Severi-Brauer varieties over F1

M-A. Knus

ETH Zürich

Banff, September 17, 2009

A report on joint work with Jean-Pierre Tignol,

Louvain-La-Neuve

Workshop on Linear Algebraic Groups and Related Structures

Outline (Keywords)

I Objects over the field F1 of characteristic 1

I Twisted Objects

I Clifford algebras

I Examples

Objects over the field F1 of characteristic 1

I Tits’ Geometry over the field F1 of characteristic 1

I Recent developments

Tits’ Geometry over the field F1 of characteristic 1

The idea of a «field F1 of characteristic 1», that is of «a field

with one element» first showed up in the paper of Tits (1957)

Aim of the paper:

Associate geometries to all simple complex groups (i.e., to all

Dynkin diagrams).

”Abordant par une voie nouvelle le problème de l’interprétation

géométrique des groupes simples complexes, j’ai été conduit à

associer à chacun de ces groupes G une «géométrie» Γ(G)

ayant un groupe d’automorphismes isomorphe à G.”

Tits’ Geometries

G = G(E ; Gr0, . . . , Grn−1 ; ι; A

)

• E : set distributed in n families Gri .

• ι: incidence relation on E , symmetric and reflexive.

• A: group of automorphisms of the incidence structure.

n : Index of the geometry

Axioms:

1. The group A acts transitively on each Gri .

2. No Gi is empty.

3. Two distinct elements of a same family are never incident.

4. For all pairs of families Gri and Grj , A acts transitively on

the couples of incident elements (a, a′), a ∈ Gri , a′ ∈ Grj .

5. One can choose in each family Gri one element ai such

that the ai are pairwise incident.

Geometry of Type An−1

G(P) = G(P; Gr0(P), . . . , Grn−1(P); ι, A

)P = Pn−1(F ) Projective space of dimension n-1 over F .

Grk−1(P): Grassmannian of (k-1)-dimensional linear

varieties contained in P, 1 ≤ k ≤ n.

ι: Usual incidence relation.

A = PGLn(F ).

Geometry of Type Dn

G(Q) = G(Q; Gr0(Q), . . . , Grn−2(Q), C+(Q), C−(Q); ι; A

)Q ⊂ P2n−1(F ): (2n-2)-dimensional quadric defined by

a 2n-dimensional hyperbolic quadratic form.

Grk−1(Q) Grassmannian of (k-1)-dimensional linear

varieties contained in Q, 1 ≤ k ≤ n.

Grn−1(Q) = C(Q) = C+(Q) t C−(Q)

(For v1, v2 ∈ C(Q), v1 ≡ v2 ⇐⇒ dim(v1 ∩ v2) ≡ n − 1 mod 2)

ι Usual incidence with the convention that two linear varieties

of dimension n-1 belonging to different classes of C(Q)

are incident if their intersection has dimension n-2.

A = PGO+2n(F ).

Geometries associated to Chevalley groups and Weylgroups

Theorem(Tits). Let D be any Dynkin diagram. Let GF (D) be

the Chevalley group over F attached to D and let W (D) be the

corresponding Weyl group. There exists unique geometries

GF (D) and GW (D) such that the automorphism groups of the

geometries are resp. GF (D) and W (D).

Tits calls the geometries GW (D) attached to Weyl groups

Geometries over the «field F1 of characteristic 1».

”Nous désignerons par K = K1 le « corps de caractéristique 1»

formé du seul élément 1 = 0 (19). Il est naturel d’appeler

espace projectif à n dimensions sur K , un ensemble Pn of n + 1

points dont tous les sous-ensembles sont considérés comme

des variétés linéaires {...}.

(19) K1 n’est généralement pas considéré comme un corps.”

Geometry of Type An−1 over F1

Pn−1(F1)def⇐⇒ n-element set X .

(q-1)-dimensional subvariety of Pn−1(F1)def⇐⇒ (q)-element subset of X

Grk−1(X ):= {(k − 1)-dimensional linear varieties ⊂ Pn−1(F1) }.

=⇒

Projective geometry of dimension n-1 over F1

A = Sn

Geometry of Type Dn over F1

A quadric of dimension 2(n − 1) over F1 is a set Y of 2n points

distributed in pairs (a1, b1), . . . , (an, bn)

A linear subvariety of dimension q − 1 of the quadric Y is a

q-element subset of Y not containing any pair (ai , bi).

Grk−1(Y )= {(k − 1)-dimensional linear subvarieties ⊂ Y }

Grn−1(Y ) = C(Y ) = C+(Y ) t C−(Y )

=⇒

Geometry of the quadric Y over F1

A = Sn−12 o Sn

Why are these geometries called

Geometries over the "field F1 of characteristic 1" ?

|Pn−1(Fq)| = qn − 1/q − 1 = 1 + q + q2 + · · ·+ qn−1

q = 1 =⇒ |Pn−1(F1)| = n.

References on recent developments

can be found on the site

located at http://matrix.cmi.ua.ac.be/fun/

and kept by Lieven Le Bruyn.

Our contribution

I Profinite group action, usually an absolute Galois group.

=⇒Algebraic structures, in parallel to the geometric structures.

I F -structures lying over corresponding structures over F1

(D. Saltman: ”Triality, cocycles, crossed products,

involutions, Clifford algebras and invariants”).

I A natural construction generalizing the Clifford algebra of a

quadratic form.

For simplicity we shall restrict to fields of characteristic not 2

and consider only situations of type An−1 and Dn.

Our contribution

I Profinite group action, usually an absolute Galois group.

=⇒Algebraic structures, in parallel to the geometric structures.

I F -structures lying over corresponding structures over F1

(D. Saltman: ”Triality, cocycles, crossed products,

involutions, Clifford algebras and invariants”).

I A natural construction generalizing the Clifford algebra of a

quadratic form.

For simplicity we shall restrict to fields of characteristic not 2

and consider only situations of type An−1 and Dn.

Our contribution

I Profinite group action, usually an absolute Galois group.

=⇒Algebraic structures, in parallel to the geometric structures.

I F -structures lying over corresponding structures over F1

(D. