The underlying topological space of a blue scheme is sober.
Proof. Since the topology of a blue scheme is defined by open affine covers, a blue schemeis sober if all of its affine open subsets are sober. Thus assume X = SpecB is an affine bluescheme. A basis of the topology of closed subsets ofX is formed by
Va = { p⊂ B prime ideal| a∈ p }
wherea ranges through all elements ofB. Given an irreducible closed subsetV, we defineη =
⋂p∈V p, which is an ideal ofB.
We claim thatη is a prime ideal. Letab∈ η. Since everyp ∈V containsη and thereforeab,we haveV ⊂Vab =Va∪Vb. Thus
V = Vab∩V = (Va∪Vb)∩V = (Va∩V)∪ (Vb∩V).
SinceV is irreducible, eitherV =Va∩V or V =Vb∩V, i.e.V ⊂Va or V ⊂Vb This means thateithera∈ η or b∈ η, which shows thatη is a prime ideal.
The closed subsetV is the intersection of allVa with a∈ p for all p ∈V. Sinceη is definedas intersection of allp ∈V, it is contained in allVa that containV. Thusη ∈V.
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We show thatη is the unique generic point ofV. The closure{η} of η consists of all primeideals that containη, and thusV ⊂ {η}. Thusη is a generic point ofV. If η′ is another genericpoint ofV, thenη′ is contained in every prime idealp ∈V. Thusη′ = η, andη is unique.
Definition 1.2. A topological space isfinite if it has finitely many points. A topological spaceis locally finite if it has an open covering by finite topological spaces.
These notions find application to blue schemes of (locally) finite type as introduced in Sec-tion 1.3.
Lemma 1.3. Let X be a locally finite and sober topological space. Let x∈ X. Then the set{x}is locally closed in X.
Proof. Since this is a local question, we may assume thatX is finite. DefineV =⋃
x/∈{y} {y},which is a finite union of closed subsets, which does not contain x. ThusU = X−V is an openneighbourhood ofx. If x∈ {y}, i.e.y∈U , andy∈ {x}, thenx= y sinceX is sober. ThereforeU ∩{x}= {x}, which verifies that{x} is locally closed.
In the following we consider a topological spaceX as a poset by the rulex≤ y if and onlyif y∈ {x} for x,y∈ X.
Lemma 1.4. Let X be a locally finite topological space, and U a subset of X.Then U is open(closed) if and only if for all x≤ y, y∈U implies x∈U (x∈U implies y∈U).
Proof. We prove only the statement about closed subsets. The statement about open subsets iscomplementary and can be easily deduced by formal negation of the following.
Since this is a local question, we may assume thatX is finite. If U is closed, thenx≤ y andx∈U impliesy∈ {x} ⊂U .
Conversely, ifx≤ y andx∈U impliesy∈U for all x,y∈ X, then we have that for allx∈Uits closure{x} = {y ∈ X|x≤ y} is a subset ofU . SinceU is finite, {U} = ⋃
x∈U {x} is theclosure ofU , and it is contained inU . ThusU is closed.
Proposition 1.5. Let X and Y be topological spaces. A continuous map f: X→ Y is order-preserving. If X is locally finite, then an order-preservingmap f : X→Y is continuous.
Proof. Let f : X→Y be continuous andx≤ y in X. The setf−1({ f (x)}) is closed and containsx. Thusy∈ f−1({ f (x)}), which means thatf (x)≤ f (y). This shows thatf is order-preserving.
Let X be locally finite andf : X→Y order-preserving. LetV be a closed subset ofY. Wehave to show thatf−1(V) is a closed subset ofX. We apply the characterization of closedsubsets from Lemma 1.4: letx∈ f−1(V) andx≤ y. Since f is order-preserving,f (x) ≤ f (y).This means thatf (y) ∈ { f (x)} ⊂V and thusy∈ f−1(V).
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Example 1.6.The previous lemma and proposition show that the underlyingtopological spaceof a locally finite blue scheme is completely determined by its associated poset. We will illus-trate locally finite schemesX by diagrams whose points are pointsx∈ X and with lines from alower pointx to a higher point pointy if x< y and their is no intermediatez, i.e.x< z< y. Forexample, the underlying topological space ofA1
F1= SpecF1[T] consists of the prime ideals(0)
and(T), the latter one being a specialization of the former one. Similarly, A2F1
= SpecF1[S,T]
has four points(0), (S), (T) and (S,T). The projective lineP1F1
= A1F1
∐Gm,F1
A1F1
has two
closed points[0 : 1] and [1 : 0] and one generic point[1 : 1]. Similarly, the points ofP2F1
cor-respond to all combinations[x0 : x1 : x2] with xi = 0 or 1 with exception ofx0 = x1 = x2 = 0.These blue schemes can be illustrated as in Figure 1.
(T)
(0)
(T)(S)
(S,T)
(0) [1 : 1]
[0 : 1] [1 : 0]
[1 : 0 : 0]
[1 : 1 : 0]
[1 : 1 : 1]
[0 : 1 : 1]
[0 : 0 : 1][0 : 1 : 0]
[1 : 0 : 1]
Figure 1: The blue schemeA1F1
, A2F1
, P1F1
andP2F1
(from left to right)
1.3 Closed immersions
An important tool to describe all points of a blue scheme are closed immersions into known blueschemes. We generalize the notion of closed immersions as introduced in [14] to blue schemes.
Definition 1.7. A morphismϕ : X→Y of blue schemes is aclosed immersionif ϕ is a homeo-morphism onto its image and for every affine open subsetU of Y, the inverse imageV =ϕ−1(U)is affine inX andϕ#(U) : Γ(OY,U)→ Γ(OX,V) is surjective. Aclosed subscheme of Yis ablue schemeX together with a closed immersionX→Y.
Remark 1.8. In contrast to usual scheme theory, it is in general not true that the image ofa closed immersionϕ : X → Y is a closed subset ofY. Consider, for instance, the diago-nal embedding∆ : A1
F1→ A1
F1×F1 A
1F1
= A2F1
, which corresponds to the blueprint morphismF1[T1,T2]→ F1[T] that maps bothT1 andT2 to T. Then the inverse image of the 0-ideal is the0-ideal, and the inverse image of the ideal(T) is (T1,T2). But the set{(0),(T1,T2)} is not closedin A2
F1as illustrated in Figure 2.
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Figure 2: The diagonal embedding∆ : A1F1→ A2
F1
Lemma 1.9. Let f : B→C be a surjective morphism of blueprints. Then f∗ : SpecC→ SpecBis a closed immersion.
Proof. PutX = SpecB, Y = SpecC andϕ= f ∗ : Y→X. We first show thatϕ is injective. Sincef : B→ C is surjective,f ( f−1(p)) = p for all p ⊂ C. If p andp′ are prime ideals ofC withϕ(p) = ϕ(p′), thenp= f ( f−1(p)) = f ( f−1(p′)) = p′. Thusϕ is injective.
For to show thatϕ is a homeomorphism onto its image, we have to verify that every opensubsetV of Y is the inverse imageϕ−1(U) of some open subsetU of X. It suffices to verifythis for basic opens. LetVa = {p ∈Y|a /∈ p} for somea∈C. Then there is ab∈ B such thatf (b) = a and thusVa = ϕ−1(Ub) for Ub = {q ∈ X|b /∈ q}. Henceϕ is a homeomorphism ontoits image.
Affine opens ofX = SpecB are of the formU ≃ Spec(S−1B) for some multiplicative subsetSof B. The inverse imageV = ϕ−1(U) is then of the formV ≃Spec( f (S)−1C), and thus affine.Since f : B→C is surjective, also the induced mapS−1 f : S−1B→ f (S)−1C is surjective. Thusϕ is a closed immersion.
If A is a monoid (with 0), then we considerA as the blueprintB= A�〈 /0〉. SinceA�〈 /0〉 →A�R is surjective for any pre-additionR on A, we have the following immediate consequenceof the previous lemma.
Corollary 1.10. If B = A�R is a representation of the blueprint B, thenSpecB⊂ SpecA.
Let f : B→C be a morphism of blueprints. We say thatC is finitely generated over B (as ablueprint)or that f is of finite typeif f factorizes through a surjective morphismB[T1, . . . ,Tn]→C for somen∈ N. If C is finitely generated over a blue field, thenC has finitely many primeideals and thus SpecC is finite.
Let ϕ : X→ Sbe a morphism of blue schemes. We say thatX is locally of finite type over S(as a blue scheme)if for every affine open subsetU of X that is mapped to an affine open subsetV of S the morphismϕ#(V) : Γ(OS,V)→ Γ(OX,U) between sections is of finite type. We saythatX is of finite type over S (as a blue scheme)if X is locally finitely generated and compact.
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If X is (locally) of finite type over a blue fieldκ, i.e.X→ Specκ is (locally) of finite type, thenX is (locally) finite.
Example 1.11.We can apply Corollary 1.10 to describe the topological space of affine blueschemes of finite type overF1. It is easily seen that the prime ideals of the free blueprintF1[T1, . . . .Tn] are of the formpI = (Ti)i∈I whereI is an arbitrary subset ofn = {1, . . . ,n}. EveryblueprintB that is finitely generated overF1 has a representationB= F1[T1, . . . ,Tn]�R, then ev-ery prime ideal ofB is also of the formpI (where it may happen thatTi ≡ Tj if the representationof B is not proper).
More precisely,pI is a prime ideal ofB = F1[T1, . . . ,Tn]�R if and only if for all additiverelations∑ai ≡ ∑b j in B, either all termsai andb j are contained inpI or at least two of themare not contained inpI .
SinceAnF1
is finite, SpecB is so, too, and the topology of SpecB is completely determinedby the inclusion relation of prime ideals ofB.
1.4 Reduced blueprints and closed subschemes
In this section, we extend the notions of reduced rings and closed subschemes to the context ofblueprints and blue schemes. Since all proofs have straightforward generalizations, we forgoto spell them out and restrict ourselves to state the facts that are needed in this paper.
Definition 1.12. Let B be a blueprint andI ⊂ B an ideal. Theradical Rad(I) of I is the inter-section
⋂p of all prime idealsp of B that containI . Thenilradical Nil(B) of B is the radical
Rad(0) of the 0-ideal ofB.
Remark 1.13. If B is a ring, then Rad(I) equals the set√
I = { a∈ B | an ∈ I for somen> 0 }.
The inclusion√
I ⊂ Rad(I) holds for all blueprints and Rad(I)⊂√
I holds true ifB is with−1.The latter inclusion is, however, not true in general as the following example shows.
Let B = F1[S,T,U ]�〈S≡ T +U〉 and I = (S2,T2) = {S2b,T2b|b ∈ B}. Then Rad(I) =(S,T,U) while
√I = {Sb,Tb|b∈ B} does not containU .
If, however,I is the 0-ideal, then the equality√
0= Rad(0) holds true for all blueprints.
Lemma 1.14.Let B be a blueprint. Then the following conditions are equivalent.
(i) Nil (B) = 0;
(ii)√
0= 0;
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(iii) 0 is a prime ideal of B.
If B satisfies these conditions, then B is said to bereduced.
We defineBred=B/Nil(B) as the quotient ofBby its nilradical, which is a reduced blueprint.Every morphism fromB into a reduced blueprint factors uniquely through the quotient mapB→ Bred.
Lemma 1.15.The universal morphism f: B→Bred induces a homeomorphism f∗ : SpecBred→SpecB between the underlying topological spaces of the spectra of B and Bred.
Proposition 1.16. Let X be a blue scheme with structure sheafOX. Then the following condi-tions are equivalent.
(i) OX(U) is reduced for every open subset U of X.
(ii) OX(Ui) is reduced for all i∈ I where{Ui}i∈I is an affine open cover of X.
If X satisfies these conditions, then X is said to bereduced.
Corollary 1.17. A blueprint B is reduced if and only its spectrumSpecB is reduced.
Let X be a blue scheme. We define the reduced blue schemeXred as the underlying topolog-ical space ofX together with the structure sheafO red
X that is defined byO redX (U) = OX(U)red.
It comes together with a closed immersionXred→ X, which is a homeomorphism between theunderlying topological spaces.
More generally, there is for every closed subsetV of X a reduced closed subschemeY of Xsuch that the inclusionY→ X has set theoretic imageV and such that every morphismZ→ Xfrom a reduced schemeZ to X with image inV factors uniquely throughY → X. We callY the(reduced) subscheme of X with support V.
Remark 1.18. Note thatY is not the smallest subscheme ofX with supportV since in general,there quotients of blueprints that are the quotient by an ideal. For example consider the bluefield {0}∪µn whereµn is a group of ordern together with the surjective blueprint morphism{0}∪µn→ {0}∪µm wherem is a divisor ofn. Then Spec({0}∪µm)→ Spec({0}∪µn) is aclosed immersion of reduced schemes with the same topological space, which consists of onepoint.
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1.5 Mixed characteristics
A major tool for our studies of the topological space of a blueschemeX are morphismsSpeck→ X from the spectrum of a semifieldk into X, whose image is a pointx of X. Inparticular, the characteristics thatk can assume are an important invariant ofx. Note that unlikefields (in the usual sense), a blue field might admit morphismsinto fields of different charac-teristics. For instance, the blue fieldF1 = {0,1} embeds into every field. In this section, weinvestigate the behaviour of blue fields and their characteristics.
Definition 1.19. Let B be a blueprint. The(arithmetic) characteristiccharB of B is the charac-teristic of the ringB+
Z.
We apply the convention that the characteristic of the zero ring {0} is 1. Thus a ring is ofcharacteristic 1 if and only it is the zero ring. As the examples below show, there are, however,non-trivial blueprints of characteristic 1.
With this definition, the arithmetic characteristic of a blueprintB= A�R is finite (i.e. notequal to 0) if and only if there is an additive relation of the form
∑ai +1+ · · ·+1︸ ︷︷ ︸n-times
≡ ∑ai
in R with n> 0, and charB is equal to the smallest suchn.A prime semifieldis a semifield that does not contain any proper sub-semifield.Prime
semifields are close to prime fields, which areQ orFp wherep is a prime. Indeed,Fp are primesemifields since they do not contain any smaller semifield. The rational numbersQ containthe smaller prime semifieldQ≥0 of non-negative rational numbers. There is only one moreprime semifield, which is the idempotent semifieldB1 = {0,1}�〈1+1≡ 1〉 (cf. [22] and [11,p. 13]). Note that semifields, and, more generally, semirings B that containB1 are idempotent,i.e. a+a≡ a for all a∈ B.
Every semifieldk contains a unique prime semifield, which is generated by 1 as asemifield.If k containsFp, then chark= pandk is a field since it is with−1. If k containsB1, then chark=1. If k containsQ≥0, thenk+
Zis either a field of characteristic 0 or the zero ring{0}. Thus the
characteristic ofk is either 0 or 1. In the former case,k→ k+Z
is a morphism into a field ofcharacteristic 0. To see that the latter case occurs, consider the examplek=Q≥0(T)�〈T +1≡T〉 whereQ≥0(T) are all rational functionsP(T)/Q(T) whereP(T) andQ(T) are polynomialswith non-negative rational coefficients. Indeed,k+
Z= {0} since 1≡ (T +1)−T ≡ 0; it is not
hard to see thatk containsQ≥0 as constant polynomials.
Definition 1.20. Let B be a blueprint. An integerp is called apotential characteristic of Bifthere is a semifieldk of characteristicp and a morphismB→ k. We say thatB is of mixed
18
characteristicsif B has more than one potential characteristic, and thatB is of indefinite char-acteristicif all primes p, 0 and 1 are potential characteristics ofB. A blueprintB is almost ofindefinite characteristic, if all but finitely many primesp are potential characteristics ofB.
We investigate the potential characteristics of semifields.
Lemma 1.21.Let k be a semifield. Then there is a morphism k→ B1 if and only if k is withoutan additive inverse−1 of 1. Consequently,chark is the only potential characteristic of k, unlessk is of arithmetic characteristic0, but without−1. In this case, k has potential characteristics0 and1.
Proof. Let k be a semifield. Then the mapf : k→ B1 that sends 0 to 0 and every other elementto 1 is multiplicative. Ifk is with−1, then 1+(−1)≡ 0 in k, but 1+1≡/ 0 inB1; thus f is not amorphism in this case. Ifk is without−1, then for every relation∑ai ≡∑b j in k neither sum isempty. Since∑1≡ ∑1 holds true inB1 if neither sum is empty,f is a morphism of semifields.This proves the first statement of the lemma.
Trivially, the arithmetic characteristic of a semifieldk is a potential characteristic ofk. If kcontains−1, thenk is a field and has a unique characteristic. Since there is no morphism froman idempotent semiring into a cancellative semifield, semifields of characteristic 1 have onlypotential characteristic 1. The only case left out, is the case thatk is of arithmetic characteristic0, but is without−1. Then there is a morphismk→ B1, and thusk has potential characteristic0 and 1.
Let B be a blueprint of arithmetic characteristicn> 1. Since every morphismB→ k into asemifieldk factorizes throughB+, which is with−1= n−1, the semifieldk is a field and everypotential characteristicp of B is a divisor ofn. This generalizes trivially to the casesn = 0andn = 1. The reverse implication is not true since 1 divides all other characteristics. Evenif we excludep = 1 as potential characteristic, the reverse implication does also not hold forblueprints of arithmetic characteristic 0, as the exampleB=Q and, more general, every properlocalization ofZ, witnesses. However, it is true for blueprints of finite arithmetic characteristic.
Lemma 1.22.Let B be a blueprint of characteristic n≥ 1. If p is a prime divisor of n, then p isa potential characteristic of B.
Proof. If pdividesn, thenn>1 andp=1+ · · ·+1 generates a proper ideal inB+Z
. ThusB+Z/(p)
is a ring of characteristicp and, in particular, not the zero ring. Therefore, there is a morphismB+Z/(p)→ k into a fieldk of characteristicp. The compositionB→ B+
Z→ B+
Z/(p)→ k verifies
that p is a potential characteristic ofB.
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If G is an abelian semigroup, then we denote byB[G] the(blue) semigroup algebra of G overB, which is the blueprintA�R with A= B×G andR = 〈∑(ai ,1)≡∑(b j ,1)|∑ai ≡∑b j in B〉.Since there are morphismsB→ B[G], which mapsb to (b,1), andB[G]→ B, which maps(b,g)to b, the potential characteristics ofB andB[G] are the same. Thus every blue field of the formF1[G] is of indefinite characteristic. More generally, we have thefollowing. Recall from [25]thatF1n (for n≥ 1) is the blue field(0∪µn)�R whereµn is a cyclic group withn elements andR is generated by the relations∑ζ∈H ζ ≡ 0 whereH varies through all non-trivial subgroups ofµn.
Lemma 1.23.Let G be an abelian semigroup and n≥ 1. ThenF1n[G] has all potential charac-teristics but1 unless n= 1, in which caseF1[G] is of indefinite characteristic.
Proof. Since there is a morphismF1n[G]→ F1n that maps all elements ofG to 1, it suffices toshow thatF1n is of indefinite characteristic. Letζn be a primitive root of unity. ThenF1n embedsintoQ[ζn] and thus 0 is a potential characteristic ofF1n. Let p be a prime that does not dividen.ThenF1n embeds into the algebraic closureFpof Fp, andp is a potential characteristic ofF1n.
The last case is thatp is a prime that dividesn. Then we can define a unique multiplicativemap f : F1n→ Fp whose kernel consists of thoseζ ∈ F1n whose multiplicative order is divisibleby p. We have to verify that this map induces a map between the pre-additions. It is enoughto verify this on generators of the pre-addition ofF1n. Let H be a non-trivial subgroup ofµn
whose order is not divisible byp. ThenH is mapped injectively onto the non-trivial subgroupf (H) of Fp
×, and we have∑ζ∈H f (ζ) = ∑ζ ′∈ f (H) ζ
′ = 0 in Fp. If H is a subgroup ofµn whose
order is divisible byp, then the kernel of the restrictionf : H → Fp×
is of some orderpk withk≥ 1. Thus
∑ζ∈H
f (ζ) = ∑ζ ′∈ f (H)
(ζ ′+ · · ·+ ζ ′︸ ︷︷ ︸pk-times
) = 0.
This shows thatf : F1n→ Fp is a morphism of blueprints and thatp is a potential characteristicof F1n. If n 6= 1, thenF+
1n = Z[ζn] is with −1 whereζn is a primitiven-th root of unity. Thusthere is no blueprint morphismF1n→ k into a semi-field of characteristic 1 unlessn= 1.
To conclude this section, we transfer the terminology from algebra to geometry.
Definition 1.24. Let X be a blue scheme,x a point ofX andκ(x) be the residue field ofx. The(arithmetic) characteristicchar(x) of x is the arithmetic characteristic ofκ(x). We say thatp isa potential characteristic of xif p is a potential characteristic ofκ(x), and we say thatx is ofmixedor of indefinite characteristicsif κ(x) is so.
By a monoidal scheme, we mean aM0-scheme in the sense of [25]. A monoidal scheme ischaracterized by its coordinate blueprints, which are blueprints with trivial pre-addition.
20
Corollary 1.25. Let X be a monoidal scheme. Then every point of X is of indefinite character-istic.
Proof. This follow immediately from Lemma 1.23 since the residue field of a point in a monoidalscheme is of the formF1[G] for some abelian groupG.
Example 1.26.We give two examples to demonstrate certain effects of potential characteristicsunder specialization. LetB1 = F1[T]�〈T +T ≡ 0〉. ThenB1 has two prime idealsx0 = (0)andxT = (T). The residue fieldκ(x0) = F1[T±1]�〈T +T ≡ 0〉 ≃ F2[T±1] has only potentialcharacteristic 2 since 1+1≡ T−1(T+T)≡ 0, while the residue fieldκ(xT) = F1 is of indefinitecharacteristic.
The blueprintB2 = F1[T]�〈1+1 = T〉 has also two prime idealsx0 = (0) andxT = (T).The residue fieldκ(x0) = F1[T±1]�〈1+1≡ T〉 has all potential characteristics except for 2since 1+1≡ T is invertible, while the residue fieldκ(xT) = F1�〈1+1≡ 0〉 = F2 has onlycharacteristic 2. We illustrate the spectra ofB1 and B2 together with their residue fields inFigure 3.
κ(xT) = F1[T]�〈T +T ≡ 0〉
κ(x0) = F2[T±1]
κ(xT) = F2
κ(x0) = F1[T±1]�〈1+1≡ T〉
Figure 3: The spectra ofB1 andB2 together with their respective residue fields
1.6 Fibres and image of morphisms from (semiring) schemes
The fibre of a morphismϕ : Y→ X of schemes over a pointx∈ X is defined as the fibre product{x}×+X Y. The canonical morphism{x}×+X Y→ Y is an embedding of topological spaces. Inthis section, we extend this result to blue schemes. Recall from [25, Prop. 3.27] that the categoryof blue schemes contains fibre products.
Let ϕ : Y→ X be a morphism of blue schemes andx∈ X. Thefibre ofϕ over xis the blueschemeϕ−1(x) = {x}×+X Y and thetopological fibre ofϕ over x is the subspaceϕ−1(x)top ={y ∈ Y|ϕ(y) = x} of Y. The following lemma justifies the notation sinceϕ−1(x)top is indeedcanonically homeomorphic to the underlying topological space ofϕ−1(x).
Lemma 1.27. Let ϕ : Y→ X be a morphism of blue schemes and x∈ X. Then the canonicalmorphismϕ−1(x)→Y is a homeomorphism ontoϕ−1(x)top.
21
Proof. Since the diagram
ϕ−1(x) //
��
Y
ϕ
��{x} // X
commutes, the image ofϕ−1(x)→ Y is contained inϕ−1(x)top. Given a pointy ∈ ϕ−1(x)top,consider the canonical morphism Specκ(y) → Y with imagey, and the induced morphismSpecκ(y)→Specκ(x) of residue fields, which has image{x}⊂X. The universal property of thetensor product implies that both morphisms factorize through a morphism Specκ(y)→ ϕ−1(x),which shows that the canonical mapϕ−1(x)→ ϕ−1(x)top is surjective.
We have to show thatϕ−1(x)→ ϕ−1(x)top is open. Since this is a local question, we mayassume thatX = SpecB andY = SpecC are affine blue schemes with coordinate blueprintsB andC. Thenϕ−1(x) = Spec(κ(x)⊗B C) andκ(x) = S−1B/p(S−1B) whereS= B− p andp= x∈ SpecB. Let f = Γ(ϕ,X) : B→C. Then
κ(x) ⊗B C =(S−1B / p(S−1B)
)⊗B C ≃
(S−1B / p(S−1B)
)⊗S−1B S−1B ⊗B C
≃(S−1B / p(S−1B)
)⊗S−1B f (S)−1C ≃ f (S)−1C / f (p)( f (S)−1C),
which is the quotient of a localization ofC. Note that the last two isomorphisms follow easilyfrom the universal property of the tensor product combined with the universal property of local-izations and quotients, completely analogous to the case ofrings. This proves thatϕ−1(x)→Yis a topological embedding.
Proposition 1.28.Let X be a blue scheme and x∈X. LetαX : X+→X andβX : X+Z→X be the
canonical morphisms. Then the canonical morphismsSpecκ(x)+→ α(x)−1 andSpecκ(x)+Z→
β(x)−1 are isomorphisms.
Proof. We prove the proposition only forαX. The proof forβX is completely analogous. Sincethe statement is local aroundx, we may assume thatX is affine with coordinate blueprintB, andx= p is a prime ideal ofB. ThenX+ = SpecB+, and we have to show that the canonical mapκ(x)⊗B B+→ κ(x)+ is an isomorphism. Note that the canonical mapB→ B+ is injective, sowe may considerB as a subset ofB+. Let S= B−p. The same calculation as in the proof ofLemma 1.27 shows that
thatB+/I+ ≃ (B/I)+ whereI+ is the ideal ofB+ that is generated by the image ofI in B+. Weapply this to derive
S−1B+ / (p(S−1B))+ ≃(
S−1B / p(S−1B))+
= κ(x)+,
which finishes the proof of Specκ(x)+ ≃ α(x)−1.
The potential characteristics of the points of a blue schemeare closely related to the fibresof the canonical morphism from its semiring scheme as the following lemma shows.
Lemma 1.29.The canonical morphismαX : X+→X is surjective. The potential characteristicsof a point x∈X correspond to the potential characteristics of the pointsy in the fibre ofαX overx.
Proof. The morphismαX : X+→ X is surjective for the following reason. The canonical mor-phismB→ B+ is injective. In particular,κ(x)→ κ(x)+ is injective for every pointx of X. Thismeans thatκ(x)+ is non-trivial, and thusα−1(x)≃ κ(x)+ non-empty. This shows the first claimof the lemma.
If x = α(y) for somey in the fibre ofαX over x, then there is a morphismκ(x)→ κ(y)between the residue fields, and the potential characteristics of the semifieldκ(y) are potentialcharacteristics of the blue fieldκ(x). On the other hand, ifκ(x)→ k is a map into a semifieldk of characteristicp, then this defines a morphism Speck→ X with imagex, which factorsthroughX+. Thus the mapκ(x)→ k factors throughκ(x)→ κ(y) for somey in the fibre ofαX
overx. Thus the latter claim of the lemma.
Remark 1.30. By the previous lemma, every pointx of a blue schemeX lies in the image ofsomeαX,k : X+×+
Nk→ X wherek is a semifield, which can be chosen to be an algebraically
closed field if it is not an idempotent semifield. This shows that the geometry of a blue scheme isdominated by algebraic geometry over algebraically closedfields and idempotent geometry, bywhich I mean geometry that is associated to idempotent semirings. There are various (different)viewpoints on this: idempotent analysis as considered by Kolokoltsov and Maslov, et al. (see,for instance, [27]), tropical geometry as considered Itenberg, Mikhalkin, et al. (see, for instance,[18], [28] and, in particular, [29, Chapter 2]) and idempotent geometry that attempts to mimicF1-geometry (see [11], [22] and [32]). These theories might find a common background in thetheory of blue schemes.
Lemma 1.31. Let B be a cancellative blueprint and I⊂ B be an ideal of B. Then the quotientB/I is cancellative.
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Proof. We first establish the following claim: two elementsa,b∈ B define the same classa≡ bin B/I if and only if there are elementsck,dl ∈ I such thata+∑ck≡ b+∑dl in B. Per definition,a≡ b if and only if there is a sequence of the form
a ≡ ∑c1,k ∼IN ∑d1,k ≡ ∑c2,k ∼I
N · · · ∼IN ∑dn,k ≡ b
where∑ck ∼IN ∑dk if for all k eitherck = dk or ck,dk ∈ I (cf. [25, Def. 2.11]). If we add up all
additive relations in this sequence, we obtain
a+∑ci,k ≡ b+∑di,k.
SinceB is cancellative, we can cancel all termsci,k ≡ di,k that appear on both sides, and stayover with a relation of the form
a+∑ ck ≡ b+∑ dk
with ck, dk ∈ I . This shows one direction of the claim. To prove the reverse direction, considera relation of the forma+∑ck ≡ b+∑dl with ck,dl ∈ I . Then we have
a ≡ a+∑0 ∼IN a+∑ck ≡ b+∑dl ∼I
N b+∑0 ≡ b,
which shows thata≡ b in B/I .With this fact at hand, we can prove thatB/I is cancellative. Consider a relation of the form
∑ai +c0 ≡ ∑b j +d0
in B wherec0≡ d0 in B/I . We have to show that∑ai ≡ ∑b j in B/I . By the above fact,c0≡ d0if and only if there areck,dl ∈ I such thatc0+∑ck ≡ d0+∑dl . Adding this equation to theabove equation, with left and right hand side reversed, yields
∑ai +c0+d0+∑dl ≡ ∑b j +d0+c0+∑ck.
SinceB is cancellative, we can cancel the termc0+d0 on both sides and obtain the sequence
∑ai ≡∑ai +∑dl ≡ ∑b j +∑ck ≡ ∑bj
in B/I , which proves thatB/I is cancellative.
Lemma 1.32. If X is cancellative, then the canonical morphismβX : X+Z→ X is surjective.
24
Proof. This is a local question, so we may assume thatX =SpecB for a cancellative blueprintB.Since localizing preserves cancellative blueprints (see [25, section 1.13]),Bp is also cancellativefor every prime idealp of B. The residue field atx= p is κ(x) = Bp/pBp, which is cancellativeby the Lemma 1.31. This it is a subblueprint of the (non-zero)ring κ(x)+
Z. Thus the canonical
morphism Specκ(x)+Z→Specκ(x) →X has image{x} and factors throughX+
Zby the universal
property of the schemeX+Z
.
Remark 1.33. Every sesquiad (see [16]) can be seen as a cancellative blueprint. A prime idealof a sesquiads is the intersection of a prime ideal of the prime ideal of its universal ring with thesesquiad. The previous lemma shows that the sesquiad prime ideals coincide with its blueprintprime ideals.
While the points of potential characteristicp 6= 1 are governed by usual scheme theory, thepoints of potential characteristic 1 in a fibreα−1(x) are of a particularly simple shape.
Lemma 1.34. Let x∈ X be a point with potential characteristic1. Thenα−1X (x) is irreducible
with generic pointη, which is the only point ofα−1(x) with potential characteristic1. If X iscancellative, thenη has also potential characteristic0.
Proof. Letκ be the residue field ofx. Sincex has potential characteristic 1, there is a morphisminto a semifieldk of characteristic 1. By the universal property ofκ→ κ+, this morphismfactors through a morphismf : κ+→ k. Every element ofκ+ is of the form∑ai whereai areunits ofκ.
Consider the case thatf (∑ai) = 0. Unless the sum is trivial, it is of the forma+∑a′j forsome unita of κ. Thenb′ = f (∑a′j) is an additive inverse ofb = f (a) in k. Since units aremapped to units,b is a unit ofk and therefore 1= b−1b has the additive inverse−1 = b−1b′.But this is not possible in a semifield of characteristic 1. Therefore we conclude that∑ai hasto be the trivial sum and that the kernel off : κ+→ k is 0. This shows that 0 is a prime ideal,thatα−1(x) is irreducible with generic pointη = 0 and thatη is the only point ofα−1(x) withpotential characteristic 1.
By Lemma 1.32, the canonical morphismβ : X+Z→ X+ is surjective ifX+ is cancellative,
which is the case ifX is cancellative. Therefore every pointx of X+ with potential characteristic1 has at least one other potential characteristic, which must be 0 sinceκ(x) is a semifield without−1 (cf. Lemma 1.21). This proves the last claim of the lemma.
1.7 The topology of fibre products
In this section, we investigate the topological space of theproduct of two blue schemes. Thecanonical projections of the fibre product of blue schemes are continuous, and thus induce a
25
universal continuous map into the product of the underlyingtopological spaces. In contrast tothe product of two varieties over an algebraically closed field, which surjects onto the productof the underlying topological spaces, the product of two blue schemes injects into the productof the underlying topological spaces.
Proposition 1.35.Let X1→X0 and X2→X0 be morphisms of blue schemes. Then the canonicalmap
τ : X1 ×X0 X2 −→ X1 ×topX0
X2
is an embedding of topological spaces.
Proof. We have to show thatτ is a homeomorphism onto its image. Since the claim of theproposition is a local question, we may assume thatXi = SpecBi are affine withBi = Ai�Ri .Then there are morphismsj1 : B0→ B1 and j2 : B0→B2, and we haveX1×X0 X2 = SpecB1⊗B0
B2 with
B1 ⊗B0 B2 = A1×A2�〈R1×{1}, {1}×R2, (a0a1,a2)≡ (a1,a0a2) | ai ∈ Bi 〉.
Note that this is not a proper representation ofB1⊗B0 B2. Since we are only concerned withtopological properties of SpecB1⊗B0 B2, this is legitimate (cf. Section 1.1).
We begin to show injectivity ofτ . Let p be a prime ideal ofB. Thenτ(p) = (p0,p1,p2)wherepi = ι−1
i (p) is a prime ideal ofBi andιi : Bi→ B1⊗B0 B2 is the canonical map that sendsa to j1(a)⊗1= 1⊗ j2(a) if i = 0, that sendsa to a⊗1 if i = 1 and that sendsa to 1⊗a if i = 2.Since fora1⊗a2 = (a1⊗1) · (1⊗a2) ∈ p eithera1⊗1∈ p∩ ι1(B1) or 1⊗a2 ∈ p∩ ι2(B2), theprime idealp equals the set{a1⊗a2|ai ∈ pi}. Thusp is uniquely determined byτ(p).
We show thatτ is a homeomorphism onto its image. Given a basic openU =Ua1×topX0
Ua2
of X1×topX0
X2 whereai ∈ Bi andUai = {pi ∈ Xi |ai /∈ pi} is the according basic open ofXi fori = 1,2. Then
τ−1(U) = { p ∈ X1 ×X0 X2 | a1⊗1 /∈ p and 1⊗a2 /∈ p }= { p ∈ X1 ×X0 X2 | a1⊗a2 /∈ p } = Ua1⊗a2,
which is a basic open ofX1×X0 X2. Thusτ is continuous. Since every basic open ofX1×X0 X2
is of the formUa1⊗a2 for somea1 ∈ B1 anda2 ∈ B2, τ is is indeed a homeomorphism onto itsimage.
Therefore, we can regardX1×X0 X2 as a subspace ofX1×topX0
X2, and we denote a pointx ofX1×X0 X2 by the coordinates(x1,x2) of τ(x) wherex1 ∈ X1 andx2 ∈ X2.
In the rest of this section, we investigate the image ofτ in the caseX0 = SpecF1.
26
Lemma 1.36.Let B1 and B2 be blueprints. Then p is a potential characteristic of B1⊗F1 B2 ifand only if p is a potential characteristic of both B1 and B2. Consequently, B1⊗F1 B2 = {0} ifand only if B1 and B2 have no potential characteristic in common.
Proof. Since there are canonical mapsBi→B1⊗F1 B2 for i = 1,2, every potential characteristicof B1⊗F1 B2 is a potential characteristic of bothB1 andB2.
Conversely, letp be a common potential characteristic ofB1 andB2. In casep 6= 1, there aremorphismsBi → ki into fieldsk1 andk2 of characteristicp. The compositumk of k1 andk2 is afield of characteristicp that containsk1 andk2 as subfields. This yields morphismsfi : Bi → kand thus a morphismf : B1⊗F1 B2→ k (note that there is a unique mapF1→ k, which factorizesthrough f1 and f2). Thusp is a potential characteristic ofB1⊗F1 B2.
If p = 1, thenB1 andB2 are both without−1, and there are morphismsfi : Bi → B1 byLemma 1.21. Therefore there is a morphismB1⊗F1 B2→ B1, and 1 is a potential characteristicof B1⊗F1 B2.
This shows in particular thatB1⊗F1 B2 6= {0} if B1 andB2 have a potential characteristic incommon. If there is no morphismB1⊗F1 B2→ k into a semifieldk, then there is no morphismB1⊗F1 B2→ κ into any blue fieldκ. This means that SpecB1⊗F1 B2 is the empty scheme andB1⊗F1 B2 is {0}.
Example 1.37.While B1 andB2 possess all potential characteristics ofB1⊗κ0 B1 for an arbi-trary blue fieldκ0, the contrary is not true in general.
For instance, consider the tensor productF12⊗F1[T±1]F12 with respect to the two morphisms
f1 : F1[T±1]→ F12 with f1(T) = 1 and f2 : F1[T±1]→ F12 with f2(T) = −1. We have thatF12⊗F1 F12 = (F12)inv = F12, which is represented by{0⊗0,1⊗1,1⊗(−1)}. The tensor prod-uctF12⊗F1[T±1] F12 is a quotient ofF12, and we have
1⊗(−1) = 1⊗(1 · f2(T)) = (1 · f1(T))⊗1 = 1⊗1.
Thus 1⊗1 is its own additive inverse andF12⊗F1[T±1] F12 = F2, the field with two elements.
WhileF12 andF1[T±1] have both indefinite characteristic,F2 has characteristic 2.
Theorem 1.38.Let X1 and X2 be blue schemes. Then the embeddingτ : X1×F1 X2→X1×topX2is a homeomorphism onto the subspace
{ (x1,x2) ∈ X1×topX2 | x1 and x2 have a common potential characteristic}.
Proof. Note that since the underlying topological space ofX0 = SpecF1 is the one-point space,we haveX1×top
X0X2 = X1×topX2. Sinceτ is an embedding (cf. 1.35), we have only to show that
the image ofτ is as described in the theorem.
27
Let x = (x1,x2) ∈ X1×top X2. Write κi for κ(xi) andκ for κ1⊗F1 κ2. If x ∈ X1×F1 X2,thenκ = κ(x). The canonical morphismκ→ κ+ witnesses thatp = charκ+ is a potentialcharacteristic ofκ. By Lemma 1.36, the potential characteristics ofκ (or, equivalently,x)correspond to the potential characteristic ofκ1 andκ2 (or, equivalently,x1 andx2). Therefore,x1 andx2 have a potential characteristic in common.
If, conversely,x1 andx2 have a common potential characteristicp, thenp is also a potentialcharacteristic ofκ by Lemma 1.36. This means that there exists a morphismκ → k into asemifieldk. The morphism Speck→ Specκ has imagex= (x1,x2), and thus(x1,x2) ∈ X1×F1
X2.
For later reference, we state the following fact, which follows from the local definition ofthe fibre product. We use the shorthand notationΓX for the global sectionsΓ(X,OX) of X.
Lemma 1.39.Let X→ Z and Y→ Z be two morphisms of blue schemes. Then
Γ(X×Z Y) ≃ ΓX ⊗ΓZ ΓY.
1.8 Relative additive closures
Let f : B→C be a morphism. Theadditive closure of B in C w.r.t. to fis the subblueprint
f+(B) = { c∈C | c≡∑ f (ai) for ai ∈ B}
of C. Note that this is indeed a subblueprint ofC since forc,d ∈ f+(B), i.e. c≡ ∑ f (ai) andd≡ ∑ f (b j), the productcd= ∑ f (aib j) is an element off+(B).
If B is a subblueprint ofC andι : B →C the inclusion, then we callι+(B) briefly theadditiveclosure of B in C. The subblueprintB is additively closed in Cif B= ι+(B).
We list some immediate properties of relative additive closures. Letf : B→C be a blueprintmorphism. Thenf+(B) is additively closed inC. More precisely,f+(B) is the smallest addi-tively closed subblueprintB′ of C such that the morphismf : B→C factors throughB′ →C. IfC is a semiring, thenf+(B) is isomorphic to the universal semiringf (B)+ associated withf (B)(considered as a subblueprint ofC).
Lemma 1.40.For any commutative diagram
Bg //
f��
B
f��
Cg // C,
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there exists a unique blueprint morphism f+(g) : f+(B)→ f+(B) such that the diagram
Bg //
f��
ww♥♥♥♥♥♥♥♥ B
f
��
ww♥♥♥♥♥♥♥♥♥
f+(B)f+(g)
//� t
''❖❖❖❖
❖❖❖❖
f+(B)� t
''❖❖❖❖
❖❖❖❖
Cg // C,
commutes.
Proof. The uniqueness off+(g) follows from the injectivity of f+(B) → C. For c ∈ f+(B)define f+(g)(c) asg(c), which is a priori an element ofC. Sincec≡ ∑ f (ai) for certainai ∈ B,we have that ˜g(c) ≡ ∑ g( f (ai)) ≡ ∑ f (g(ai)), thusg(C) is indeed an element off+(B). Thisshows thatf+(g) : f+(B)→ f+(B) is a blueprint morphism with the desired property.
Let B be a blueprint andι : B→ Binv the base extension fromF1 to F12 where we writeBinv = B⊗F1 F12. Theinverse closure of Bis the subblueprintB= ι+(B) of Binv. A blueprintBis inverse closedif B≃ B.
Note that since(Binv)inv = Binv, the inverse closureB of B is inverse closed. The previouslemma extends the associationB 7→ B naturally to a functor(−)ˆ : Bl pr→Bl pr whose essen-tial image are the inverse closed blueprints. Further note that the inverse closureB of B equalsthe intersection ofBinv with B+
canc insideB+Z
. In particular note thatB is cancellative.
Lemma 1.41.Let B1 and B2 be blueprints. Then(B1⊗F1 B2)ˆ ≃ B1⊗F1 B2.
Proof. Since(B1⊗F1 B2)canc= B1,canc⊗F1 B2,cancandB= (Bcanc)ˆ, we can assume thatB1 andB2 are cancellative. Therefore, we can consider(B1⊗F1 B2)ˆ andB1⊗F1 B2 as subblueprints of(B1⊗F1 B2)inv = B1,inv⊗F1 B2,inv that both containB1⊗F1 B2 as a subblueprint.
Let a⊗b be an element ofB1⊗F1 B2. we have to show that its additive inverse−(a⊗b) iscontained in(B1⊗F1 B2)ˆ if and only if it is contained inB1⊗F1 B2.
Assume that−(a⊗b) is contained in(B1⊗F1 B2)ˆ. Then there is an additive relation of theform a⊗b+∑ck⊗dk ≡ 0 in B1⊗F1 B2. By the definition of the tensor productB1⊗F1 B2, thismust come from an additive relation of the forma+∑ ck ≡ 0 in B1 or an additive relation ofthe formb+∑ dk ≡ 0 in B2. Thus−a∈ B1 or−b∈ B2. In either case,(−a)⊗b=−(a⊗b) =a⊗ (−b) is an element ofB1⊗F1 B2.
Assume that−(a⊗b) is contained inB1⊗F1 B2. By symmetry of the argument, we mayassume that−a∈ B1, i.e. we have an additive relationa+∑ck≡ 0 in B1. Thusa⊗b+∑ck⊗bin B1⊗B2, which shows that−(a⊗b) is an element of(B1⊗F1 B2)ˆ.
If Z = SpecB is an affine blue scheme, then we defineZ = SpecB. It comes together with amorphismγZ : Z→ Z induced by the blueprint morphismB→ B.
29
Remark 1.42. The inverse closure(−)ˆ of a blueprint does not behave well with localizations.It seems that there is no (meaningful) extension of(−)ˆ from affine blue schemes to all blueschemes. To illustrate the incompatibility with localizations, consider the subblueprintB =F1[T] of C = F1[T,S]�〈ST+S≡ 0〉, which is additively closed inC. Let q be the ideal ofCthat is generated byT. ThenCq = F1[T,S±1]�〈ST+S≡ 0〉 ≃ F12[S±1]. The additive closure ofB in Cq (w.r.t. the canonical morphismf : B →C→Cq) is f+(B) ≃ F12 while the localizationBp at the prime idealp= q∩B of B is equal toB= F1[T] itself.
1.9 The unit field and the unit scheme
The units of a ring form naturally a group. In certain cases like polynomial rings over fieldsor discrete valuation rings of positive characteristics, the unit group together with 0 forms afield; but, for a general ring, this is not true. However, the unit group together with 0 and therestriction of the (pre-)addition of the ring has always thestructure of a blue field, which leadsto the following definition.
Let B= A�R be a blueprint. Theunit field of Bis the blue fieldB⋆ = A×∪{0}�R⋆ whereR⋆ = R|A×∪{0} is the restriction ofR to the submonoidA×∪{0} of A. It comes together witha canonical inclusionu : B⋆→ B of blueprints.
Let X be a blue scheme andB= ΓX its global sections. By [25, Lemma 3.25], there exists acanonical morphismX→ SpecB that factors every morphism fromX to an affine blue schemein a unique way. Theunit scheme of Xis the blue schemeX⋆ = SpecB⋆ together with themorphism
υ : X −→ SpecBu∗−→ SpecB⋆ = X⋆.
The blue fieldF⋆(X) = B⋆ is called theunit field of X. The unit schemeX⋆ consists of one pointη, which is corresponds to the unique prime ideal{0} of the unit fieldF⋆(X).
For a pointx of X, we writeF⋆(x) for the unit field of the reduced closed subschemexof X whose support is the closure ofx. We callF⋆(x) theunit field at x. There is a canonicalmorphismψ : F⋆(x)→Γx→OX,x→ κ(x) into the residue field ofx, which is, in general, neitherinjective nor surjective. If, however,X is a reduced scheme that consists of only one pointx,thenψ : F⋆(x)→ κ(x) is an isomorphism. This means, in particular, that
F⋆(X) = F⋆(X⋆) = F⋆(η) = κ(η)
whereη is the unique point ofX⋆.Note that since a morphismf : B→C of blueprints sends 0 to 0 and units to units, it induces
a morphismf ⋆ : B⋆→C⋆ between the unit fields. Thus taking the unit field is an idempotentendofunctor of the category of blueprints whose essential image is the full subcategory of bluefields. Similarly, taking the unit scheme of a blue scheme is an idempotent endofunctor of the
30
category of blue schemes. Note further that the category of unit schemes is dual to the categoryof blue fields since unit schemes are affine.
A blueprintB is generated by its unitsif u+(B⋆) = B for u : B⋆→ B. This is equivalent tosaying thatu+(B⋆) = B induces an isomorphismu+ : (B⋆)+→B+ of semirings. A blue schemeX is generated by its unitsif υ : X→ X⋆ induces an isomorphismυ+ : X+→ (X⋆)+ of semiringschemes.
2 The Tits category
In this section, we will introduce Tits morphisms between blue schemes, which will be thetechnical core of the theory of Tits-Weyl models of algebraic groups. As a first task, we intro-duce the rank space of a blue scheme. With this, we are prepared to define Tits morphisms andto investigate their relationship to morphisms (in the usual sense), which we also calllocallyalgebraic morphisms.
2.1 The rank space
Let X be a blue scheme andx a point ofX. In the following, we will understand byx the closureof x in X together with its structure as a reduced closed subscheme (see Section 1.4).
Definition 2.1. A point x of X is pseudo-Hopfif x is almost of indefinite characteristic,x isaffine,xinv is generated by its units andx+
Zis a flat scheme.
Remark 2.2. If x is pseudo-Hopf andF = F⋆(x) is the unit field ofx, thenΓx+Z
is a quotient ofthe Hopf algebraZ[F×], namely, by the ideal
I = {∑ai−∑b j | ∑ai ≡∑b j in F }.
Recall from Section 1.8 that for an affine blue schemeZ = SpecB, we haveZ = SpecBtogether withγZ : Z→ Z. If x∈ X is a point such thatx is affine, then this yields the morphism
ρx : xγx−→ x
ιx−→ X.
Definition 2.3. Let X be a blue scheme andx a point ofX. Therank rkx of x is the dimensionof the schemex+
QoverQ.
Let X be connected. Then therank of X is
rkX = inf { rkx | x is pseudo-Hopf}
31
if X has a pseudo-Hopf point, and rkX = 0 otherwise. LetZ (X) be the set of all pseudo-Hopfpoints ofX whose rank equals rkX. Thepre-rank space of Xis
X∼ =∐
x∈Z (X)
x
and therank space of XisXrk =
∐
x∈Z (X)
x⋆.
If X =∐
Xi is the disjoint union of connected schemesXi , then
X∼ =∐
X∼i and Xrk =∐
Xrki
are the pre-rank space and the rank space ofX.
We describe some immediate consequences of these definitions. The canonical morphismsρx : x→ x→ X define a morphismρX : X∼→ X and the canonical morphismsυx : x→ x
⋆into
the unit scheme define a morphismυX : X∼→ Xrk. Thus we obtain for every blue schemeXthe diagram
Xrk υX←− X∼ρX−→ X.
By the definition of pseudo-Hopf points,υX : X∼ → Xrk induces an isomorphismυ+X,Z :
X∼,+Z→ Xrk,+
Zof schemes where we use the shorthand notationsX∼,+
Z= (X∼)+
ZandXrk,+ =
(Xrk)+Z
. Thus we obtain a commutative diagram
Xrk,+Z
βXrk��
X∼,+Z
ρ+X,Z //
βX∼��
X+Z
βX
��Xrk X∼
υXoo ρX // X .
In the following, we identifyXrk,+Z
with X∼,+Z
via υ+X,Z, which allows us to considerρ+X,Z as a
morphism fromXrk,+Z
to X+Z
. If υX : X∼→ Xrk is an isomorphism, then we say thatthe rankspace Xrk lifts to X and we may define ˜ρX : Xrk→ X asρX ◦υ−1
X . If additionallyρX is a closedimmersion, then we say thatthe rank space Xrk embeds into X.
We turn to an investigation of the rank spaces. For this, we introduce the notion of blueschemes of pure rank.
32
Definition 2.4. A blue schemeX is of pure rankif it is discrete and reduced, if all points arepseudo-Hopf and ifx→ x is an isomorphism for allx∈ X. We denote the full subcategory ofSchF1 whose objects are blue schemes of pure rank by Schrk
F1.
If X is of pure rank, then everyx∈X has all all potential characteristics with the possible ex-ception of 1 since{x}+
Z= x+
Zis a flat non-empty scheme. A scheme of pure rank is cancellative
since for every connected component{x}, the blueprintΓx≃ Γx= Γ{x} is cancellative.
Proposition 2.5.
(i) The rank space of a blue scheme is of pure rank.
(ii) If X is a scheme of pure rank, then Xrk lifts to X andρX : Xrk→ X is an isomorphism.
Proof. We show (i). LetX be a blue scheme. Since its rank space is the disjoint union ofspectraof blue fields,Xrk is discrete and reduced. Before we show thatx is pseudo-Hopf, we show thatx→ x is an isomorphism for allx∈ Xrk. By definition of the rank space, there is a pseudo-Hopfpoint y∈ X such that{x} = y
⋆. If we denoteΓy by B, thenx= SpecB⋆ and we have to show
that the natural morphismB⋆→ (B⋆)ˆ is an isomorphism. In the case thatB⋆ is with −1, theunit field B⋆ is with inverses and equals its additive closure in(B⋆)inv. In case thatB⋆ is without−1, B⋆ does not contain the additive inverse of any elementb. ThusB⋆ equals the image ofB⋆
in (B⋆)inv, which is the same asB⋆canc, and if∑ai ≡ 0 in B, thenai ≡ 0 for all i. This means that
(B⋆)ˆ ≃ (B⋆canc)ˆ ≃ B⋆
canc≃ B⋆.We show that every pointx of Xrk is pseudo-Hopf. Clearly,x = {x} is affine for every
x∈ Xrk. Let y be a pseudo-Hopf point ofX such that{x}= y⋆. Thenx+
Z≃ y
+Z is a non-empty
flat scheme andx is almost of indefinite characteristic. The blueprintΓxinv = Γxinv is generatedby its units sinceΓxinv ⊂ (Γx+inv)
⋆. Thusx is pseudo-Hopf, which finishes the proof of (i).We show (ii). IfX is of pure rank, then every pointx is pseudo-Hopf of minimal rank in its
component, i.e.Z (X) = X. SinceX is discrete and reduced,F⋆(x)≃OX({x})≃ κ(x) is a bluefield for all x∈ X. Sincex≃ x, we have isomorphisms
x⋆ ∼←− x
∼−→ x∼−→ SpecOX({x})
and, consequently,Xrk ≃ X∼ ≃ X. This completes the proof of the proposition.
We give a series of examples of blue schemes and their rank spaces.
Example 2.6(Tori). The key example of rank spaces are tori overF1. Let X = Grm,F1
be the
spectrum ofB = F1[T±11 , . . . ,T±1
r ]. ThenX consists of one pointη, namely, the 0-ideal ofB,andη is of indefinite characteristic,η = SpecB is affine,Binv is generated by its units since
33
B⋆inv = Binv, andB+
Z= Z[T±1
1 , . . . ,T±1r ] is a freeZ-module. Thereforeη is pseudo-Hopf and we
have isomorphismsXrk ≃ X∼ ≃ X.Note that the rank ofX is r, which equals the rank of the group scheme+Gr
m,Z. This is a firstinstance for the meaning of the rank of a blue scheme. We will see later that, more generally,the rank of a “Tits-Weyl model” of a reductive group scheme equals the reductive rank of thegroup scheme (see Theorem 3.14).
Example 2.7(Monoidal schemes). If X is a monoidal scheme, then every pointx of X is ofindefinite characteristic andΓx+
Zis a freeZ-module. The schemexinv is generated by its units if
and only ifx= {x}, i.e. if and only ifx is a closed point ofX. In this case,x= {x} is an affineblue scheme. Thus the pseudo-Hopf points ofX are its closed points. Therefore, the rank spaceof a monoidal schemeX lifts to X. If X is locally of finite type, thenXrk embeds intoX.
The closed points that belong to the rank space, i.e. that areof minimal rank, are easilydetermined since the rank of a pointx of a monoidal schemeX equals the free rank of the unitgroupO
×X,x of the stalkOX,x atx. For example, the projective spacePn
F1hasn+1 closed points,
which are all of rank 0. Thus(PnF1)rk consists ofn+ 1 points, which are all isomorphic to
SpecF1.
Example 2.8(Semiring schemes). If X is a semiring scheme, then none of its points is almostof indefinite characteristic. Thus both the pre-rank space and the rank space ofX are empty.
Example 2.9. The following are four examples that demonstrate certain effects that can occurfor blue schemes and their rank spaces. The first example shows that pseudo-Hopf points arein general not closed, a fact that we have to consider in case of F1-models of adjoint groups.Let B= F1[T]�〈T ≡ 1+1〉 andX = SpecB. ThenX has two pointsη = (0) andx= (T). Theclosed subschemex is isomorphic toF2, which is of characteristic 2 and not a freeZ-module.The closed subschemeη is B itself and thus affine. The pointη has all potential characteristicsexcept for 2. The unit field ofBinv is B⋆
inv = F1, thusB+Z≃Z≃ (B⋆)+
Z, which showsη is pseudo-
Hopf. ThusX is of rank 0 andZ (X) = {η} is not closed inX. The morphismρX : X∼→ X isan isomorphism, but the rank spaceXrk does not lift toX.
The second example extends the first example in a way such thatthe morphismρX is nolonger injective. LetB = F1[S,T]�〈S+T ≡ 1+1〉 andX = SpecB. ThenX has four pointsη = (0), x= (S), y= (T) andz= (S,T). For similar reasons as in the first example, the pseudo-Hopf points ofX arex andy, which are both of rank 0. ThusZ (X) = {x,y} andX∼ = x∪ y.Both, the closed point ofx and the closed point ofy are mapped toz. ThusρX is not injective.
The third example presents a blue scheme in which one pseudo-Hopf point lies in the closureof another pseudo-Hopf point. LetB = F12[S,T]�〈T2 ≡ 1,S≡ T +1〉 andX = SpecB. ThenX has two pointsη = (0) andx = (S). We havex = SpecF12, which means thatx is pseudo-Hopf of rank 0. The pointη has all potential characteristics except for 1,η = SpecB is affine,
34
B⋆inv = F12[T]�〈T2 ≡ 1〉 = F12[µ2] is generated by its units (whereµ2 is the cyclic group with
two elements) and its extension toZ is the flat ringB+Z= Z[µ2]. Thusη is also pseudo-Hopf of
rank 0. This means thatZ (X) = {η,x} andX∼= η∪x≃X∪SpecF12 does not map injectivelyto X. The rank space ofX is Xrk ≃ SpecF12[µ2]∪SpecF12.
The forth example shows that in generalυ+X : X∼,+→Xrk,+ is not an isomorphism. LetB=F1[S,T±1]�〈T ≡ S+1+1〉. ThenX = SpecB has two pointsx= (S) andη = (0). The schemex is the spectrum ofF1[T±1]�〈T ≡ 1+1〉, whose base extension toZ is the localizationZ(2),which is not a flat ring. The pointη is easily seen to be pseudo-Hopf. ThusX∼ = η = SpecBandXrk = η
⋆= SpecF1[T±1]. The embeddingN[T±1]→ N[S,T±1]�〈T ≡ S+ 1+ 1〉 is not
surjective, thusυ+X : X∼,+→ Xrk,+ is not an isomorphism.
2.2 Tits morphisms
Definition 2.10. Let X andY be blue schemes. ATits morphismϕ : X→Y is a pair(ϕrk,ϕ+)whereϕrk : Xrk→Yrk is a morphism between the rank spaces ofX andY andϕ+ : X+→Y+ isa morphism between the universal semiring schemes ofX andY such that the diagram
Xrk,+Z
ϕrk,+Z //
ρ+X,Z��
Yrk,+Z
ρ+Y,Z��
X+Z
ϕ+Z // Y+
Z
commutes.If ϕ : X→Y andψ : Y→ Z are two Tits morphisms, then the compositionψ ◦ϕ : X→ Z is
defined as the pair(ψrk ◦ϕrk,ψ+ ◦ϕ+). TheTits categoryis the category SchT whose objectsare blue schemes and whose morphisms are Tits morphisms.
To make a clear distinction between Tits morphisms between blue schemes and morphismsin the usual sense, we will often refer to the latter kind of morphism aslocally algebraic mor-phisms(cf. [25, Thm. 3.23] for the fact that locally algebraic morphisms are locally algebraic).
Remark 2.11. For a wide class of blue schemesX, the base extensionυ+X : X∼,+→ Xrk,+ ofυX : X∼→ Xrk is already an isomorphism and we can considerρ+X as a morphism fromXrk,+ toX+. If this is the case forX andY, then a pair(ϕrk,ϕ+) as above is a Tits morphism if and onlyif the diagram
Xrk,+ ϕrk,+//
ρ+X ��
Yrk,+
ρ+Y��X+ ϕ+
// Y+
35
commutes. In fact, all of the blue schemes that we will encounter in the rest of the paper, willbe of this sort.
The Tits category comes together with two important functors: the base extension(−)+ :SchT → Sch+ to semiring schemes, which sends a blue schemeX to X+ and a Tits morphismϕ : X→Y toϕ+ : X+→Y+; and the extension(−)rk : SchT → Schrk
F1to blue schemes of pure
rank, which sends a blue schemeX to its rank spaceXrk and a Tits morphismϕ : X → Y toϕrk : Xrk→Yrk.
The former functor allows us to define the base extensions(−)+k : SchT → Sch+k for everysemiringk or, more generally, the base extension−⊗+ S : SchT → Sch+S for every semiringschemeS.
The latter functor allows us to define theWeyl extensionW : SchT →Sets from the Titscategory to the category of sets that associates to each blueschemeX the underlying set ofits rank spaceXrk and to each Tits morphismϕ : X→Y the underlying map of the morphismϕ : Xrk→Yrk between the rank spaces.
Both locally algebraic morphisms and Tits morphisms between two blue schemesX andY have an base extension to semiring scheme morphisms betweenX+ andY+. The classesof semiring scheme morphisms betweenX+ andY+ that are the respective base extensions oflocally algebraic morphisms and of Tits morphisms betweenX andY are, in general, different.For example, the natural embeddingιN : Gm,N→ A1
N descends to a locally algebraic morphismιF1 : Gm,F1 → A1
F1, but there is no Tits morphism ˜ι : Gm,F1 → A1
F1with ι+ = ιN. As we will
see in the following, Tits morphisms are more flexible in other aspects, which will allow us todescend the group laws of many group schemes to “F1-models” of the group scheme, which isnot the case for locally algebraic morphisms.
In the following, we will investigate the case that a locallyalgebraic morphismϕ : X→Yof blue schemes defines a Tits morphism. Namely, ifϕ mapsZ (X) to Z (Y), then we candefine a morphismϕ∼ : X∼→ Y∼ by ϕ∼|x = (ϕ|x)ˆ for x∈ Z (X). This defines a morphismϕ∼ between the pre-rank spaces ofX andY sincey= ϕ(x) ∈Z (Y) and thus the morphismϕrestricts to a morphismϕ|x : x→ y to which we can apply the functor(−)ˆ. This definition ofϕ∼ behaves well with composition, i.e. ifϕ is as above andψ : Y→ Z is a locally algebraicmorphism that mapsZ (Y) to Z (Z), then(ψ ◦ϕ)∼ = ψ∼ ◦ϕ∼.
Let ϕ : X→Y be a locally algebraic morphism of blue schemes that mapsZ (X) to Z (Y)andϕ∼ : X∼→Y∼ the corresponding morphism between the pre-rank spaces ofX andY. Thenapplying the functor(−)rk to the connected components ofX∼ yields a morphismϕrk : Xrk→Yrk between the rank spaces ofX andY.
We say that a locally algebraic morphismϕ : X→Y that mapsZ (X) to Z (Y) is Titsor thatϕ is a locally algebraic Tits morphism. We denote the category of blue schemes together withlocally algebraic Tits morphisms by SchF1,T . The following proposition justifies the terminol-
36
ogy.
Proposition 2.12. Letϕ : X→ Y be a locally algebraic morphism of blue schemes that mapsZ (X) to Z (Y). Then the pair(ϕrk,ϕ+) is a Tits morphism from X to Y.
Proof. The base extension of the commutative diagram
X∼ϕ∼ //
ρX��
Y∼
ρY��
Xϕ // Y
to semiring schemes yields the commutative diagram
Xrk,+Z
ϕrk,+Z //
ρ+X,Z��
Yrk,+Z
ρ+Y,Z��
X+Z
ϕ+Z // Y+
Z,
which proves the lemma.
Let X andY be two blue schemes. We define theset YT(X) of X-rational Tits points of Yas the set HomT (X,Y) of Tits morphisms fromX to Y. We denote the set of locally algebraicmorphismsX→Y of blue schemes by Hom(X,Y).
Since the rank space of a semiring scheme is empty, we have thefollowing immediateconsequence of the previous proposition.
Corollary 2.13. Let X be a semiring scheme and Y a blue scheme. LetαY : Y+→Y the baseextension morphism. Then the map
HomT (X,Y) −→ Hom(X,Y)(ϕrk,ϕ+) 7−→ αY ◦ϕ+
is a bijection. This means in particular thatSch+N
embeds as a full subcategory intoSchT .
Since the rank space of a blue schemeX of pure rank is isomorphic toX itself, a Titsmorphismϕ : X → Y between two blue schemesX andY of pure rank is determined by themorphismϕrk : X→Y. Therefore, also Schrk
F1is a full subcategory of SchT .
37
Example 2.14(F1-rational Tits points of monoidal schemes). Given a monoidal schemeX, thenfor all its pointsx, the schemesx= x andx
⋆are also monoidal. Sincex is generated by its units
if and only if x= x⋆, the rank spaceXrk lifts to X (if X is locally of finite type,Xrk embeds into
X). Thus a Tits morphismϕ : Y→ X from a schemeY of pure rank is already determined byϕrk : Yrk→ Xrk. This holds, in particular, forY = ∗F1. Since a blue field that is a monoid admitsprecisely one morphism toF1, theF1-rational Tits points ofX correspond to the points of therank spaceXrk. These correspond, in turn, to the setZ (X) of pseudo-Hopf points of minimalrank inX, which is the image of ˜ρX : Xrk→ X.
Note that in case of a connected monoidal scheme, the set ofF1-rational Tits points coin-cides with the sets ofF1-rational points as defined [24].
The following proposition characterizes those semiring scheme morphisms that are baseextensions of Tits morphisms. Note that ifX is a blue scheme andx∈Z (X) is a pseudo-Hopfpoint of minimal rank, thenΓx+
Z= Γx
+Z = F⋆(x)+
Z. In particular, the cancellative blue fieldF⋆(x)
is a subblueprint ofΓx+Z
.
Proposition 2.15. Let X and Y be two blue schemes andϕ+ : X+ → Y+ a morphism. Thenthere exists a morphismϕrk : Xrk→Yrk between the rank spaces of X and Y such that(ϕrk,ϕ+)is a Tits morphism from X to Y if and only if there is a mapϕ0 : Z (X)→Z (Y) such that forall x ∈Z (X) and y= ϕ0(x),
(i) ϕ+(ρ+X(
x+))⊂ ρ+Y
(y+)
and
(ii) the blueprint morphism fx = Γ(ϕ+|x+)+Z : Γy+Z→ Γx+
ZmapsF⋆(y)⊂ Γy+
ZtoF⋆(x)⊂ Γx+
Z.
If ρ+Y : Yrk,+→Y+ is injective, thenϕrk is uniquely determined byϕ+.
Proof. For every pointx∈Z (X), the schemex⋆
consists of one point, which denote by ˜x. Theassociationx→ x is a bijection betweenZ (X) and the points ofXrk. Similarly, we denote by ˜ythe point ofYrk that corresponds toy∈Z (Y).
Given a Tits morphism(ϕrk,ϕ+) from X to Y, defineϕ0(x) = y if ϕrk(x) = y. Evidently,this map satisfies (i) and (ii).
Given a morphismϕ+ : X+ → Y+ and a mapϕ0 : Z (X)→ Z (Y) that satisfies (i) and(ii), we defineϕrk(x) = y (as a map) ifϕ0(x) = y. The morphism(ϕrk)# between the structuresheaves is determined by the blueprint morphismsΓ(ϕrk|{x}) = fx|F⋆(y) : F⋆(y)→ F⋆(x). The
pair (ϕrk,ϕ+) is clearly a Tits morphism fromX toY.Assume thatρ+Y : Yrk,+→Y+ is injective. Then the mapϕ0 : Z (X)→ Z (Y) is uniquely
determined by the condition that there must be ay∈ Z (Y) for everyx∈ Z (X) such thatϕ+
restricts to a morphismϕ+|x+ : x+→ y+. This determinesϕrk : x 7→ y as a map. Ifϕrk : Xrk→
38
Yrk can be extended to a morphism, then property (ii) of the proposition applied the schememorphism(ϕ+|x+)+Z : x+
Z→ y+
Zshows that the morphismϕrk is uniquely determined byϕ+.
This shows the additional statement of the proposition.
3 Tits monoids
In this section, we introduce the notion of a Tits monoid as a monoid in the Tits category. Westart with a reminder on groups and monoids in Cartesian categories. Then we show that theTits category SchT as well as some other categories and functors between them are Cartesian.This allows us to introduce the objects that will be in the focus of our attention for the rest of thepaper: Tits-Weyl models of smooth affine group schemesG of finite type. Roughly speaking, aTits-Weyl model ofG is a Tits monoidG such thatG+
Zis isomorphic toG as a group scheme
and such thatW (G) is isomorphic to the Weyl group ofG .
3.1 Reminder on Cartesian categories
A Cartesian categoryis a categoryC that contains finite products and a terminal object∗C .A Cartesian functoris a (covariant) functor between Cartesian categories thatcommutes withfinite products and sends terminal objects to terminal objects. The importance of Cartesiancategories is that they admit to define group objects, and theimportance of Cartesian functors isthat they send group objects to groups objects. In the following, we will expose some facts on(semi-)group objects. All this is general knowledge and we stay away from proving facts. Formore details, see, for instance, [24, Section 1].
Semigroups
Let C be a Cartesian category. Asemigroup inC is a pair(G,µ) whereG is an object inC andµ : G×G→G is a morphism such that the diagram
G×G×Gµ×id //
id׵��
G×G
µ��
G×Gµ // G
commutes. We often suppressµ from the notation and say thatG is a semigroup object inC .We callµ thesemigroup law of G.
39
An (both-sided) identityfor a semigroupG is a morphismǫ : ∗C →G such that the diagrams
G×∗C(id,ǫ)
//
pr1((◗◗
◗◗◗◗
◗◗◗◗
◗◗◗◗
G×G
µ
��G
and ∗C ×G(ǫ,id)
//
pr2((◗◗
◗◗◗◗
◗◗◗◗
◗◗◗◗
G×G
µ
��G
commute. An identity forG is unique. IfG is with an identity, we say thatG is amonoid inC
and thatµ is itsmonoid law.A group in C is a monoid(G,µ) with identity ǫ : ∗C → G that has aninversion, i.e. a
morphismι : G→G such that the diagrams
G ∆ //
��
G×G(id,ι)
// G×G
µ
��∗C ǫ // G
and G ∆ //
��
G×G(ι,id)
// G×G
µ
��∗C ǫ // G
commute. An inversion is unique. IfG is a group, we callµ its group law.A pair (G,µ) is a semigroup (monoid / group) inC if and only if HomC (X,G) together with
the composition induced byµ is a semigroup (monoid / group) inSetsfor all objectsX in C .Let F : C → D be a Cartesian functor and(G,µ) a semigroup inC . Then(F (G),F (µ))
is a semigroup inD , andF maps an identity to an identity and an inversion to an inversion.For every objectX in C , the map
HomC (X,G) −→ HomD(F (X),F (G))
is a semigroup homomorphism, which maps an identity to an identity and inverses to inversesif they exist.
A homomorphism of semigroups(G1,µ1) and(G2,µ2) in C is a morphismϕ : G1→ G2such that the diagram
G1×G1µ1 //
(ϕ,ϕ)��
G1
ϕ
��G2×G2
µ2 // G2
commutes. IfG1 is with an identityǫ1 andG2 is with an identityǫ2, then a semigroup homo-morphismϕ : G1→G2 is calledunital (or monoid homomorphism) if the diagram
G1
ϕ
��∗C
ǫ133❤❤❤❤❤❤❤❤❤❤❤❤❤
ǫ2 ++❱❱❱❱❱❱❱
❱❱❱❱❱❱
G2
40
commutes. IfG2 is a group, then every semigroup homomorphismϕ : G1→ G2 is unital. ACartesian functorF : C →D sends (unital) semigroup homomorphisms to (unital) semigrouphomomorphisms.
Monoid and group actions
Let (G,µ) be a monoid with identityǫ : ∗C → G andX an object inC . A (unitary left) actionof G on X inC is a morphismθ : G×X→ X such that the diagrams
G×G×X(id,θ)
//
(m,id)��
G×X
�
G×X θ // X
and ∗C ×X(ǫ,id)
//
pr2((❘❘
❘❘❘❘
❘❘❘❘
❘❘❘❘
❘G×X
�
X
commute. LetF : C →D be a Cartesian functor. ThenF sends an actionθ of G onX in C toan actionF (θ) of F (G) onF (X) in D . If θ is unitary, thenF (θ) is unitary. IfG is a monoid,then we call a unitary actionθ : G×X→ X also amonoid action, if G is a group, then we callθ agroup action.
Semidirect products of groups
The direct product of groups(G1,m1) and (G2,m2) in a Cartesian categoryC is the productG1×G2 together with the pairm= (m1,m2) as group law, which is easily seen to define agroup object.
Let (N,mN) and (H,mH) be groups inC and letθ : H ×N→ N be a group action thatrespects the group lawmN of N, i.e. if we define thechange of factors alongθ as
χθ : H×N(∆,id)
// H×H×N(id,θ)
// H×Nχ // N×H,
then the diagram
H×N×N(id,mN) //
(χθ,id)
��
H×Nθ
**❯❯❯❯❯
❯❯❯❯❯❯
N
N×H×N(id,θ)
// N×NmN
44✐✐✐✐✐✐✐✐✐✐✐
commutes. Then the morphism
mθ : N×H×N×H(id,χθ,id) // N×N×H×H
(mN,mH) // N×H
41
is a group law forG= N×H. We say thatG is thesemidirect product of N with H w.r.t.θ andwrite G= N⋊θ H. The group objectN is a normal subgroup ofG with quotient groupH, andH is a subgroup ofG that acts onN by conjugation. The conjugationH×N→ N equalsθ. Ifθ : H×N→ N is the canonical projection to the second factor ofH×N, thenN⋊θ H is equalto the direct product ofN andH (as a group).
If F : C → D is a Cartesian functor and ifG= N⋊θ H in C , thenF (G) = F (N)⋊F (θ)
F (H) in D .
3.2 The Cartesian categories and functors of interest
In this section, we show that the Tits category SchT is Cartesian, which allows us to considermonoids and group objects in this category. We will further investigate certain Cartesian func-tors to and from SchT .
In order to prove that SchT is Cartesian, we have to verify that certain constructions behavewell with products.
Lemma 3.1. Let X and Y be two blue schemes. Then(X×Y)⋆ ≃ X⋆×Y⋆.
lemma by establishing an isomorphism between the corresponding blueprints of global section.By Lemma 1.39,Γ(X×Y) = ΓX⊗F1 ΓY. Let ΓX = AX�RX andΓY = AY�RY be properrepresentations of the global sections ofX andY, respectively. Then
ΓX⊗F1 ΓY = AX×AY�R
for R = 〈RX×{1},{1}×RY〉 and
(ΓX⊗F1 ΓY)⋆ = {0}∪ (AX×AY)×�R
′
for R ′ = R|{0}∪(AX×AY)×. Since(AX×AY)× = A×X ×A×Y , the above expression equals
Lemma 3.2. Let B1 and B2 be two blueprints and B= B1⊗F1 B2 their tensor product. Assumethat both B+1,Z and B+2,Z are non-zero and free asZ-modules. Then the canonical inclusion
u+Z : (B⋆)+Z −→ B+Z
is an isomorphism if and only if the canonical inclusions u+i,Z : (B⋆
i )+Z→ B+
i,Z are isomorphismsfor i = 1,2.
42
Proof. Since(B⋆)+Z= (B⋆
1)+Z⊗+
Z(B⋆
2)+Z
(by the previous lemma) andB+Z= B+
1,Z⊗+Z B+2,Z, the
inclusionu+Z
is clearly an isomorphism if bothu+1,Z andu+2,Z are so.
Assume thatu+Z
is an isomorphism. SinceB+Z= B+
1,Z⊗+Z B+2,Z is non-zero and free, the
isomorphic blueprint(B⋆)+Z= (B⋆
1)+Z⊗+
Z(B⋆
2)+Z
is non-zero and free. Thus both factors(B⋆1)
+Z
and(B⋆2)
+Z
are non-zero and free. Therefore we obtain a commutative diagram
(B⋆1)
+Z⊗+
Z(B⋆
2)+Z
∼u+Z
// B+1,Z⊗+Z B+
2,Z
(B⋆i )
+Z
� �u+i,Z //
?�
OO
B+i,Z
?�
OO
of inclusions of freeZ-modules fori = 1,2 where the morphisms on the top is an isomorphism.If we choose a basis(ai) for (B⋆
1)+Z
and a basis(b j) for (B⋆2)
+Z
, then(ai ⊗ b j) is a basis for(B⋆
1)+Z⊗+
Z(B⋆
2)+Z= B+
1,Z⊗+Z B+2,Z. Thus(ai) is a basis forB+
1,Z and(b j) is a basis forB+2,Z, which
proves thatu+1,Z andu+2,Z are isomorphisms.
Proposition 3.3. Let X1 and X2 be two blue schemes. Then there are canonical identifications
Z (X1×X2) = Z (X1)×Z (X2), (X1×X2)∼ = X∼1 ×X∼2 and (X1×X2)
rk = Xrk1 ×Xrk
2
such thatZ (X1×X2)
� � //
pri��
X1×X2
pri��
Z (Xi)� � // Xi
commutes as a diagram inSets and
Xrk1 ×Xrk
2
pri��
X∼1 ×X∼2υX1×X2oo
ρX1×X2 //
pri��
X1×X2
pri��
Xrki X∼i
υXiooρXi // Xi
commutes as a diagram inSchF1 for i = 1,2.
Proof. If X1 =∐
X1,k and X2 =∐
X2,l are the respective decompositions ofX1 and X2 intoconnected components, thenX1×X2 =
∐X1,k×X2,l is the decomposition ofX1×X2 into con-
nected components. Since these decompositions are compatible with the canonical projectionspri : X1×X2→ Xi, we can assume for the proof thatX1, X2 andX1×X2 are connected.
43
Recall thatZ (X) are the pseudo-Hopf points of a (connected) blue schemeX that are ofminimal rank, i.e. of rank equal to rkX. By Lemma 1.36, the point(x1,x2) ∈ X1×X2 is ofalmost indefinite characteristic if and only if bothx1 ∈ X1 andx2 ∈ X2 are points that are ofalmost indefinite characteristic. Conversely, ifx1∈X1 andx2∈X2 are points of almost indefinitecharacteristic, then the point(x1,x2) exists inX1×X2 by Theorem 1.38. The closed subscheme
(x1,x2) is affine if and only if bothx1 andx2 are affine. By Lemma 3.2, the scheme(x1,x2)+
Z
is flat and non-empty if and only if bothx1+Z
andx2+Z
are flat and non-empty. This shows that(x1,x2) is pseudo-Hopf if and only if bothx1 andx2 are pseudo-Hopf. To complete the proofof Z (X1×X2) = Z (X1)×Z (X2), note that(x1,x2) is of minimal rank if and only if bothx1andx2 are of minimal rank. Since pri(x1,x2) = xi , it is clear from the preceding that the firstdiagram of the proposition is commutative.
SinceZ (X1×X2) = Z (X1)×Z (X2), we have an isomorphism
(X1×X2)∼ =
∐
(x1,x2)∈Z (X1×X2)
ˆ(x1,x2) ≃( ∐
x1∈Z (X1)
x1
)×( ∐
x2∈Z (X2)
x2
)= X∼1 ×X∼2
by Lemma 1.41. It is obvious that this identification makes the right square of the seconddiagram in the proposition commutative.
By the preceding and Lemma 3.1, we have canonical isomorphisms
(X1×X2)rk = ((X1×X2)
∼)⋆ ≃ (X∼1 ×X∼2 )⋆ ≃ (X∼1 )⋆× (X∼2 )⋆ = Xrk1 ×Xrk
2 .
It is obvious that the left square of the second diagram of theproposition commutes.
As a side product of the equalityZ (X1×X2) = Z (X1)×Z (X2), we have the followingfact.
Corollary 3.4. Let X1 and X2 be connected blue schemes. Thenrk(X1×X2) = rkX1+ rkX2.
For brevity, we will denote SpecB by ∗B, which should emphasize that∗B is the terminalobject in SchB, the category of blue schemes with base scheme∗B = SpecB. In particular,∗F1
is the terminal object of SchF1. Note that∗B is the terminal object of both SchB and Sch+B if Bis a semiring.
Theorem 3.5. The categorySchT is Cartesian. Its terminal object is∗F1 and the product oftwo blue schemes inSchT is represented by the product inSchF1.
Proof. We begin to show that∗F1 is terminal. First note that∗F1 is of pure rank, i.e.∗rkF1= ∗F1,
and that∗+F1
= ∗N. Let X be a blue scheme. Then there are a unique morphismϕrk : Xrk→∗rkF1
44
and a unique morphismϕ+ : X+ → ∗N. Thus uniqueness is clear. It is easily verified that(ϕrk,ϕ+) is a Tits morphism.
To prove that the product of two blue schemesX1 andX2 in SchF1 together with the canonicalprojections pri : X1×X2→ Xi (which are Tits by Proposition 3.3) represents the product inSchT , consider two Tits morphismsϕ1 : Y→ X1 andϕ2 : Y→ X2 for a blue schemeY, i.e.ϕi = (ϕrk
i ,ϕ+i ) for i = 1,2. We defineϕrk asϕrk
1 ×ϕrk2 : Yrk→ Xrk
1 ×Xrk2 andϕ+ asϕ+
1 ×ϕ+2 :
Y+→X+1 ×X+
2 . We have to show that the pairϕ=(ϕrk,ϕ+) is a Tits morphismϕ :Y→X1×X2.Once this is shown, it is clear thatϕi = pri ◦ϕ and thatϕ is unique with this property.
To verify thatϕ is Tits, consider fori = 1,2 the diagram
Y+Z
ϕ+Z //
◗◗◗◗
◗◗◗◗
◗◗◗◗
◗◗◗◗
◗
ϕ+i,Z
((◗◗◗◗
◗◗◗◗
◗◗◗◗
◗◗◗◗
◗
X+1,Z×+ X+
2,Z
pri
��
Yrk,+Z
ρ+Y,Z
99ssssssssssss ϕrk,+Z //
ϕrk,+i,Z
&&◆◆◆◆
◆◆◆◆
◆◆◆◆
◆◆◆◆
◆◆◆◆
◆◆◆◆
◆◆◆◆
◆◆◆◆
Xrk,+1,Z ×+ Xrk,+
2,Z
pri
��
ρ+X,Z
55❧❧❧❧❧❧❧❧❧❧❧❧❧
X+i,Z
Xrk,+i,Z
ρ+Xi ,Z
55❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧
which we know to commute up to the top square. The top square commutes because bothρ+X,Z◦ϕ
rk,+Z
andϕ+Z◦ρ+Y,Z equal the canonical morphismsYrk,+
Z→X+
1,Z×+X+2,Z that is associated
to the morphismsρ+Xi ,Z◦ϕrk,+
i,Z = ϕ+i,Z ◦ρ+Y,Z : Y→ X+
i,Z for i = 1,2. This shows thatϕ is Tits.
Proposition 3.6. The categorySchF1,T is Cartesian. Its terminal object is∗F1 and the productof two blue schemes inSchF1,T is represented by the product inSchF1.
Proof. Since∗F1 is of pure rank, the unique morphismϕ : X→∗F1 is Tits for each blue schemeX by Proposition 2.12. Thus∗F1 is a terminal object in SchF1,T .
We show that the productX1×X2 of two blue schemesX1 andX2 in SchF1 represents theproduct in SchF1,T . First note that the canonical projectionsπi : X1×X2→ Xi are Tits byProposition 3.3. Letϕ1 : Y→ X1 andϕ2 : Y→ X2 be two locally algebraic Tits morphisms andϕ= ϕ1×ϕ2 : Y→ X1×X2 the canonical morphism. Ify∈Z (Y), thenϕ(y) = (ϕ1(y),ϕ2(y))is an element ofZ (X1)×Z (X2) = Z (X1×x2). Thusϕ is Tits. This shows thatX1×X2 is theproduct ofX1 andX2 in SchF1,T ,
45
Proposition 3.7. The categorySchrkF1
is Cartesian. Its terminal object is∗F1 and the product of
two blue schemes inSchrkF1
is represented by the product inSchF1.
Proof. Since SchrkF1is a full subcategory of SchF1, it suffices to show that the terminal object
of SchF1 and the product of two schemes of pure rank (taken in SchF1) are in SchrkF1. The
terminal object∗F1 is of pure rank. IfX1 andX2 are of pure rank, i.e. discrete, reduced and ofalmost indefinite characteristic, thenX1×X2 is also discrete, reduced and of almost indefinitecharacteristic. This proves the proposition.
We collect the results of this section in the following theorem, which gives an overview ofthe Cartesian categories and the Cartesian functors between them, which will be of importancefor the rest of this paper. Before we can state it, we fix some notation. We denote byι : Schrk
F1→
SchF1,T andι : SchF1,T → SchF1 the inclusions as subcategories.The functorT : SchF1,T → SchT is the identity on objects and sends a locally algebraic
Tits morphismϕ : X→Y to the Tits morphism(ϕrk,ϕ+) : X→Y (cf. Proposition 2.12). Sincea morphismf : B→C of blueprints is uniquely determined by the morphismf+ : B+→C+ ofsemirings and since morphisms of blue schemes are locally algebraic (see [25, Thm. 3.23]), amorphismϕ : X→Y of blue schemes is uniquely determined by its base extensionϕ+ : X+→Y+. This means thatT : SchF1,T → SchT is faithful and that we can, in fact, consider SchF1,T
as a subcategory of SchT .The functorW : SchT → Sets is the Weyl extension, which factors through Schrk
F1(cf.
Section 2.2). For any semiringk, the base extension(−)+k : SchT → Sch+k to semiring schemesoverk factors through Sch+
N.
Theorem 3.8.The diagram
SchrkF1
id //� t
ι
&&◆◆◆◆
◆◆◆◆
◆
SchrkF1
W // Sets
SchF1,T� � T //� t
ι
''◆◆◆◆
◆◆◆◆
SchT
(−)rk99rrrrrrrr
(−)+
&&▲▲▲▲
▲▲▲▲
SchF1
(−)+ // Sch+N
(−)+k // Sch+k
is an essentially commutative diagram of Cartesian categories and Cartesian functors where kis an arbitrary semiring.
Proof. All categories are Cartesian: the terminal object ofSets is the one-point set∗ and theproduct is the Cartesian product of sets; the terminal object of SchF1 is∗F1 and the product is thefibre product−×F1−; for any semiringk, the terminal object of Sch+k is ∗k and the product is
46
the fibre product−×+k −; that SchT , SchF1,T and SchrkF1are Cartesian is subject of in Theorem
3.5, Proposition 3.6 and Proposition 3.7, respectively.All functors are Cartesian: since the terminal object and the product in SchT , SchF1,T and
SchrkF1
coincides with the terminal object and the product in SchF1, the inclusionsι : SchrkF1→
SchF1,T , ι : SchF1,T → SchF1 andT : SchF1,T → SchT are Cartesian; the identity functorid : Schrk
F1→Schrk
F1is evidently Cartesian;(−)rk : SchT →Schrk
F1and is Cartesian by Proposition
3.3 and since∗F1 is of pure rank;(−)+ : SchT →Sch+N
and(−)+ : SchF1→Sch+N
are Cartesiansince(X×Y)+ = X+×+Y+; (−)+k : Sch+
N→ Sch+k is Cartesian since(X×+Y)+k = X+
k ×+k Y+k ;
andW : SchrkF1→Sets is Cartesian since the underlying set of∗F1 is the one-point set∗ and
since every point of a scheme of pure rank is of almost indefinite characteristic and thereforeW (X1×X2) is the Cartesian product ofX1,X2 ∈ Schrk
F1by Theorem 1.38.
The composition(−)rk◦T ◦ ι : SchrkF1→Schrk
F1is isomorphic to the identity functor because
Xrk ≃ X for a blue scheme of pure rank andϕrk = ϕ for a morphism between blue schemesof pure rank. The functors(−)+ ◦T : SchF1,T → Sch+
Nand(−)+ ◦ ι : SchF1,T → Sch+
Nare
isomorphic because in both cases a blue schemeX is sent toX+ and a locally algebraic Titsmorphismϕ : X→Y is sent toϕ+ : X+→Y+. This finishes the proof of the theorem.
3.3 Tits-Weyl models
Definition 3.9. A Tits monoidis a monoid in SchT . TheWeyl monoidof a Tits monoid(G,µ)is the monoid(W (G),W (µ)) in Sets. In case thatW (G) is a group, we call it also theWeylgroup of G.
Often, we will suppress the semigroup law from the notation if it is not necessarily needed.A Tits modelof a group schemeG is a Tits monoidG whose base extensionG+
Zto schemes is
isomorphic toG as a group scheme.
Remark 3.10. Note that the definition of a Tits model given here differs from that in [16].While we define a Tits model of a group scheme to be a monoid in SchT whose base extensionis isomorphic to the group scheme, the definition of a Tits model of a Chevalley group schemeG in [16] means a cancellative blue schemeG such thatG+
Z≃ G (as schemes) and such that the
number of morphisms∗F1→G coincides with the number of elements in the Weyl group ofG .The notion of a Tits-Weyl model as defined below will combine these two aspects in a certainway.
Let (G,µ) be a Tits monoid with identityǫ : ∗F1→G. Let ebe the image point ofǫrk in Grk
ande= {e}= Specκ(e) the closed subscheme ofGrk with supporte. We calle theWeyl kernelof G.
47
Lemma 3.11.Let (G,µ) be a Tits monoid ande its Weyl kernel. The semigroup lawµrk of Grk
restricts to a semigroup lawµe of e, which turnse into a commutative group inSchrkF1
.
Proof. Let ǫ : ∗F1 → G be the identity ofG. Sinceǫrk is a both-sided identity forµrk, thesemigroup law ofGrk restricts to a semigroup lawµrk
e : e× e→ e. Sincee is of pure rank,µe : e× e→ e is a Tits morphism and thus a semigroup law fore in Schrk
F1.
We verify that this semigroup law is indeed a commutative group law. Its identity is therestriction ofǫrk to ǫe : ∗F1→ e. Consider the comultiplicationm= Γµe : κ(e)→ κ(e)⊗F1κ(e).It sends an elementa∈ κ(e) to an elementb⊗c= m(a) of κ(e)⊗F1 κ(e). The coidentity yieldsa commutative diagram
κ(e)m
rr❞❞❞❞❞❞❞❞❞❞❞
❞❞❞❞❞❞❞❞
❞❞❞❞❞❞❞❞
❞❞
m
,,❩❩❩❩❩❩❩❩❩❩
❩❩❩❩❩❩❩❩
❩❩❩❩❩❩❩❩
❩❩❩
id
��
κ(e)⊗F1 κ(e) Γǫe⊗id++❲❲❲❲
❲❲❲❲❲❲
❲❲κ(e)⊗F1 κ(e),id⊗Γǫe
ss❣❣❣❣❣❣❣❣
❣❣❣❣
F1⊗F1 κ(e) ≃ κ(e) κ(e)⊗F1 F1≃
which means that 1⊗a= 1⊗c anda⊗1= b⊗1. Thus,m(a) = a⊗a, which implies thatµe iscommutative. The inverseιe of µe is defined by the morphismΓιe : κ(e)→ κ(e) that sends 0 to0 anda to a−1 if a 6= 0.
Lemma 3.12. Let G be a Tits monoid that is locally of finite type ande its Weyl kernel. Thenthe group schemee+
Zis diagonalizable, i.e. a closed subgroup of a split torus over Z.
Proof. Sincee+Z
is affine, the claim of the lemma means that the global sections B= κ(e)+Z
ofe+Z
are a quotient ofZ[T±11 , . . . ,T±1
n ] by some ideal wheren∈ N. The global sectionsκ(e) of eform a blue field, andB is generated by the image of the multiplicative groupκ(e)×in B. SinceB is a finitely generated algebra, it is already generated by a finitely generated subgroupH ofκ(e)×. In other words,B is a quotient of the group ringZ[H]. By the structure theorem forfinitely generated abelian groups,H is the quotient of a finitely generated free abelian group ofsome rankn. ThusZ[H], and thereforeB, is a quotient ofZ[T±1
1 , . . . ,T±1n ].
This shows thate+Z
is a closed subscheme of a torus. Thereforee+Z
can neither have aunipotent component nor a semisimple component. As a flat commutative group scheme itmust be an extension of a constant group scheme by a torus. Since e is commutative,e+
Zis
commutative and therefore diagonalizable.
The Weyl kernele of G is the identity component ofGrk, i.e. the connected componentthat contains the image of the identityǫrk : ∗F1→ Grk. Thuse+
Zequals the identity component
(Grk,+Z
)0 of Grk,+Z
, which is a normal subgroup ofGrk,+Z
, and we can consider the quotient group
48
Grk,+Z
/e+Z
. In the following, we are interested in comparing this quotient to the Weyl group ofG.
We recall some notions from the theory of group schemes. LetG be a group scheme offinite type. For a torusT of G , we denote its centralizer byC(T) and its normalizer byN(T).We defineW (T) = N(T)/C(T), theWeyl group relative to T, which is quasi-finite, etale andseparated over SpecZ (cf. [2, ]). This means, in particular, thatΓW (T) is a flatZ-module offinite rank, or, in other words, thatW (T) is a finite group scheme. Since SpecZ has no non-trivial connected finite etale extensions (cf. [12, Section 6.4]),W (T) is indeed a constant groupscheme overZ.
A maximal torus ofG is a subgroupT of G that is a torus such that for every geometricpoint s : Speck→ Z of SpecZ, the torusTs is maximal inGs (cf. [2, XII.1.3]). A maximal torusT of G is always split (cf. [12, Section 6.4]). Note that in general, G does not have a maximaltorus. If T is a maximal torus ofG , then the rank ofT is called thereductive rank ofG andC(T) is called aCartan subgroup ofG . In case of a maximal torus, we callW (T) simply theWeyl group ofG . If G is affine smooth andT a maximal torus ofG , then the geometric fibreW (T)s is the Weyl group ofGs, which is also called thegeometric Weyl group (overk). SinceW (T) is a constant group scheme, the groupW (T)(R) of R-rational points does not depend onthe chosen ringR. We call this group theordinary Weyl group ofG .
Let G be a Tits model ofG , i.e. we identifyG+Z
with G , and lete be the Weyl kernel ofG.A consequence of Lemma 3.12 is thate+
Zcontains a unique maximal torusT (cf. [2, XII.1.12]).
We callT thecanonical torus ofG (with respect to G). Thene+Z
is contained in the centralizer
C(T) of T in G . Sincee+Z
is a normal subgroup ofGrk,+Z
andT is the unique maximal torus of
e+Z
, the subgroupGrk,+Z
normalizesT in G , which means thatGrk,+Z
embeds intoN(T). Thus we
obtain a morphismΨe : Grk,+Z
/e+Z→W (T) of group schemes.
Definition 3.13. Let G be an affine smooth group scheme of finite type. ATits-Weyl model ofG is a Tits modelG of G such that the canonical torusT is a maximal torus ofG and such thatΨe : Grk,+
Z/e+
Z→W (T) is an isomorphism of group schemes wheree is the Weyl kernel ofG.
Before we can collect the first properties of a Tits-Weyl model of a group schemeG , wehave to fix some more notation. We define therank of Gas the rank of the connected blueschemeG0 (as a blue scheme). Note that the rank of each connected component ofG is equalto the rank of the identity componentG0 of G since each connected component ofG is a torsorof G0.
Let G be an affine smooth group scheme of finite type with maximal torusT. In general,the Weyl groupW cannot be realized as theZ-rational points of a finite subgroup ofG . Thisis an obstacle to realizeW as theF1-points of a group scheme overF1 as suggested by Tits inhis ’56 paper [33] (for more explanation on this, cf. [24, Problem B] and [10]). However, in
49
caseG is a split reductive group scheme, Tits describes himself inhis paper [34] from ’66 acertain extensionW of W, called theextended Weyl groupor Tits group, which can be realizedas theZ-valued points of a finite flat group schemeW (T) of G . Namely,W (T) is definedasN(T)(Z)-translates of the 2-torsion subgroupT[2] of T whereN(T) is the normalizer ofT.This yields a short exact sequence of group schemes
1 −→ T[2] −→ W (T) −→ W (T) −→ 1,
and thus an isomorphismW≃ W (Z)/T(Z) sinceT(Z) = T[2](Z) is a 2-torsion group.Let G be a Tits monoid andSa blue scheme. SinceG is a monoid in SchT , the setGT(S) =
HomT (S,G) of S-rational Tits points ofG is a monoid inSets. If S= SpecB, we also writeGT(B) for GT(S).
Theorem 3.14.Let G be an affine smooth group scheme of finite type. IfG has a Tits-Weylmodel G, then the following properties hold true.
(i) The Weyl groupW (G) is canonically isomorphic to the ordinary Weyl group W ofG .
(ii) The rank of G is equal to the reductive rank ofG .
(iii) The group GT(F1) of F1-rational Tits points of G is a subgroup ofW (G).
(iv) If G is a split reductive group scheme, then GT(F12) is canonically isomorphic to theextended Weyl groupW ofG .
Proof. We prove (i). The ordinary Weyl groupW equals the groupW (T)(C) of C-rationalpoints of the geometric Weyl group overC. The isomorphisms
W (G )(C) ≃ N(T)(C)/C(T)(C) ≃ Grk(C)/e(C)
show that the elements ofW (G )(C) stay in one-to-one correspondence with the connectedcomponents ofGrk, which in turn is the underlying set ofW (G). It is clear that the groupstructures coincide.
We prove (ii). Lete be the Weyl kernel ofG. The rank ofG equals the dimension of thevarietye+
QoverQ. By Lemma 3.12,e+
Qis a closed subgroup of a split torus, which means that
it is an extension ofTQ by a finite group scheme whereT is the maximal torus ofe+Z
. Thereforethe dimension ofe+
Qequals the rank ofT, which is the reductive rank ofG sinceT is a maximal
torus ofG .We prove (iii). We denote as usual SpecF1 by ∗F1. A Tits morphismϕ : ∗F1 → G is de-
termined by the set theoretical image ofϕrk : ∗F1 → Grk since there is at most one morphism
50
from a blue field, i.e. from the residue field of the image point, to F1. Note that necessarilyϕ+ = ϕrk,+. ThusGT(F1) is a subset ofW (G) and it inherits its semigroup structure fromW (G). SinceW (G) is a finite group,GT(F1) is also a group.
We prove (iv). IfG is a split reductive group, then the subgroupsT, e+Z
andC(T) coincide,
and consequently alsoN(T) andGrk,+Z
coincide. We write brieflyN for N(T). For a pointx ofGrk, the scheme{x}+
Zis a translatenT of T by some elementn∈ N(Z). The schemenT is iso-
morphic toT, i.e. nT is isomorphic to the spectrum ofZ[T±11 , . . . ,T±1
r ], wherer is the rank ofT. Its largest blue subfield isF12[T±1
1 , . . . ,T±1r ]. The mapκ(x)→ κ(x)+
Z≃Z[T±1
1 , . . . ,T±1r ] fac-
torizes throughκ(x)inv = κ(x)⊗F1 F12 sinceκ(x)+Z
is with inverses. Sinceκ(x)inv is a blue fieldwith inverses, it must be equal toF12[T±1
stay in one-to-one correspondence with the morphismsκ(x)+Z→Z, i.e. withnT(Z) = nT[2](Z).
Note that similar to the case ofF1-rational points, a Tits morphismϕ : SpecF12 → G isdetermined byϕrk. This means that everyF12-rational Tits pointϕ : SpecF12→G with imagex is given by a morphismκ(x)→ F12 of blueprints. ThereforeGT(F12) is isomorphic to thesubgroup ofG(Z) that is generated by the translatesnT[2](Z) wheren ranges throughN(Z).This subgroup is by definition the extended Weyl groupW of G . This finishes the proof of thetheorem.
3.4 Groups of pure rank
In this section, we will explain first examples of Tits models, namely, of constant group schemesand split tori. All these examples will be of pure rank, thus the group law will be indeed a locallyalgebraic morphism of blue schemes, which makes the description particularly easy. The Titsmonoids appearing in this section are indeed group objects in SchrkF1
. In case of a torus, or, moregenerally, of a semidirect product of a torus by a constant group scheme satisfying a certaincondition, the described Tits model is a Tits-Weyl model.
Constant groups
Let G be a finite group. Then the constant group schemeGZ that is associated toG is definedas the scheme
GZ = Spec∏g∈G
Z
51
together with the multiplicationµZ : GZ×GZ→GZ that is defined by the comultiplication
ΓµZ : ∏g∈G
Z −→ ∏g∈G
Z⊗+Z ∏
g∈G
Z
(ag)g∈G 7−→ ∑g1,g2∈G
ag1g2eg1⊗eg2
whereeh is the element(ag)g∈G of ∏g∈GZ with ag = 1 if g= h andag = 0 otherwise.This group scheme descends to a group objectGF1 in Schrk
F1. Namely, define the scheme
GF1 as Spec∏g∈GF1, which is obviously of pure rank. Then, we have indeed canonical isomor-phisms(GF1)Z ≃ (GF1)
+Z≃GZ, which justifies our notation. The group lawµZ descends to the
group lawµF1 : GF1×GF1→GF1 that is defined by the comultiplication
ΓµF1 : ∏g∈G
F1 −→ ∏g∈G
F1⊗F1 ∏g∈G
F1 =(∏g∈G
F1)×(∏g∈G
F1)�R
(ag)g∈G 7−→ (ag1g2)g1,g2∈G
whereR is the pre-addition that is generated by the relations(a,0) ≡ (0,0) ≡ (0,a) for a ∈∏g∈GF1.
The morphismµF1 : GF1×GF1 → GF1 is indeed a group law: its identity is the morphismǫF1 : ∗F1→GF1 given by
ΓǫF1 : ∏g∈G
F1 −→ F1
(ag)g∈G 7−→ ae
wheree is the identity element ofG and its inverse is the morphismιF1 : GF1→GF1 given by
ΓǫF1 : ∏g∈G
F1 −→ ∏g∈G
F1.
(ag)g∈G 7−→ (ag−1)g∈G
This shows thatGF1 together withµF1 is a group object in SchrkF1and therefore in SchT . In
particular,GF1 is a Tits model ofGZ.The Weyl kernele of GF1 is its identity componentGF1,0 = SpecF1. Thus the canonical
torus ofGF1 equals the identity componentGZ,0 of GZ, which is a maximal torus ofGZ. Both,
its centralizer and its normalizer is the whole group schemeGZ ≃ Grk,+Z
. Thus the morphism
Ψe : Grk,+Z
/e+Z→W (T) is an isomorphism only for the trivial group scheme∗F1.
52
Split tori
We proceed with the description of a Tits-Weyl model of the split torus+Grm,Z of rankr, which
is SpecZ[T±11 , . . . ,T±1
r ]+ as a scheme. Its group lawµ+Grm,Z
: +Grm,Z×+Gr
m,Z→+Grm,Z is given
by the comultiplication
Γµ+Grm,Z
: Z[T±11 , . . . ,T±1
r ]+ −→ Z[(T ′1)±1, . . . ,(T ′r )
±1,(T ′′1 )±1, . . . ,(T′′r )
±1]+
that mapsTi to T ′i ⊗T ′′i for i = 1, . . . , r.This group scheme has the Tits model(Gr
m,F1,µ) whereGr
m,F1= SpecF1[T
±11 , . . . ,T±1
r ] andµ : Gr
m,F1×Gr
m,F1→Gr
m,F1given by the morphism
Γµ : F1[T±11 , . . . ,T±1
r ] −→ F1[(T′1)±1, . . . ,(T′r )
±1,(T ′′1 )±1, . . . ,(T′′r )
±1]
that mapsTi to T ′i ⊗ T ′′i for i = 1, . . . , r. Note thatGrm,F1
has precisely one point, which isof indefinite characteristic, and thatGr
m,F1is reduced. This means thatGr
m,F1is of pure rank
and thatµ is Tits. Its identity is the morphismǫ : ∗F1 → Grm,F1
given by the morphismΓǫ :
F1[T±11 , . . . ,T±1
r ]→ F1 that maps all elementsa 6= 0 to 1 inF1. Its inverse is the morphismι : Gr
m,F1→Gr
m,F1given by the morphismΓι : F1[T
±11 , . . . ,T±1
r ]→ F1[T±11 , . . . ,T±1
r ] that maps
Ti to T−1i for i = 1, . . . , r. ThusGr
m,F1is a group object in SchrkF1
and therefore in SchT .The Weyl kernel ofGr
m,F1is Gr
m,F1itself. The canonical torusT of Gr
m,F1is +Gr
m,Z, whichis further its own normalizerN. ThusT is a maximal torus ofGr
m,F1and the morphismΨe :
(Grm,F1
)rk,+Z
/e+Z→W (T) is an isomorphism of group schemes. This shows thatGr
m,F1is a Tits-
Weyl model of+Grm,Z. Its Weyl group is the trivial group and consequently(Gr
m,F1)T (F1) is
the trivial group. Since the rank of+Grm,Z is r, the group(Gr
m,F1)T (F12) is (Z/2Z)r .
Semi-direct products of split tori by constant group schemes
Group schemesN of the form+Grm,Z ⋊θ GZ appear as normalizers of split maximal tori in
reductive group schemes and will be of a particular interestin the following. We will describegroups in SchrkF1
that base extend to the group, but we can already conclude forabstract reasons
that a model ofN exists in SchrkF1if θ descends to a morphism in Schrk
F1. More precisely, the
conjugation actionθ : GZ×+Z
+Grm,Z→ +Gr
m,Z restricts to morphisms
θg : {g} ×+Z
+Grm,Z ≃ {g} ×+
Z+Gr
m,Z ×+Z {g−1} µ◦(µ,id)−→ +Gr
m,Z
for everyg ∈ G. This yields blueprint morphismsΓθg : Z[T±11 , . . . ,T±1
r ]→ Z[T±11 , . . . ,T±1
r ].
If the imagesθg(Ti) are of the form∏rj=1T
ei, j (g)j for certain exponentsei, j(g) ∈ Z for all i, j =
53
1, . . . , r andg ∈ G, then the actionθ descends to an actionθ of GF1 on Grm,F1
. Thus we can
form the semidirect productN =Grm,F1
⋊θ GF1 in SchrkF1
, which is an group scheme whose baseextension to rings isN. By definition,+Gr
m,Z is normal inN. Thus if the centralizer ofT is T
itself, thenN is a Tits-Weyl model ofN. We summarize this in the following statement.
Proposition 3.15. Let G be a group andθ : GZ×+Z
+Grm,Z→ +Gr
m,Z be a group action that is
defined by integers ei, j(g) as above. Thenθ descends to a group actionθ : GF1×Grm,F1→Gr
m,F1
andN =Grm,F1
⋊θ GF1 is a group inSchrkF1
whose base extension toZ is N+Z= +Gr
m,Z⋊θ GZ.If for every g∈G different from the neutral element e∈G, the matrix A(g)= (ai, j(g))i, j=1,...,r
is different from the identity matrix, thenN is a Tits-Weyl model of N.
4 Tits-Weyl models of Chevalley groups
In this section, we prove for a wide class of Chevalley groupsthat they have a Tits-Weyl model.Namely, for special linear groups, general linear groups, symplectic groups, special orthogonalgroups and all Chevalley groups of adjoint type. As a first step, we establish Tits-Weyl modelfor the special linear groups. Tits-Weyl models for all other groups of the above list but theadjoint Chevalley groups can be obtained by a general principle for subgroups of the speciallinear groups, which is formulated in Theorem 4.7, a centralresult of this section. Finally, wefind Tits-Weyl models of adjoint Chevalley groups by a close examination of explicit formulasfor their adjoint representation over algebraically closed fields.
The precise meaning of the term Chevalley group varies within the literature. The originalworks of Chevalley refer to simple groups (cf. [8]) and, later, to semisimple groups (cf. [9]).When we refer to a Chevalley group in this text, we mean, in a more loose sense, a split reductivegroup scheme. But note that in fact almost all of the Chevalley groups that occur in the followingare semisimple. As a general reference for background on Chevalley groups and split reductivegroup schemes, see SGA3 ([1],[2],[3]), Demazure and Gabriel’s book ([17]) or Conrad’s lecturenotes ([12]). There are plenteous more compact and readableaccounts of root systems andChevalley bases of Chevalley groups (for instance, cf. [7]).
4.1 The special linear group
In this section, we describe a Tits-Weyl model SLn of the special linear group SL+n,Z.
To begin with, consider a closed subscheme of+AnZ of the formX =SpecZ[T1, . . . ,Tn]
+/IwhereI is an ideal ofZ[T1, . . . ,Tn]
+. The set
RI ={
∑ai ≡∑b j
∣∣∣∑ai−∑b j ∈ I
}
54
is a pre-addition forF1[T1, . . . ,Tn] and defines a blueprintB= F1[T1, . . . ,Tn]�RI . We call theblue schemeX = SpecB an F1-model of the schemeX . It satisfiesX+
Z≃X . Further, the
canonical morphismF1[T1, . . . ,Tn]→ B of blueprints defines a closed embeddingι : X→ AnF1
,andι+
Zis equal to the embedding ofX = X+
Zas closed subscheme of+An
Z.The underlying topological space ofX is a subspace of the underlying topological space of
AnF1
. Recall from Example 1.11 that the prime ideals ofF1[T1, . . . ,Tn] are of the formpI = (Ti)i∈I
whereI ranges through all subsets ofn = {1, . . . ,n}. Thus the underlying topological space ofAnF1
is finite and completely determined by the rulepI ≤ pI ′ if and only if I ⊂ I ′ (cf. section 1.2).In particular, this applies to the special linear group SL+
(which expresses the condition that the determinant of a matrix (ai, j) equals 1) and whereµ+Z
isdefined by the comultiplication
m+Z
= Γµ+Z
: Z[SLn]+ −→
(Z[SLn]
+)⊗+
Z
(Z[SLn]
+).
Ti, j 7−→n
∑k=1
T ′i,k⊗T ′′k, j
Thus SL+n,Z is a closed subscheme of+An2
Z , and therefore has anF1-model SLn = SpecF1[Ti, j ]�RI .
Before we describe the group law for SLn in SchT , we determine the rank space of SLn.Since SLn is a closed subscheme ofAn2
, each point of SLn is of the formpI = (Ti, j)(i, j)∈I
whereI is a subset ofn2. We writepσ = pI(σ) for I(σ) = n−{(i,σ(i))}i∈n whereσ ∈ Sn is apermutation.
Proposition 4.1. The underlying set ofSLn is
{ pI | I ⊂ I(σ) for someσ ∈ Sn }.The rank ofSLn is n−1 and the set of pseudo-Hopf points of minimal rank is
Z (SLn) = { pσ | σ ∈ Sn },which equals the set of closed points ofSLn. The residue field ofpσ is
κ(pσ) = F1[T±1i,σ(i)]�〈
n
∏i=1
Ti,σ(i) ≡ 1〉
55
if σ is an element of the alternating group An and
κ(pσ) = F12[T±1i,σ(i)]�〈
n
∏i=1
Ti,σ(i)+1≡ 0〉
if σ ∈ Sn−An. The rank space is
SLrkn =
∐
σ∈Sn
Specκ(pσ)
and embeds intoSLn.
Proof. The pre-additionR of the global sectionsF1[SLn] = F1[Ti, j ]�R of SLn is generated bythe relation
∑σ∈An
n
∏i=1
Ti,σ(i) ≡ ∑σ∈Sn−An
n
∏i=1
Ti,σ(i) + 1. (1)
ThuspI is a prime ideal if and only if there is at least oneσ ∈ Sn such that∏ni=1Ti,σ(i) /∈ pI . In
other words, a prime idealpI of F1[Ti, j ]i, j∈n generates a prime ideal ofF1[SLn] if and only ifthere is aσ ∈ Sn such thatpI ⊂ pσ. Consequently, the closed points of SLn are the prime idealspσ for σ ∈ Sn.
We determine the pseudo-Hopf points of SLn. For a pointpI , the coordinate blueprint of theclosed subschemepI of SLn is ΓpI = F1[Ti, j |i, j ∈ n]�〈R〉 whose pre-additionR is generatedby the relation (1) together with the relationsTi, j ≡ 0 for (i, j) ∈ I . If I = I(σ) for someσ ∈ Sn,then Γpσ = κ(pσ) and in the relation (1) survive only the “1” and one other termif Ti, j issubstituted by 0 for all(i, j) ∈ I , i.e. it looks like
n
∏i=1
Ti,σ(i) ≡ 1 orn
∏i=1
Ti,σ(i) + 1 ≡ 0
depending on the sign ofσ. In both cases,Ti,σ(i) is invertible inΓpσ for i = 1, . . . ,n. Thusκ(pσ)
is as claimed in the proposition. Further, it is clear thatpσ is affine, thatpσ⋆ = pσ, thatpσ+Z is afreeZ-module and thatpσ is of indefinite characteristic. Thuspσ is pseudo-Hopf. Note thatpσ
is of rankr, independently ofσ.If I is properly contained inI(σ) for someσ ∈Sn, then there are at least two terms besides to
the “1” in relation (1) that are not trivial whenTi, j is substituted by 0 for all(i, j)∈ I . Therefore,none of theTi, j is invertible andpI
⋆ = F1. This shows thatpI is not pseudo-Hopf in this case.We conclude that the rank of SLn is r and thatZ (SLn) = {pσ|σ ∈ Sn}, which equals the
set of closed points of SLn. Therefore,υSLn : SL∼n → SLrkn is an isomorphism, and SLrk
n embedsinto the finite blue scheme SLn (cf. the comments in Section 2.1). This also proves the form ofthe rank space as claimed in the proposition.
56
Let e be the Weyl kernel of SLn. Then the canonical torusT equalse+Z
, which is the diagonaltorus of SL+n,Z. ThusT is a maximal torus of SLn,Z, which equals its own centralizer. The
normalizer ofT is the subgroupN = (SLrkn )
+Z
of monomial matrices.
Theorem 4.2.
(i) The group lawµN : N×N→N descends to a unique group lawµrk : SLrkn ×SLrk
n → SLrkn
in SchrkF1
.
(ii) The group lawµ+Z
of SL+n,Z descends to a unique monoid lawµ+ : SL+n,N×SL+n,N→SL+n,Nin Sch+
N.
(iii) The pairµ = (µrk,µ+) is a Tits morphismµ : SLn×SLn→ SLn that makesSLn a Tits-Weyl model ofSL+n,Z.
(iv) The groupSLTn (F1) of F1-rational Tits points is isomorphic to the alternating group An.
(v) For a semiring B, the monoidSLn(B) is the monoid of all matrices n×n-matrices(ai, j)with coefficients ai, j ∈ B that satisfy the determinant condition(1).
Proof. We prove (i). As a scheme,N =∐
σ∈SnSpecκ(pσ)+
Z, andκ(pσ)+
Z= Z[Ti,σ(i)]i=1,...,n/I
where the idealI is generated by∏ni=1Ti,σ(i)+(−1)signσ. The group lawµN : N×+ N→ N is
given by the ring homomorphism
ΓµN : ∏σ∈Sn
κ(pσ)+Z −→(
∏τ∈Sn
κ(pτ )+Z
)⊗+Z
(∏τ ′∈Sn
κ(pτ′)+Z
).
Ti,σ(i) 7−→ ∑ττ ′=σ
Ti,τ (i)⊗Tτ (i),σ(i)
This descends to a morphismµrk : SLrkn ×SLrk
n → SLrkn that is defined by
Γµrk : ∏σ∈Sn
κ(pσ) −→ ∏τ∈Sn
κ(pτ ) ⊗F1 ∏τ ′∈Sn
κ(pτ′) = ∏
τ ,τ ′∈Sn
κ(pτ )×κ(pτ ′)�R
Ti,σ(i) 7−→ (aτ ,τ ′)τ ,τ ′∈Sn
whereR is the pre-addition that defines the tensor product and whereaτ ,τ ′ = (Ti,,τ (i),Tτ (i),σ(i))
57
if ττ ′ = σ andaτ ,τ ′ = 0 otherwise. This means that the diagram
∏σ∈Snκ(pσ)+
Z
µN //
(∏τ∈Sn
κ(pτ )+Z
)⊗+
Z
(∏τ ′∈Sn
κ(pτ′)+Z
)
∏σ∈Snκ(pσ)
µrk//
OO
∏τ∈Snκ(pτ ) ⊗F1 ∏τ ′∈Sn
κ(pτ′)
OO
commutes. Since∏σ∈Snκ(pσ) is cancellative, the vertical arrows are inclusions. Consequently,
µrk is uniquely determined byµN. It is easily seen thatµrk is a group law in SchrkF1with identity
ǫrk : ∗F1→ SLrkn given by
Γǫrk : ∏σ∈Sn
κ(pσ) −→ F1
(aσ)σ∈Sn 7−→ ae
wheree∈ Sn is the trivial permutation and with inverseιrk : SLrkn → SLrk
n given by
Γιrk : ∏σ∈Sn
κ(pσ) −→ ∏σ∈Sn
κ(pσ)
Ti,σ(i) 7−→ T−1σ(i),i
where we understand the elementTi,σ(i) of κ(pσ) as the element(aσ′) of ∏σ∈Snκ(pσ) with
aσ′ = Ti,σ(i) if σ′ = σ andaσ′ = 0 otherwise. This shows (i).We continue with (ii). The group lawµ+
Z: SL+n,Z×+SL+n,Z→ SL+n,Z is defined by the ring
homomorphism
Γµ+Z
: Z[Ti, j ]i, j∈n/ I −→(Z[T ′i, j ]i, j∈n/ I ′
)⊗+
Z
(Z[T ′′i, j ]i, j∈n/ I ′′
)
Ti, j 7−→n
∑k=1
T ′i,k⊗T ′′k, j
where the idealsI , I ′ and I ′′ are generated by the relation that expresses that the determinantequals 1 (as explained in the beginning of this section). Sinceµ+
Zcan be defined without the use
of additive inverses, it descends to a morphismµ+ : SL+n ×SL+n → SL+n . Uniqueness follows,as in the case ofµrk, because SL+n is cancellative. It is easily seen thatµ+ is a semigroup law inSch+
Nwith identity ǫ+ : ∗N→ SL+n that is given by the blueprint morphismΓǫ+ : N[SLn]→ N
that mapsTi, j to 1 if i = j and to 0 ifi 6= j. This shows (ii). Note thatµ+ does not have an inverse
58
since the inverse ofµ+Z
involves additive inverses of theTi, j . Note further that(µ+)+Z= µ+
Z,
which justifies the notation.We proceed with (iii). It is clear from the definitions ofµrk andµ+ that the diagram
N×+ Nµ
rk,+Z
=µN //
ρ+SLn,Z×ρ+SLn,Z
��
N
ρ+SLn,Z��
SL+n,Z×+SL+n,Zµ+Z // SL+n,Z
commutes. Thusµ= (µrk,µ+) is a Tits morphism that is a semigroup law for SLn in SchT withidentityǫ= (ǫrk,ǫ+). This shows that SLn is a Tits model of SL+n,Z. We already reasoned that the
canonical torusT = e+Z
of SLn is the diagonal torus of SL+n,Z, which is a maximal torus and its
own centralizer, and thatN = SLrkn,Z is its normalizer. Thus the morphismΨe : SLrk
n /e+Z→N/T
is an isomorphism of group schemes, which shows that SLn is a Tits-Weyl model of SL+n,Z. Thisproves (iii).
We proceed with (iv). A morphism∗F1→ SLrkn is determined by its image pointpσ and a
morphismκ(pσ)→ F1, which is necessarily unique. The latter morphism exists ifσ ∈ An sincein this caseκ(pσ) is a monoid (cf. Proposition 4.1). In case,σ ∈ Sn−An, the residue fieldκ(pσ)contains−1 and does not admit a blueprint morphism toF1.
We show (v). LetB be a semiring. A morphism SpecB→ SLn is given by a blueprintmorphismf : F1[Ti, j ]�R→B whereR is the pre-addition generated by the relation (1). Such amorphism is determined by the imagesai, j = f (Ti, j) ∈ B of the generatorsTi, j , and a family ofelements(ai, j) occurs as images of a blueprint morphismf if and only if theAi, j satisfy relation(1). It is clear that the multiplication onSLn(B) that is induced by the monoid law of SLn is theusual matrix multiplication. This concludes the proof of the theorem.
4.2 The cube lemma
In the rest of this part of the paper, we will establish Tits-Weyl models of subgroupsG of SL+n,Z.To show that the semigroup law of SLn restricts to a givenF1-modelG of G , we will often needto prove the existence of a morphismh1 : X1→Y1 that completes a commuting diagram of the
59
form
X′2h′2 //
fX,2��
Y′2
fY,2
��
X′1
g′X66♥♥♥♥♥♥♥♥♥♥♥ h′1 //
fX,1
��
Y′1g′Y
77♥♥♥♥♥♥♥♥♥♥♥
fY,1
��
X2h2 // Y2
X1
gX66❧❧❧❧❧❧❧❧❧❧❧
Y1gY
66♠♠♠♠♠♠♠♠♠♠♠
(2)
to a commuting cube
X′2h′2 //
fX,2��
Y′2
fY,2
��
X′1
g′X66♥♥♥♥♥♥♥♥♥♥♥ h′1 //
fX,1
��
Y′1g′Y
77♥♥♥♥♥♥♥♥♥♥♥
fY,1
��
X2h2 // Y2
X1h1 //
gX66❧❧❧❧❧❧❧❧❧❧❧
Y1gY
66♠♠♠♠♠♠♠♠♠♠♠
(3)
of morphisms. In this section, we provide the necessary hypotheses that yield the morphism inquestion for the categoriesSets, Top and SchF1.
Lemma 4.3 (The cube lemma for sets). Consider a commutative diagram of the form(2) inthe categorySets. If fX,1 : X′1→ X1 is surjective and gY injective, then there exists a uniquemap h1 : X1→Y1 such that the resulting cube(3) commutes.
Proof. Let x∈ X1. Then there is ax′ ∈ X′1 such thatfX,1(x′) = x. Defineh1(x) = fY,1◦h′1(x′).
We verify that the definition ofh1 does not depend on the choice ofx′. Let x′1 andx′2 be twoelements ofX′1 with fX,1(x′1) = fX,1(x′2) = x. Then
gY ◦ fY,1◦h′1(x′i) = fY,2◦g′Y ◦h′1(x
′i)
= fY,2◦h′2◦g′X(x′i)
= h2◦ fX,2◦g′X(x′i)
= h2◦gX ◦ fX,1(x′i)
= h2◦gX(x)
is the same element inY2 for i = 1,2. SincegY is injective, fY,1 ◦h′1(x′1) = fY,1 ◦h′1(x
′2) in Y1,
which means that the definition ofh1(x) does not depend on the choice ofx′ in f−1X,1(x).
60
By definition ofh1, the diagram
X′1h′1 //
fX,1��
Y′1fY,1��
X1h1 // Y1
commutes. The same calculation as above shows thatgY ◦ h1(x) = gY ◦ fY,1 ◦ h′1(x′) equals
h2◦gX(x), which means that the diagram
X2h2 // Y2
X1h1 //
gX::tttttt
Y1gY
::✉✉✉✉✉✉
commutes. This proves the lemma.
Lemma 4.4 (The cube lemma for topological spaces). Consider a commutative diagram ofthe form(2) in the categoryTop. If fX,1 : X′1→X1 is surjective and gY an immersion, then thereexists a unique continuous map h1 : X1→Y1 such that the resulting cube(3) commutes.
Proof. By the cube lemma for sets (Lemma 4.3), a unique maph1 : X1→ Y1 exists such thatthe cube (3) commutes. We have to show thath1 is continuous. LetU be an open subset ofY1.Then there is an open subsetU ′ of Y2 such thatg−1
Y (U ′) = U sincegY is an immersion. Thenthe subset
h−11 (U) = h−1
1 ◦g−1Y (U ′) = g−1
X ◦h−12 (U ′)
of X1 is open as an inverse image ofU ′ under a continuous map. This proves the lemma.
A quasi-submersion of blue schemesis a morphismf : X→Y that is surjective and satisfiesfor every affine open subsetU of Y thatV = f−1(U) is affine and thatf #(U) : Γ(OY,U)→Γ(OX,V) is an inclusion as a subblueprint, i.e.f #(U) is injective and the pre-addition ofΓ(OY,U) is the restriction of the pre-addition ofΓ(OX,V) to Γ(OY,U) (cf. [25, section 2.1]).
Lemma 4.5. If f : B →C is an inclusion of a subblueprint B of C, then f∗ : SpecC→ SpecB isa quasi-submersion.
Proof. Every affine openU of X = SpecB is of the formU = SpecS−1B for some finitelygenerated multiplicative subsetS of B. The inverse imagef ∗(U) is isomorphic to SpecS−1Cand therefore affine. We have to show that the induced blueprint morphismg : S−1B→ S−1C isan inclusion of a subblueprint.
61
We show injectivity ofg. If g(as)= g(a′
s′ ), i.e. f (a)s =
f (a′)s′ , for a,b∈B ands,s′ ∈S, then there
is at ∈ Ssuch thatts′ f (a) = ts f(a′) in C. SinceB is a subblueprint ofC, we havets′a= tsa′ inB, which shows thatas =
a′s′ in B. Thusg : S−1B→ S−1C is injective.
We show thatS−1B is a subblueprint ofS−1C. Consider an additive relation∑ f (ai)si≡∑ f (b j )
r j
in S−1C. Then there is at ∈Ssuch that∑ tsi f (ai)≡∑ tr j f (b j) in C wheresi = (∏k6=i sk) ·(∏l r l )
and r j = (∏k sk) · (∏l 6= j r l). SinceB is a subblueprint ofC, we have∑ tsiai ≡ ∑ tr jb j in B,
which means that∑ aisi≡ ∑ b j
r jin S−1B. This shows thatg : S−1B→ S−1C is an inclusion of a
subblueprint and finishes the proof of the lemma.
Lemma 4.6 (The cube lemma for blue schemes). Consider a commutative diagram of theform (2) in the categorySchF1. Suppose that fX,1 : X′1→ X1 is a quasi-submersion and gY is aclosed immersion. Then there exists a unique morphism h1 : X1→Y1 of blue schemes such thatthe resulting cube(3) commutes.
Proof. By the cube lemma for topological spaces (Lemma 4.4), a unique continuous maph1 :X1→Y1 exists such that the cube (3) commutes. We have to show that there exists a morphismΓh1 : OY1→ OX1 between the structure sheaves ofX1 andY1. Since a morphism of sheaves canbe defined locally, we may assume that all blue schemes in question are affine.
If we denote byΓZ the coordinate ring of the blue schemeZ, then there is a morphismΓh1 : ΓY1→ ΓX1 of blueprints that completes the commutative diagram
ΓX′2 ooΓh′2
OO
Γ fX,2
ΓY′2OO
Γ fY,2ΓX′1vv
Γg′X ❧❧❧❧❧❧❧❧❧❧❧oo
Γh′1OO
Γ fX,1
ΓY′1vv Γg′Y
♠♠♠♠♠♠♠♠♠♠♠
OOΓ fY,1
ΓX2 oo Γh2 ΓY2
ΓX1uu
ΓgX❦❦❦❦❦❦❦❦❦❦❦ ΓY1
uu ΓgY
❦❦❦❦❦❦❦❦❦❦❦
to a commuting cube. SincegY is a closed immersion,ΓgY is a surjective morphism of blueprints.Since fX,1 a quasi-submersion, we may assume further thatfX,1 : X′1→ X1 is still surjective andΓ fX,1 is an injective morphism of blueprints. Then by the cube lemma for sets (Lemma 4.3),there is a unique mapΓh1 : ΓY1→ ΓX1.
To verify thatΓh1 is a morphism of monoids, leta,b∈ ΓY1. Then there are elementsa′,b′ ∈ΓY2 such thatΓgY(a′) = a andΓgY(b′) = b. Therefore,
Γh1(ab) = ΓgY ◦Γh1(a′b′) = Γh2◦ΓgX(a
′b′) =(
Γh2◦ΓgX(a′))·(
Γh2◦ΓgX(b′)),
62
which is, tracing back the above calculation for both factors, equal toΓh1(a) ·Γh1(b). ThusΓh1is multiplicative. Similarly, the calculationΓh1(1) = ΓgY ◦Γh1(1) = Γh2 ◦ΓgX(1) = 1 showsthatΓh1 is unital.
We are left with showing thatΓh1 maps the pre-addition ofΓY1 to the pre-addition ofΓX1.Consider an arbitrary additive relation∑ai ≡ ∑b j in ΓY1. Applying the morphismΓh′1◦Γ fY,1,we see that
∑ Γh′1◦Γ fY,1(ai) ≡ ∑ Γh′1◦Γ fY,1(b j)
in ΓX′1. SinceΓ fX,1 is the inclusion ofΓX1 as a subblueprint ofΓX′1 andΓh′1◦Γ fY,1 = Γ fX,1◦Γh1, this relation restricts toΓX1, i.e.∑Γh1(ai) ≡ ∑Γh1(b j) in ΓX1. This finishes the proof ofthe lemma.
4.3 Closed subgroups of Tits-Weyl models
In this section, we formulate a criterion for subgroupsH of a group schemeG that yields aTits-Weyl model ofH . Applied toG = SL+n,Z, this will establish a variety of Tits-Weyl modelsof prominent algebraic groups that we discuss in more detailat the end of this section: generallinear groups, special orthogonal groups, symplectic groups and some of their isogenies.
If G = Spec(A�R
)is anF1-model ofG andιH : H → G a closed immersion, then we
can consider the pre-additionR ′ on A that is generated byR and all defining relations ofHin elements ofA. This defines anF1-modelH = Spec
(A�R ′
)of H together with a closed
immersionι : H →G that base extends toιH = ι+Z
.Let G be an affine smooth group scheme of finite type with anF1-modelG. We say that a
torusT ⊂ G is diagonal w.r.t. Gif for everyx∈G, the group lawµG of G restricts to morphisms
T ×+Z
x+Z−→ x+
Zand x+
Z×+
ZT −→ x+
Z.
If G is theF1-model ofG that is associated to an embeddingι : G → SL+n,Z, then a torusT ⊂ G
is diagonal w.r.t.G if and only if the imageι(T) is contained in the diagonal torus of SL+n,Z. In
particular, the canonical torus of SLn is diagonal.If G is a Tits-Weyl model ofG , then we say briefly that the canonical torusT is diagonal if
T is diagonal w.r.t.G.
Theorem 4.7. LetG be an affine smooth group scheme of finite type with a Tits Weyl model G.Assume that the canonical torus T is diagonal and that N= Grk,+
Zis the normalizer of T inG .
Then C= e+Z
is the centralizer of T wheree is the Weyl kernel of G.LetH be a smooth closed subgroup ofG andι : H →G the associatedF1-model. Assume
that T = T ∩H is a maximal torus of H. Assume further that the centralizerC of T in H iscontained in C∩H and that the normalizerN of T in H is contained in N∩H . Then thefollowing holds true.
63
(i) The set of pseudo-Hopf points of H isZ (H) = ι−1(Z (G)). Thus the closed immersionι : H →G is Tits.
(ii) We haveC=C∩H = e+Z
andN = N∩H = Hrk,+Z
.
(iii) The monoid lawµG of G restricts uniquely to a monoid lawµH of H. The pair(H,µH)is a Tits-Weyl model ofH whose canonical torusT is diagonal. The Tits morphism(ιrk, ι+) : H →G is a monoid homomorphism inSchT .
(iv) If Grk lifts to G, then Hrk lifts to H.
Proof. First note that sinceΨe : Grk,+Z
/e+Z
∼→ N/C = W is an isomorphism andGrk,+Z
= N, thecentralizerC equals indeede+
Z. The proof of (i)–(iv) is somewhat interwoven and doesn’t follow
the order of the statements.Let x ∈ H andy = ι(x) be the image inG. Thenx+
Z= y+
Z∩H . Therefore, both left and
right action ofT on y+Z
restricts to an action ofT on x+Z
. This shows thatT is diagonal w.r.t.H.We show that there is an elemente in Hrk such thatιrk(e) = e. Let e∈Z (G) be the pseudo-
Hopf point such thate = e⋆. ThenT = e+
Z. SinceT ⊂ T maps toH, there is a point ˜e in H
such thatι(e) = e. As a closed subgroup ofT = e+Z
, the group schemee+Z is diagonalizable
and its global sections are isomorphic to a group ring. Thuse+Z is flat, einv is generated byits units,e is affine and ˜e is almost of indefinite characteristic sincee is so and the morphismT → e factorizes throughe. Thuse is pseudo-Hopf of rankr = rkT. SinceT acts onx+
Zfor all
pseudo-Hopf points ofH, the rank of a pseudo-Hopf point is at leastr. Thuse∈Z (H), whichdefines a pointe ∈ Hrk.
We show thatC=C∩H = e+Z
. Sincee+Z=C, we havee+
Z= e+
Z∩H =C∩H . Sincee is
an abelian group in SchF1, e+Z
centralizesT. ThusC= e+Z
.We show thatN = N∩H . By the hypothesis of the theorem, we already know thatN ⊂
N∩H . By the second isomorphism theorem for groups,N/C is a subgroup of the constantgroup schemeN/C. ThusN is isomorphic to a finite disjoint union of copies ofC as a scheme.Therefore,N is flat and we can investigateN by considering complex points. Letn ∈ (N∩H )(C). ThennT(C)n−1 is contained in bothH (C) andT(C), whose intersection isT(C).Thusn is contained in the normalizer ofT(C) in H (C), which isN(C).
We show thatN = Hrk,+Z
andZ (H) = ι−1(Z (G)). SinceN = N∩H , the image ofN→H is ι−1(Z (G)). SinceN is the disjoint union of schemes isomorphic toC, every point ofι−1(Z (G)) is pseudo-Hopf of rankr for the same reason as the one that showed thate ispseudo-Hopf of rankr. Thusι−1(Z (G))⊂Z (H) andN⊂ Hrk,+
Z.
To show the reverse conclusion, consider an arbitrary pseudo-Hopf pointx of H. Since wehave a left-right double action
T ×+Z x+Z ×+Z T −→ x+Z
64
of T on x+Z
, the rank ofx is at leastr. If the rank rkx= dimx+Q
is r, thenT(C)pT(C) = pT(C)for anyp∈ x(C). This means that for allt1, t2∈ T(C), there is at3∈ T(C) such thatt1pt2 = pt3.Multiplying the latter equation witht−1
2 from the right yieldst1p = pt3t−12 , which shows that
T(C)p= pT(C). This means thatp∈ N(C) = H (C)∩N(C), i.e.x∈Z (H).This finishes the proof of (ii). Further we have proven thatι mapsZ (H) to Z (G), which
implies thatι : H→G is Tits. This finishes the proof of (i).We turn to the proof of (iii). We show thatµ+G : G+×+ G+→ G+ descends to a morphism
µH : H+×+ H+→ H+. SinceH → G is a closed immersion, we have a surjectionΓG։ ΓHand a surjectionΓG+
։ ΓH+. By Lemma 1.9, the morphismH+→G+ is a closed immersion.SinceH is cancellative,ΓH+ is also cancellative andΓH+ → ΓH+
Zis an inclusion as a sub-
blueprint. By Lemma 4.5, the morphismH+Z→H+ is a quasi-submersion. The same is true for
H+Z×+
ZH+Z→ H+×+ H+. Therefore we can apply the cube lemma for blue schemes (Lemma
4.6) to the commutative diagram
G ×+Z
GµG //
����
G
����
H ×+Z
H
'�
44✐✐✐✐✐✐✐✐✐✐✐✐µH //
����
H)
66♠♠♠♠♠♠♠♠♠♠♠♠
����
G+×+ G+µ+
G // G+ ,
H+×+ H+&�
33❤❤❤❤❤❤❤❤❤❤❤H+
(�
55❧❧❧❧❧❧❧❧❧❧
which shows thatµ+G descends to a unique morphismµ+H : H+×+ H+→ H+. It easy to verifythatµ+H is associative. Thatǫ+G : ∗N→G+ descends to an identityǫ+H : ∗N→H+ of H+ is shownsimilarly.
To show thatµrkG : Grk×Grk→ Grk of the rank space descends to a morphismµrk
H : Hrk×Hrk→ Hrk is more subtle since, in general,Hrk→Grk is not a closed immersion.
By Lemma 3.12, the Weyl kernele of G is diagonalizable. ThusΓe+Z
is a group ringZ[Λ]for an abelian groupΛ and the group law ofe+
Zcomes from the multiplication ofΛ. Therefore
the unit field ofe+Z
is F⋆(e+Z) = F12[Λ] = F1[Λ]inv. Let e∈ Z (G) be the pseudo-Hopf point
of G with e = e⋆. Thene is generated by its units, which means that eithere ≃ SpecF1[Λ] or
e≃ F12[Λ]. The analogous statement is true fore ∈ Hrk. Sincee+Z→ e+
Zis a closed immersion,
Γe+Z→ Γe+
Zis surjective, and so isΓeinv → Γeinv. By Lemma 1.9, the morphismeinv → einv
is a closed immersion. Since for every pointx ∈ Hrk andy = ιrk(x) ∈ Grk, the schemex+Z
isisomorphic toe+
Zandy is isomorphic toe+
Z, the same argument as above shows thatHrk
inv→Grkinv
is a closed immersion.SinceHrk
inv is cancellative,ΓHrkinv → ΓN is a subblueprint, and by Lemma 4.5,N→Hrk
inv is aquasi-submersion. The same holds forN×+
ZN→Hrk
inv×Hrkinv. Therefore we can apply the cube
65
lemma for blue schemes (Lemma 4.6) to the commutative diagram
N×+Z
NµN //
����
N
����
N×+Z
N'�
44✐✐✐✐✐✐✐✐✐✐✐✐✐ µN //
����
N)
66♠♠♠♠♠♠♠♠♠♠♠♠♠
����
Grkinv×Grk
inv
µrkG,inv // Grk
inv,
Hrkinv×Hrk
inv
'�
44✐✐✐✐✐✐✐✐✐✐Hrk
inv
)
66♠♠♠♠♠♠♠♠♠♠
which shows thatµrkG,inv descends to a unique morphismµHrk
inv: Hrk
inv×Hrkinv→Hrk
inv.
We show thatµHrkinv
descends to a morphismµrkH : Hrk×Hrk → Hrk. Let B = ΓH∼ and
C = Γ(H∼×H∼
). Then B+ andC+ are cancellative and embed as subblueprints into the
rings B+Z≃ ΓN andC+
Z≃ Γ(N×+
ZN), rspectively. By Lemma 4.5, the canonical morphism
N×+Z
N→ H∼,+×+ H∼,+ is a quasi-submersion. SinceD+ ≃ D+canc for an arbitrary blueprint
D, the semiringB+ is canonical isomorphic to the coordinate ring of∐
x∈Z (H) x+canc. SinceGis a Tits-Weyl model ofG , the morphism
∐y∈Z (G) y→ G is necessarily a closed immersion.
Therefore∐
x∈Z (H) x→ H is also a closed immersion. SinceΓH+ → Γx+ → Γx+canc= x+
issurjective, the induced morphismH∼,+ → H+ is a closed immersion. This shows that thecommuting diagram
H ×+Z
HµH //
����
H
����
N×+Z
N&�
33❤❤❤❤❤❤❤❤❤❤❤❤❤❤ µN //
����
N)
66♠♠♠♠♠♠♠♠♠♠♠♠♠
����
H+×+N
H+µ+
H // H+
H∼,+×+ H∼,+%�
33❣❣❣❣❣❣❣❣❣❣
H∼,+(�
55❧❧❧❧❧❧❧❧❧❧
satisfies the hypotheses of the cube lemma for schemes, whichyields thatµ+H descends to amorphismµ∼,+ : H∼,+×+ H∼,+→ H∼,+. With B andC as above, we have that
B⋆ = ΓHrk, B⋆inv = ΓHrk
inv, B+ = ΓH∼,+ and B+Z = ΓN,
and analogous identities forC⋆, C⋆inv, C+ andC+
Z. The morphismsµHrk
inv, µ∼,+ andµN yield a
66
commutative diagram
B⋆inv� _
��
ΓµHrk
inv // C⋆inv� _
��
B⋆(�
55❧❧❧❧❧❧❧❧❧❧❧� _
��
C⋆(�
55❧❧❧❧❧❧❧❧❧❧❧� _
��
B+Z
ΓµN // C+Z
B+)
66♠♠♠♠♠♠♠♠♠♠♠ Γµ∼,+ // C+)
66♠♠♠♠♠♠♠♠♠♠♠
of cancellative blueprints. SinceC⋆ is the intersection ofC⋆inv with C+ insideC+
H can be easily derived from the associativity ofµrkG by using the
commutativity of certain diagrams. Similarly to the existence ofµrkH , one shows that there are
an identityǫrkH : ∗F1→ Hrk and an inversionιrkH : Hrk→ Hrk which turnHrk into a group objectin Schrk
F1.
Moreover, it is easy to see that the pairsµH = (µrkH ,µ
+H) andǫH = (ǫrkH ,ǫ
+H) are Tits mor-
phisms that giveH the structure of a Tits monoid. The Weyl kernel ofH is e and the canonicaltorus isT, which is a maximal torus ofH by hypothesis. Sincee+
Z= C andHrk,+
Z= N, the
morphismΨe : Hrk,+Z
/e+Z
∼→ N/C is an isomorphism of group schemes. This shows thatHis a Tits-Weyl model ofH . It is clear by the definition ofµH that ι : H → G is a monoidhomomorphism in SchT . This shows (iii).
We show (iv). Assume thatGrk lifts to G. The existence of the inverse ofυHinv : H∼inv→Hrkinv
follows an application of the cube lemma to
N id //
����
N
����
N(�
ιrk,+Z
55❦❦❦❦❦❦❦❦❦❦❦❦❦ id //
����
N'�
ιrk,+Z
55❥❥❥❥❥❥❥❥❥❥❥❥❥
����
Grkinv
υ−1G,inv // G∼inv .
Hrkinv
)
ιrk66♠♠♠♠♠♠♠♠♠♠
H∼inv
) ι∼
66♠♠♠♠♠♠♠♠♠♠
Since forx∈Z (X), the subschemex of H∼ is with−1 if and only if the subschemex⋆
of Hrk
is with−1, it is clear thatυ−1Hinv
comes from an isomorphismυ−1H : Hrk→ H∼. This completes
the proof of the theorem.
67
4.3.1 The general linear group
As a first application of Theorem 4.7, we establish Tits-Weylmodels of general linear groups.The Tits-Weyl models GLn of GL+
n,Z is of importance for all other Chevalley group schemessince one can consider them as closed subgroups of a general linear group. The standard wayto embed GL+n,Z as a closed subgroup in SL+
n+1,Z is by sending an invertiblen×n-matrix A tothe(n+1)×(n+1)-matrix whose upper leftn×n block equalsA, whose coefficient at the verylower right equals(detA)−1 and whose other entries are 0.
In other words, ifZ[SLn]+ = Z[Ti, j |i, j ∈ (n+1)]+/I is the coordinate ring of SL+n+1,Z
whereI is the ideal generated by∑σ∈Sn+1
(sign(σ) ·∏n+1
i=1 Ti,σ(i)
)−1 (cf. Section 4.1), then the
closed subscheme GL+n,Z of SL+n+1,Z is defined by the ideal generated byTi,n+1 andTn+1,i for
i = 1, . . . ,n and byTn+1,n+1 ·∑σ∈Sn
(sign(σ) ·∏n
i=1Ti,σ(i)
)−1.
It is clear that this embedding satisfies the hypothesis of Theorem 4.7. Thus we obtainthe Tits-Weyl model GLn of GL+
n,Z. We describe the points of the blue scheme GLn. Recallfrom Proposition 4.1 that the points of SLn+1 are of the formpI = (Ti, j |(i, j) ∈ I) for someI ∈ (n+1)× (n+1) such that there is a permutationσ ∈ Sn+1 with I ⊂ I(σ). By the definitionof GLn, it is clear that a pointpI of SLn is a point of GLn, if and only if pI containsTi,n+1 andTn+1,i for i = 1, . . . ,n, but does not containTn+1,n+1. This means that a pointpI of SLn+1 is inGLn if and only if I ⊂ I(σ) for a permutationσ ∈ Sn+1 that fixesn+1.
Since a permutationσ ∈ Sn+1 fixesn+1 if and only if it lies in the image of the standardembeddingι : Sn → Sn+1, a pointpI is contained in GLn if and only if I ⊂ I(ι(σ)) for someσ ∈ Sn. This shows, in particular, that every prime ideal of GLn is generated by a subset of{Ti, j}i, j∈n and that the rank space of GLn equals GLrkn = {pι(σ)|σ ∈ Sn}.
The residue field ofpσ depends, as in the case of SLn, on the sign ofσ: if signσ is even,thenκ(pσ) ≃ F1[T
±1i,σ(i)]; if signσ is odd, thenκ(pσ) ≃ F12[T±1
i,σ(i)]. Thus GLTn(F1) is equal to
the alternating group insideW (GLn) = Sn. The rank of GLn is n and the extended Weyl groupequalsGT(F12)≃ (Z/2Z)n⋊Sn.
We specialize Theorem 4.7 for subgroups ofG = GL+n,Z. Let diag(GL+
n,Z) be the diagonal
torus of GL+n,Z and mon(GL+n,Z) the group of monomial matrices, which is the normalizer of the
diagonal torus.
Corollary 4.8. LetG be a smooth closed subgroup ofGL+n,Z and G the correspondingF1-model.
Assume that T= diag(GL+n,Z)∩G is a maximal torus ofG whose normalizer N is contained in
mon(GL+n,Z). Then the following holds true.
(i) The monoid law ofGLn restricts to G and makes G a Tits-Weyl model ofG .
(ii) The closed embedding G→GLn is Tits and a homomorphism of semigroups inSchT .
68
(iii) The rank space Grk lifts to G and equals the intersectionGLrkn ∩G.
(iv) The canonical torus T of G is diagonal and equalse+Z
wheree is the Weyl kernel of G. Its
normalizer in G is Grk,+Z
.
4.3.2 Other groups of typeAn
It is interesting to reconsider SL+n,Z as a closed subgroup of GL+n,Z. The Tits-Weyl model asso-
ciated to this embedding is indeed isomorphic to the Tits-Weyl model SLn that we described inTheorem 4.2. The embedding SLn→ GLn that we obtain from Corollary 4.8 is a homeomor-phism between the underlying topological spaces.
Corollary 4.8 yields a Tits-Weyl model of the adjoint group schemeG of typeAn as follows.Let Mat+n,Z be the scheme of then×n-matrices, which is isomorphic to an affine space+An2
Z .
The action ofG on Mat+n,Z by conjugation has trivial stabilizer. This defines an embedding
G →GL+n2,Z
as a closed subgroup. It is easily seen that this embedding satisfies the hypotheses
of Corollary 4.8 by using the action of SL+n,Z on Mat+n,Z by conjugation, which factors through
the action ofG on Mat+n,Z via the canonical isogeny SL+n,Z→ G . This yields a Tits-Weyl modelG of G .
Note that the construction of Tits-Weyl models of adjoint groups in the Section 4.4 alsoyields a Tits-Weyl model ofG . We compare these two models in Appendix A.2 in the casen= 1.
4.3.3 Symplectic groups
The symplectic groups Sp+2n,Z have a standard representation in the following form. LetJ be the2n×2n-matrix whose non-zero entries are concentrated on the anti-diagonal withJi,2n−i = 1 if1≤ i ≤ n andJi,2n−i = −1 if n+1≤ i ≤ 2n. Then the set Sp2n(k) of k-rational points can bedescribed as follows for every ringk: the elements of Sp2n(k) correspond to the 2n×2n-matricesA= (ai, j) with entries ink that satisfyAJAt = J, i.e. the equations
n
∑l=1
ai,la j ,2n+1−l =2n
∑l=n+1
ai,la j ,2n+1−l + δi,2n+1− j
for all 1≤ i < j ≤ 2n whereδi,2n+1− j is the Kronecker symbol. These equations describe Sp+2n,Z
as a closed subscheme of GL+2n,Z and thus yield anF1-model Sp2n. The intersection of Sp+2n,Z
with the diagonal torus of GL+2n,Z is a maximal torus of Sp+2n,Z, and its normalizer is contained
in the group of monomial matrices of GL+2n,Z. Thus Theorem 4.7 applies and shows that Sp2n
is a closed submonoid of GL2n and a Tits-Weyl model of Sp+2n,Z.
69
4.3.4 Special orthogonal groups
It requires some more thought to define a model of (special) orthogonal groups over the integers.A standard way to do it is the following (cf. [12, Appendix B] for more details). We first defineintegral models of the orthogonal groups On. Define for each ringR the quadratic form
qn(x) =m
∑i=1
xixn+1−i for n= 2m, and
qn(x) = x2m+1 +
m
∑i=1
xixn+1−i for n= 2m+1
wherex= (x1, . . . ,xn) ∈ Rn. The functor
On(R) = { g∈GLn(R) | qn(gx) = x for all x∈ Rn }
is representable by a scheme O+n,Z. It is smooth in casen is even, but not for oddn, in which
case only the base extension O+n,Z[1/2] toZ[1/2] is smooth (cf. [12, Thm. B.1.8]). For oddn, we
define thespecial orthogonal groupSO+n,Z as the kernel of the determinant det : O+
n,Z→ +Gm,Z.
For evenn, we definespecial orthogonal groupSO+n,Z as the kernel of the Dickson invariant
Dq : O+n,Z→ (Z/2Z)Z. Then the scheme SO+n,Z is smooth for alln≥ 1 (cf. [12, Thm. B.1.8]).
We describe a maximal torus and its normalizer of these groups and show that we can applyTheorem 4.7 in the following.
The Tits-Weyl model ofSOn for odd n
We consider the case of oddn= 2m+1 first. A maximal torusT(R) of SOn(R) is given by thediagonal matrices with valuesλ1, . . . ,λn ∈ R on the diagonal that satisfyλn+1−i = λ−1
i for alli ∈ n and∏n
i=1λi = 1. This means thatλ1, . . . ,λm can be chosen independently fromR× andthatλm+1 = 1.
Its normalizerN(R) consists of all monomial matricesA= (ai, j) that satisfy the followingconditions. Letσ ∈ Sn be the permutation such thatai, j 6= 0 if and only if j = σ(i). ThenA∈ N(R) if and only if detA= 1, if σ(n+1− i) = n+1−σ(i) and if an+1−i,σ(n+1−i) = a−1
i,σ(i)for all i ∈ n. This means, in particular, thatσ(m+ 1) = m+ 1 and thatam+1,m+1 = signσ.The permutationσ permutes the set of pairsΛ1 = {λ1,λn}, . . . ,Λm= {λm,λm+2} and permuteseach pairΛi for i ∈ n. This means that the quotientW = N(R)/T(R), which corresponds to allpermutationsσ ∈Sn that occur, is isomorphic to a signed permutation group. A set of generatorsis determined by the involutionss1 = (1,2)(n−1,n), . . .,sm−1 = (m−1,m)(m+2,m+3) and
70
sm = (m,m+2). The Weyl groupW together with the generatorss1, . . . ,sm is a Coxeter groupof typeBm.
Since SO+n,Z ⊂ GL+n,Z is smooth, its maximal torusT is contained in the diagonal matrices
of GL+n,Z and the normalizerN of T is contained in the monomial matrices of GL+
n,Z, we can
apply Theorem 4.7 to yield a Tits-Weyl model SOn of SO+n,Z.
The Tits-Weyl model ofSOn for evenn
We consider the case of evenn = 2m. Since O+n,Z is smooth, we can ask whether O+n,Z has a
Tits-Weyl model On. This is indeed the case as we will show now. A maximal torusT(R)of On(R) is given by the diagonal matrices with valuesλ1, . . . ,λn ∈ R on the diagonal thatsatisfyλn+1−i = λ−1
i for all i ∈ n. Its normalizerN(R) in On(R) consists of all monomialmatricesA = (ai, j) that satisfy the following conditions. Letσ ∈ Sn be the permutation suchthatai, j 6= 0 if and only if j = σ(i). ThenA∈ N(R) if and only if σ(n+1− i) = n+1−σ(i)andan+1−i,σ(n+1−i) = a−1
i,σ(i) for all i ∈ n . Thus we can apply Theorem 4.7 to obtain a Tits-Weyl
model On of O+n,Z.
Note that the Weyl group of On is the same as for SOn+1: a permutationσ associated with anelement of the normalizerN permutes the set of pairsΛ1 = {λ1,λn}, . . . ,Λm = {λm,λm+1} andpermutes each pairΛi for i ∈ n. This means the quotientW = N(R)/T(R), which correspondsto all permutationsσ ∈ Sn that occur, is isomorphic to a signed permutation group.
The subgroup SO+n,Z has the same maximal torusT as O+n,Z, but its normalizer in SO+n,Zis a proper subgroupN′ of the normalizerN of T in O+
n,Z. Namely, a matrixA ∈ N(R) iscontained inN′(R) if and only if the sign of the associated permutationσ is 1. The Weyl groupW=N′(R)/T(R) is isomorphic to the subgroup of all elements of sign 1 of a signed permutationgroup. A set of generators is determined by the involutionss1 = (1,2)(n−1,n), . . . ,sm−1 =(m−1,m)(m+1,m+2) andsm = (m−1,m+1)(m,m+2). The Weyl groupW together withs1, . . . ,sm is a Coxeter group of typeDm. This shows that we can apply Theorem 4.7 to yield aTits-Weyl model SOn of SO+
n,Z.
We summarize the above results.
Theorem 4.9. All of the Chevalley groups in the following list have a Tits-Weyl model: thespecial linear groupsSL+n,Z, the general linear groupsGL+
n,Z, the adjoint Chevalley groups of
type An, the symplectic groupsSp+2n,Z, the special orthogonal groupsSO+n,Z (for all n) and the
orthogonal groups O+n,Z (for even n).
71
4.4 Adjoint Chevalley groups
In this section, we establish Tits-Weyl models of adjoint Chevalley schemes, which come fromthe action of the Chevalley group on its Lie algebrag. Namely, the choice of a Chevalley basisallows us to identify the automorphism group ofg (as a linear space) with GL+n,Z and consider
G as a subgroup of GL+n,Z. The intersection of the diagonal torus of GL+n,Z with G is a maximal
torusT of G . However, the normalizer ofT is not contained in the subgroup of monomialmatrices of GL+n,Z (unless each simple factor ofG is of typeA1), thus Theorem 4.7 does notapply to this situation. Moreover, we will see that the rank space of theF1-modelG associatedwith the embedding ofG into GL+n,Z does not lift (unless each simple factor ofG is typeAn, Dn
or En). Though the situation is more difficult, the formalism of Tits monoids applies to adjointrepresentations and we will see that there is a monoid lawµ= (µrk,µ+) for G that turnsG intoa Tits-Weyl model ofG .
Chevalley bases and the adjoint action
Let G be a adjoint Chevalley group scheme, i.e. a split semisimplegroup scheme with trivialcenter, and letg be its Lie algebra. Then its adjoint representationG → Aut(g) is a closedembedding as a subgroup (cf. [2, XVI, 1.5(a)] or [12, Thm. 5.3.5]). The choice of a Cartansubalgebrah of g yields a root systemΦ. The choice of fundamental rootsΠ⊂ Φ identifiesΦwith the corootsΦ∨, which can be seen as a subset{hr |r ∈Φ} of h, and decomposesΦ into thepositive rootsΦ+ and the negative rootsΦ−. We denote byΨ the disjoint union ofΦ andΠ.Let {er}r∈Ψ be the Chevalley basis given by the choices ofh andΠ. If r ∈ Π ⊂ Ψ, thener isthe coroothr . If r ∈ Φ⊂Ψ, then we writelr for er to avoid confusion with the coroothr . Thisleads to the decomposition
g =⊕
r∈Πhr ⊕
⊕
r∈Φlr
of g into h-invariant 1-dimensional subspaces ofg wherehr is generated byhr for r ∈Π andlris generated bylr for r ∈Φ.
The choice of the Cartan subalgebrah corresponds to the choice of a maximal torusT ofG . Let N be its normalizer andW = N/T its Weyl group. SinceG is split, the ordinary WeylgroupW is isomorphic toW (Z) = N(Z)/T(Z) andW ≃ (W)Z. If we choose an ordering onthe Chevalley basis, we obtain an isomorphism GL(g)≃GL+
n,Z wheren= #Ψ is the dimension
of g. Thus, we can realizeG as a closed subgroup of GL+n,Z. Independent of the ordering of the
Chevalley basis, the maximal torusT of G is the intersection ofG with the diagonal subgroupdiag(GL+
n,Z) of GL+n,Z.
The adjoint action ofT factors into actions on eachhr for r ∈Π andlr for r ∈Φ. The actiononh is trivial and the adjoint action ofT on lr factorizes through a character ofT. The adjoint
72
action ofN factors into an action onh and an action onl=⊕
r∈Φ lr . The action onh has kernelT, which means that it factors through the Weyl groupW . The action ofN on l restricts tonT× lr → lw(r) for eachn∈ N(Z) and each cosetw= nT(Z) ∈W. More precisely, we have
n.hr = hw(r) where hw(r) = ∑s∈Π
λws,rhs for certain integers λw
s,r and
n.lr =±lw(r)
for all r ∈Φ, n∈ N(Z) andw= nT(Z) (cf. [7, Prop. 6.4.2]).
The F1-model associated with a Chevalley basis
Let G be theF1-model associated with the closed embeddingG → GL+n,Z. ThenG is a closed
blue subscheme of GLn, and every point ofG is of the formpI for someI ⊂ n×n (cf. Section4.3.1). In this notation, the maximal torusT equals(pe)+
Zwheree is the trivial permutation. If
a point of the formpσ is contained inG, then it is a closed point since it is closed in GLn. NotethatT is diagonal w.r.t.G, i.e.T acts on(PI )
+Z
from the left and from the right for every pointpI of G. Therefore the rank of pseudo-Hopf points is at least equal to the rankr of T, which isthe same as the rank ofpe. This shows that the rank ofG is r.
Lemma 4.10.
(i) TheF1-scheme G contains the pointpI if and only if there is an algebraically closed fieldk and a matrix(ai, j) in G (k)⊂GLn(k) such that ai, j = 0 if and only if(i, j) ∈ I.
(ii) The rank of a pointpI of G equals r if and only if there is a matrix(ni, j) ∈N(C) such thatni, j = 0 if and only if(i, j) ∈ I.
Proof. We show (i). SinceG is cancellative, the morphismβ : G+Z→G is surjective (cf. Lemma
1.32). Thus there exists for every pointpI of G an algebraically closed fieldk such thatpI liesin the image ofβk : G+
k → G. SincepI is locally closed inG, the inverse imagep+I ,k under
βk is locally closed inG+k , which means thatp+I ,k contains a closed point ofG+
k . By Hilbert’sNullstellensatz, such a closed point corresponds to ak-rational pointa : k[G]+→ k, which ischaracterized byai, j = a(Ti, j) sincek[G]+ is a quotient ofk[GLn]
+ and therefore generated bytheTi, j as ak-algebra. This defines the sought matrix(ai, j) ∈G(k) of claim (i).
Part (ii) is proven by the very same argument that we used already in the proof of Theorem4.7. Namely, we can consider the double action ofT(C) on pI(C) from the left and from theright. Then the rank ofpI , which equals the complex dimension ofpI (C), is equal to the rankrof T(C) if and only if pI(C) is contained in the normalizerN(C) of T(C), i.e.pI(C) = nT(C)for somen= (ni, j) ∈ N(C), as claimed in (ii).
73
The rank space
Let n∈N(Z) andw= nT(Z). For every rootr ∈Φ, we can write the coroothw(r) as an integrallinear combinationhw(r) = ∑s∈Πλ
ws,rhs of fundamental corootshs. We writepw = pI(w) where
which is the set of all(i, j) such thatni, j = 0 if we regardn as a matrix of GL+n,Z (cf. the above
formulas in forN acting ong). Note that in general,pw is not a closed point since∣∣λw
s,r
∣∣ mightbe larger than 1, which means thatpw specializes to a point whose potential characteristics arethose primes that divide
∣∣λws,r
∣∣ (this situation occurs indeed for simple groups of typesBn, Cn,F4 andG2).
Proposition 4.11.With the notation as above, we have
Z (G) = { pw | w∈W } and Γpw⋆ ≃ F1ǫ [T
±1r,w(r)|r ∈Π]
whereǫ= 1 if w = e is the neutral element of W andǫ= 2 otherwise. In particular, N= Grk,+Z
.
Proof. To start with, we will show that the pointspw are pseudo-Hopf. The canonical torusTequals the intersection ofG with the diagonal torus diag(GL+
n,Z) inside GL+n,Z since diag(GL+n,Z)
normalizesT and the normalizerN of T in G is of the formN =⋃
n∈N(Z)nT wheren is not ofdiagonal form unlessn ∈ T(Z). If n ∈ pw(Z), thenn−1pw ⊂
(G ∩ diag(GL+
n,Z))= T. Thus
pw+Z = nT, which shows thatpw+
Z is a flat scheme.By definition, we have
Γpw =((
F1[G]�〈Tr,s≡ 0|(r,s) ∈ I(w)〉)red)
ˆ
which is a subblueprint of the coordinate ringΓnT. In particular, the equationsTr,s = λwr,s hold
in ΓnT. In Γpw, we have to readTr,s= λwr,s as
Tr,s ≡ 1+ · · ·+1︸ ︷︷ ︸λw
r,s-times
or Tr,s+1+ · · ·+1︸ ︷︷ ︸(−λw
r,s)-times
≡ 0,
depending on whetherλwr,s is positive or negative.
SinceΓpw is cancellative, we can perform calculations in the ringΓnT to obtain informationaboutΓpw. Let det(Tr,s) be the determinant of theTr,s with r,s∈Ψ. Then the defining equation
74
of GL+nZ is d · det(Tr,s) = 1 whered is the variable for the inverse of the determinant. If we
substituteTr,s by 0 if (r,s) ∈ I(w) and byλ+r,s if (r,s) ∈ Π×Π, then the determinant conditionreads as
d ·∏r∈Φ
Tr,w(r) ·det(λwr,s)r,s∈Π = ±1.
Since the submatrix(λwr,s)r,s∈Π describes the action ofw on the root systemΦ in the basis
Π, the determinant of(λwr,s)r,s∈Π equals the sign ofw. Thus we end up with an equationd ·
∏r∈Φ Tr,w(r) =±1 in ΓnT, which implies an additive relation inΓpwinv. This means thatTr,w(r)
is a unit inΓpw for all r ∈Φ. Therefore,Γpwinv is generated by its units.
Since{pw}(Q) = nT(Q), the only point inpw with potential characteristic 0 ispw itself.SinceG and pw are of finite type overF1, every other point inpw has only finitely manypotential characteristics. Sincepw is a finite space andpw+
Z is a flat scheme,pw must be almostof indefinite characteristic. This completes the proof thatpw is pseudo-Hopf.
Clearly,
N =⋃
n∈N(Z)
nT =∐
w∈W
pw+
Z =∐
w∈W
(pw
⋆)+Z
= Grk,+Z
,
thus Lemma 4.10 (ii) implies that pseudo-Hopf pointpI is of rankr = rkT if and only if pI = pw
for somew∈W. This shows thatZ (G) = {pw|w∈W}.We investigate the unit fieldsΓpw
⋆. SinceΓnT ≃ Z[T±1
r,w(r)|r ∈Π], the unit fields ofΓpw
are of the formF1ǫ [T±1r,w(r)|r ∈Π] whereǫ is either 1 or 2. Since the unit matrix of GLn(Z) is
contained inG (Z), there is a morphism∗F1→ G whose image ispe. This shows thatΓpw⋆ ≃
F1[T±1r,w(r)|r ∈Π]. If w is not the neutral element ofW, then there is a matrixn ∈ N(Z) such
thatw= wn operates non-trivially on the corootsΦ∨ ⊂ h. This means that at least one of thefundamental corootshr is mapped to a negative coroothw(r), i.e.λw
r,s< 0 for all s∈Π. Therefore
Γpw is with−1, and the unit field ofpw equalsF12[T±1r,w(r)|r ∈Π]. This finishes the proof of the
proposition.
The Tits-Weyl model
Finally, we are prepared to prove that adjoint Chevalley groups have Tits-Weyl models. Moreprecisely, we formulate the following result.
Theorem 4.12.Let G be theF1-model ofG as described above.
(i) The group lawµG descends uniquely to a monoid lawµ+ of G+.
75
(ii) The restriction ofµG to N descends uniquely to a group lawµrk of Grk.
(iii) The pairµ= (µrk,µ+) is a Tits morphism, which turns G into a Tits-Weyl model ofG .
(iv) The canonical torus is diagonal.
(v) The group GT(F1) is the trivial subgroup of W= W (G).
Proof. We prove (i). The existence and uniqueness ofµ+ : G+×+ G+ → G+ with µ+Z= µG
follows from an application of the cube lemma (Lemma 4.6) to the commutative diagram
GL+n,Z×+ZGL+
n,Z
µ+GLn,Z //
����
GL+n,Z
����
G ×+Z
G&�
33❤❤❤❤❤❤❤❤❤❤❤ µG //
����
G)
66♠♠♠♠♠♠♠♠♠♠♠
����
GL+n ×+NGL+
n
µ+GLn // GL+
n .
G+×+N
G+&�
33❤❤❤❤❤❤❤❤❤❤❤G+
(�
66❧❧❧❧❧❧❧❧❧❧
The identity ofG+ is the unique morphism that completes the diagram
∗Zǫ+GLn,Z //
����
GL+n,Z
����
∗Z)
id66♥♥♥♥♥♥♥♥♥♥♥ ǫG //
����
G(�
55❧❧❧❧❧❧❧❧❧❧❧
����
∗Nǫ+GLn // GL+
n
∗N)
id66♠♠♠♠♠♠♠♠♠♠
G+(�
55❧❧❧❧❧❧❧❧❧❧❧
to a commuting cube, which exists by the cube lemma. This proves (i).We continue with (ii). SinceN does not embed into the subgroup of monomial matrices of
GL+n,Z, we have to use a different argument as in the proof of Theorem4.7. The group lawµN
of N can be restricted to morphisms between the connected componentsµw1,w2 : n1T×n2T →(n1n2)T wheren1 andn2 range throughN(Z) andw1 andw2 are the corresponding elementsof the Weyl group. SincenT is isomorphic to the spectrum ofZ[T±1
r,w(r)]r∈Π wherew= wn, themorphismsµw1,w2 yield ring homomorphisms
Γµw1,w2 : Z[T±1r,w12(r)
]r∈Π −→ Z[(T′r,w1(r))±1,(T ′′r,w2(r)
)±1]r,s∈Π,
76
wherew12 = w1w2. These ring homomorphisms restrict to blueprint morphisms
Γµ⋆w1,w2: F12[T±1
r,w12(r)]r∈Π −→ F12[(T′r,w1(r)
)±1,(T′′r,w2(r))±1]r,s∈Π,
between the unit fields. Since the identity matrix is the neutral element ofN(Z), the morphismΓµ⋆e,e must be compatible with the mapsTr,r 7→ 1. This means thatΓµ⋆e,e is the base extensionof a morphism
Γµrke,e : Γpe
⋆ −→ Γ(pe
⋆× pe⋆)
from F1 to F12 (note that by Proposition 4.11,pe⋆
is without−1).
If one ofn1 andn2 differs frome, then it follows from Proposition 4.11 thatpw1⋆×F1
ˆpw2⋆
is
with −1. This means thatΓ( ˆpw1
⋆×F1
ˆpw2⋆)≃ F12[(T ′r,w1(r)
)±1,(T′′r,w2(r))±1]r,s∈Π. Therefore we
can define the restriction ofµrk to ˆpw1⋆×F1
ˆpw2⋆
by
Γµrkw1,w2
: Γ ˆpw12⋆ ⊂ F12[T±1
r,w12(r)]r∈Π
Γµ⋆w1,w2−→ Γ
( ˆpw1⋆×F1
ˆpw2⋆).
This defines a morphismµ : Grk×Grk→ Grk that base extends to the group lawµN of N. Theuniqueness ofµrk is clear. The associativity ofµrk follows easily from the associativity ofµN.Since the Weyl kernele= pe
⋆is without−1, the identity∗Z→N of µN descends to an identity
∗F1→ e⊂Grk of µrk. Similar arguments as above show that the inversion ofµN restricts to aninversionιrk of µrk. Thusµrk is a group law forGrk.
We proceed with (iii). SinceµN = µrk,+Z
is the restriction ofµG = µ+Z
, the pairµ= (µrk,µ+)is Tits. Sincee+
Z= T, the canonical torus is a maximal torus ofG and the morphismΨ :
Grk,+/e+Z→N/T is an isomorphism of group schemes. Thus(G,µ) is a Tits-Weyl model ofG .
Part (iv) is clear. Part (v) follows from the description of the unit fields ofpw⋆
in Proposition4.11.
5 Tits-Weyl models of subgroups
In this part of the paper, we establish Tits-Weyl models of subgroups of Chevalley groups, i.e.split reductive group schemes. Namely, we will investigateparabolic subgroups, their unipotentradicals and their Levi subgroups.
As a preliminary observation, consider a group schemeG of finite type and a torusT in G .Let C be the centralizer ofT in G andN the normalizer ofT in G . If H is a subgroup ofGthat containsT, then the centralizerC of T in H equals the intersection ofC andH , and thenormalizerN of T in H equals the intersection ofN andH .
77
This means that if we are in the situation of Theorem 4.7, i.e.if G is an affine smoothgroup scheme of finite type with Tits Weyl modelG such thatGrk,+
Zis the normalizer of the
canonical torusT, then a smooth subgroupH of G that containsT satisfies automatically allother hypotheses of Theorem 4.7.
5.1 Parabolic subgroups
LetG be a split reductive group scheme. Aparabolic subgroupof G is a smooth affine subgroupP of G such that for all algebraically closed fieldsk, the algebraic groupPk is a parabolicsubgroup ofGk.
Definition 5.1. Let G be the Tits-Weyl model of a split reductive group schemeG . A closedsubmonoidP is aparabolic submonoid of Gif it is the Tits-Weyl model ofP+
ZwhereP+
Zis a
parabolic subgroup ofG and ifPrk contains the Tits-Weyl kernel ofG.
Theorem 5.2. LetG be a reductive group scheme with Tits-Weyl model G and canonical torusT . The parabolic submonoids P of G stay in bijection with the parabolic subgroupsP of G
that contain T .
Proof. Given a parabolic submonoidP, then the parabolic subgroupP = P+Z
contains thecanonical torusT = e+
ZsincePrk contains the Weyl kernele of G. If P is a parabolic sub-
group ofG that containsT, then it is an immediate consequence of Theorem 4.7 thatP has aTits-Weyl modelP. It is clear that this two associations are inverse to each other.
5.2 Unipotent radicals
Let P be a parabolic subgroup of a reductive group schemeG . ThenP has aunipotent radicalU , i.e. the smooth closed normal subgroup such that for all algebraically closed fieldsUk isthe unipotent radical ofPk (cf. [3, XXII, 5.11.3, 5.11.4] or [12, Cor. 5.2.5] for the existence ofU ). The group schemesU that occur as unipotent radicals of parabolic subgroups of reductivegroup schemes have the following properties.
As a scheme,U is isomorphic to an affine space+AnZ. The only torus contained inU is the
imageT of the identityǫ : ∗Z→ U , which is a 0-dimensional torus. Trivially,T is a maximaltorus ofU . The centralizerC(T) and the normalizerN(T) of T in U both equalU . Therefore,the Weyl groupW = N(T)/C(T) of U is the trivial group scheme∗Z.
Let U be theF1-model of the inclusionU →P andP the Tits-Weyl model ofP. Asa consequence of the cube lemma,µ+P restricts to a monoid lawµ+U of U+. SinceT is the
intersection of the maximal torus ofP with U , U contains a pointe such thate+Z is T. Thus
e∈Z (U). If one can show thatU does not contain any other pseudo-Hopf point of rank 0, then
78
U rk = e⋆ ≃ ∗F1, and the group lawµrk
P of Prk restricts to the trivial group lawµrkU of U rk. In this
case,U together with(µrkU ,µ+U) is a Tits-Weyl model ofU .
Definition 5.3. Let G be the Tits-Weyl model of a reductive group scheme andP a parabolicsubmonoid. A submonoidU of P is theunipotent radicalof P if U+
Zis the unipotent radical of
P+Z
and ifU is a Tits-Weyl model ofU+Z
.
Remark 5.4. The uniqueness ofU is clear: ifU is the unipotent radical ofP = P+Z
, thenUmust be theF1-model associated toU →P. It is, however, not clear to me whether unipotentradicals always exist, i.e. if alwaysZ (U) = {e}.
We can prove the existence of unipotent radicals in the following special case. We call aparabolic subgroup of GL+n,Z that contains the subgroup of upper triangular matrices astandard
parabolic subgroup ofGL+n,Z.
Proposition 5.5. Let P be a standard parabolic subgroup ofGL+n,Z andU its unipotent radi-
cal. Let U → P be the associatedF1-models. Then U is the unipotent radical of P.
Proof. Everything is clear from the preceding discussion if we can show thatZ (U) containsonly one point. The unipotent radical of a standard parabolic subgroup is of the formU =Spec[Ti, j ]/I whereI is the ideal generated by the equationsTi, j = 0 for i > j, Ti, j = 1 for i = jandTi, j = 0 for certain pairs(i, j) with i < j. Let I be the subset ofn×n that contains all pairs(i, j) that did not occur in the previous relations. ThenU = SpecF1[Ti, j ](i, j)∈I ≃ A#I
F1as a blue
scheme. It is clear thatZ (U) consists of only one point, which is the maximal ideal(Ti, j)(i, j)∈Iof F1[Ti, j ](i, j)∈I (cf. Example 2.7).
5.3 Levi subgroups
Let P be a parabolic subgroup of a reductive group schemeG . Let U be theunipotent radicalof P. If P contains a maximal torusT of G , thenP has aLevi subgroupM , i.e. a reductivesubgroup that is isomorphic to the scheme theoretic quotient P/U such that for every alge-braically closed fieldk, Mk is the Levi subgroup ofPk (see [12, Thm. 4.1.7, Prop. 5.2.3] or[13, Lemmas 2.1.5 and 2.1.8] for the existence ofM ). In particular,M contains the maximaltorusT.
Definition 5.6. Let G be the Tits-Weyl model of a reductive group schemeG . Let P be aparabolic subgroup ofG. A submonoidM of P is called aLevi submonoidif M+
Zis the Levi
subgroup ofP and ifM is a Tits-Weyl model ofM+Z
.
Theorem 5.7.Let G be the Tits-Weyl model of a reductive group schemeG . Let P be a parabolicsubmonoid of G. Then P contains a unique Levi submonoid M.
79
Proof. The uniqueness follows from the uniqueness of the Levi subgroup ofP+Z
. The existencefollows from the existence of the Levi subgroup ofP+
Zand Theorem 4.7.
A Examples of Tits-Weyl models
A.1 Non-standard torus
There are different blue schemes together with a monoid law in SchT that are Tits-Weyl modelsof the torusGr
m,Z of rankr. We give one example forr = 1, i.e. a non-standard Tits-Weyl modelof the multiplicative group schemeGm,Z.
Namely, consider the blueprintB = F1[S,T±1]�〈S≡ 1+ 1〉. Its universal ring isB+Z=
Z[T±1], the coordinate ring ofGm,Z. ThusG = SpecB is anF1-model ofGm,Z. The blueschemeG consists of two points: the closed pointx= (S), which is of characteristic 2, and thegeneric pointη = (0), which has all potential characteristics except for 2. The point η is theonly pseudo-Hopf point ofG, i.e.Z (G) = {η}. The rank space ofG is Grk ≃SpecF1[T±1] andits universal semiring scheme isG+ ≃ SpecN[T±1].
The group law ofGm,Z descends to a morphismµ : G×G→ G. Namely, it is given by themorphism
Γµ : B −→ B⊗F1 B = F1[S1,S2,T±11 ,T±1
2 ]�〈S1≡ 1+1≡ S2〉
between the global sections ofG andG×G that is defined byΓµ(S) = S1 andΓµ(T) = T1⊗T2.IndeedG becomes a semigroup object in Schrk
F1without an identity: there is no morphism
B→ F1 sinceF1 contains no elementS′ that satisfiesS′ ≡ 1+1.However, the morphismµ mapsZ (G×G) to Z (G), i.e.µ is Tits. In the category SchT ,
the pair(G,µ) is a group. Since the Weyl group ofGm,Z is the trivial group andGrk consists of
one point,G is a Tits-Weyl model ofGm,Z.While it is clear thatG is not isomorphic toGm,F1 = SpecF1[T±1] in SchF1, the locally alge-
braic morphismϕ : G→Gm,F1 that is defined by the obvious inclusionF1[T±1] → F1[S,T±1]�〈S≡ 1+1〉 is Tits and an isomorphism of groups in SchT .
More generally, it can be shown that every cancellative TitsmodelG of Grm,Z is the spectrum
of a subblueprint ofZ[T±11 , . . . ,T±1
r ] that containsF1[T±11 , . . . ,T±1
r ], but not−1. Moreover, theinclusionF1[T
±11 , . . . ,T±1
r ] → ΓG is Tits and defines an isomorphismG→ Gm,F1 of groups inSchT .
80
A.2 Tits-Weyl models of typeA1
In this section, we calculate explicitly the different Tits-Weyl models that we described in themain text of the paper for groups of typeA1. Namely, we reconsider the standard model SL2of the special linear group, the Tits-Weyl model of the adjoint groupG of typeA1 given by theconjugation action on Mat2×2 and the Tits-Weyl model ofG given by the adjoint representation.
The standard model ofSL2
We reconsider the example SL2 = SpecF1[SL2] with F1[SL2] = F1[T1, . . . ,T4]�〈T1T4≡ T2T3+1〉 and make the heuristics from the introduction precise. The prime idealsp of F1[SL2] aregenerated by a subset of{T1, . . . ,T4} such that not bothT1T4 andT2T3 are contained inp. Weillustrate SL2 in Figure 4 where the encircled points are the pseudo Hopf points of minimalrank.
(0)
(T3) (T4)
(T1,T4)(T2,T3)
(T2) (T1)
Figure 4: The standard model of SL2
One sees clearly that the maximal idealp2,3 = (T2,T3) corresponds to the diagonal torusT =
{( ∗ 00 ∗)}
of the matrix group SL2(k) (wherek is a ring and∗ stays for a non-zero entry)andp1,4 = (T1,T4) corresponds to the subset
{(0 ∗∗ 0
)}of anti-diagonal matrices. The ideals
p1 = (T1), p2 = (T2), p3 = (T3) andp4 = (T4) correspond to the respective subsets{(
0 ∗∗ ∗)}
,{( ∗ 0∗ ∗)}
,{( ∗ ∗
0 ∗)}
and{( ∗ ∗∗ 0)}
, while (0) corresponds to the subset{( ∗ ∗∗ ∗
)}.
The adjoint group of type A1 via conjugation
We turn to the adjoint groupG of typeA1. Note thatG (k) = PSL2(k) if we consider an alge-braically closed fieldk. One can represent PSL2(k) by the conjugation action on 2×2-matrices.Consider
(a bc d
)∈ SL2(k) and
(e fg h
)∈Mat2×2(k). Then the product
(a bc d
)(e fg h
)(a bc d
)−1
=
(ade−ac f+bdg−bch −abe+a2 f −b2g+abhcde−c2 f +d2g−cdh −bce+ac f−bdg+adh
)
81
shows that PSL2(k) acts on the 4-dimensional affine space, generated bye, f , g andh, via thematrices
A(a,b,c,d) =
ad −ac bd −bc−ab a2 −b2 abcd −c2 d2 −cd−bc ac −bd ad
with ad−bc=1. This is a faithful representation of PSL2(k). The algebraic groupGk overk thatis associated to the group PSL2(k) = {A(a,b,c,d)|ad−bc= 1} ⊂ +A4
k descends to an integralmodelG ⊂ GL+
4,Z, which is an adjoint Chevalley group of typeA1. Let G be the associatedF1-model. Then the prime ideals ofG are generated by subsets of{Ti, j}i, j=1,...,4 whereTi, j
is the matrix coefficient at(i, j). Sincead−bc= 1, one ofad andbc has to be non-zero forA(a,b,c,d) ∈ PSL2(k). We consider the various possible combinations ofa, b, c andd beingzero or not (as above,∗ denotes a non-zero entry):
a= 0 :
(0 0 ∗ ∗0 0 ∗ 0∗ ∗ ∗ ∗∗ 0 ∗ 0
)b= 0 :
(∗ ∗ 0 00 ∗ 0 0∗ ∗ ∗ ∗0 ∗ 0 ∗
)c= 0 :
(∗ 0 ∗ 0∗ ∗ ∗ ∗0 0 ∗ 00 0 ∗ ∗
)
d = 0 :
(0 ∗ 0 ∗∗ ∗ ∗ ∗0 ∗ 0 0∗ ∗ 0 0
)a= d = 0 :
(0 0 0 ∗0 0 ∗ 00 ∗ 0 0∗ 0 0 0
)b= c= 0 :
(∗ 0 0 00 ∗ 0 00 0 ∗ 00 0 0 ∗
)
The casea,b,c,d 6= 0 corresponds to the matricesA(a,b,c,d) with no vanishing coefficient.The zero entries of each case stay for the generatorsTi, j of the prime ideals ofG. Withoutwriting out the generating sets, we see in Figure 5 that the topological space ofG is the same asthe topological space of SL2.
Figure 5: The Tits-Weyl model of typeA1 defined by the conjugation action
Since the maximal points (which are encircled in Figure 5) correspond to the diagonal andanti-diagonal matrices, respectively, they are the pseudo-Hopf points of minimal rank. Thepre-rank spaceG∼ is discrete andGrk embeds intoG.
82
The adjoint group of type A1 via the adjoint action
Let G be the Tits-Weyl model of the adjoint groupG of typeA1 that is defined by the adjointaction ofG on its Lie algebra. The roots system of typeA1 is Φ = {±a} and the set of primitiveroots isΠ = {a}. Thus a basis of the Lie algebra ofG is given by the ordered tupleΨ =(l−a,ha, la) where we chose this ordering ofΨ to obtain nice matrix representations below.SinceG is defined as a closed subscheme of GL3, every pointx of G is of the formpI forsomeI ⊂ {1,2,3}×{1,2,3}. SinceG is cancellative, the morphismβG : G → G is surjectiveby Lemma 1.32. Thus a pointpI ∈ GL3 is contained inG⊂ GL3 if and only if there is analgebraically closed fieldk and a matrix(ai, j) ∈ G (k) such thatai, j = 0 if and only if (i, j) ∈ I .This reduces the study of the topological space ofG to the study of matrices(ai, j) ∈ G (k), forwhich we can use explicit formulas.
There is a surjective group homomorphismϕ : SL2(k)→ G (k) (see [7, Section 6]). Wedescribe the image of certain elements of SL2(k) in G ⊂GL3(k)w.r.t. the basisΨ= (l−a,ha, la):
ϕ((
1 t0 1
)) =
(1 t −t2
0 1 −2t0 0 1
), ϕ(
(λ 00 λ−1
)) =
(λ−2 0 00 1 00 0 λ2
)and ϕ(
(0 1−1 0
)) =
(0 0 −10 −1 0−1 0 0
).
For the first equation, see Section 6.2 of [7], for the second equation Proposition 6.4.1 and forthe last equation Propositions 6.4.2 and 6.4.3 of [7].
The Bruhat decomposition of SL2(k) is SL2(k) = B(k)∐BwB(k) wherew=(
0 1−1 0
)andB
is the upper triangular Borel subgroup of SL2, i.e.B(k) is the set of all matrices that can writtenas a product
(λ 00 λ−1
)(1 t0 1
)with λ ∈ k× andt ∈ k. In other words, every element(ai, j) ∈ SL2(k)
can be written as a product(λ 00 λ−1
)(1 t0 1
)or as a product
(1 s0 1
)(0 1−1 0
)(λ 00 λ−1
)(1 t0 1
)with λ ∈ k×
ands, t ∈ k. Sinceϕ : SL2(k)→ G (k) is a surjective group homomorphism, we yield
G (k) =
{(λ−2 λ−2t −λ−2t2
0 1 −2t0 0 λ2
)}
λ∈k×t∈k
∐{(
λ−2s2 −s+λ−2ts2 −λ2+2st−λ−2s2t2
2λ−2s −1+2λ−2st 2t−2λ−2st2
−λ−2 −λ−2t λ−2t2
)}
λ∈k×s,t∈k
.
To find the points ofG, we have to investigate for whichλ,s, t a matrix coefficient of the abovematrices vanishes. Concerning the first matrix, we see that the following cases appear:
t = 0 :
(∗ 0 00 ∗ 00 0 ∗
)pe
t 6= 0,chark 6= 2 :
(∗ ∗ ∗0 ∗ ∗0 0 ∗
)x1 = p{(2,1),(3,1),(3,2)}
t 6= 0,chark= 2 :
(∗ ∗ ∗0 ∗ 00 0 ∗
)x′1 = p{(2,1),(2,3),(3,1),(3,2)}
83
where∗ stays for a non-zero entry and the right hand side column lists the image points inG⊂ GL3 together with the notation used in Figure 6. Recall from Section 4.1 thatpe = pI(e)wheree∈ S3 is the trivial permutation.
Concerning the second matrix, we have to consider more cases. If not both s and t arenon-zero, we obtain immediately the following list:
s= t = 0 :
(0 0 ∗0 ∗ 0∗ 0 0
)pσ
s= 0, t 6= 0,chark 6= 2 :
(0 0 ∗0 ∗ ∗∗ ∗ ∗
)x3 = p{(1,1),(1,2),(2,1)}
s= 0, t 6= 0,chark= 2 :
(0 0 ∗0 ∗ 0∗ ∗ ∗
)x′3 = p{(1,1),(1,2),(2,1),(2,3)}
s 6= 0, t = 0,chark 6= 2 :
(∗ ∗ ∗∗ ∗ 0∗ 0 0
)x4 = p{(2,3),(3,2),(3,3)}
s 6= 0, t 6= 0,chark= 2 :
(∗ ∗ ∗0 ∗ 0∗ 0 0
)x′4 = p{(2,1),(2,3),(3,2),(3,3)}
To investigate the cases of vanishing matrix coefficients with s 6= 0 6= t, consider the followingcases:
−s+λ−2s2t = 0 ⇐⇒ ts = λ2
−λ2+2st−λ−2s2t2 = 0 ⇐⇒ ts = λ2
−1+2λ−2st = 0 ⇐⇒ 2ts = λ2 (in this case chark 6= 2)
2t−2λ−2st2 = 0 ⇐⇒ ts = λ2 (if chark 6= 2)
84
This yields the following additional points ofG wheres 6= 0 6= t:
st= λ2,chark 6= 2 :
(∗ 0 0∗ ∗ 0∗ ∗ ∗
)x2 = p{(1,2),(1,3),(2,3)}
st= λ2,chark= 2 :
(∗ 0 00 ∗ 0∗ ∗ ∗
)x′2 = p{(1,2),(1,3),(2,1),(2,3)}
2st= λ2,chark 6= 2 :
(∗ ∗ ∗∗ 0 ∗∗ ∗ ∗
)x5 = p{(2,2)}
st 6= λ2 6= 2st,chark 6= 2 :
(∗ ∗ ∗∗ ∗ ∗∗ ∗ ∗
)η = p /0
st 6= λ2 6= 2st,chark= 2 :
(∗ ∗ ∗0 ∗ 0∗ ∗ ∗
)η′ = p{(2,1),(2,3)}
We summarize these calculations in Figure 6. The circled points are the pseudo-Hopf pointsof minimal rank.
x5 x4
x′3x′2x′1x1
x2 x3x′4
η ′
η
pe pσ
Figure 6: The Tits-Weyl model of typeA1 defined by the adjoint action
Remark A.1. It is clear that this Tits-Weyl model ofG differs in SchF1 from the Tits-Weylmodel that is defined by the conjugation action on 2×2-matrices (cf. Figure 5). It is, however,not clear to me whether these two models ofG are isomorphic in SchT or not.
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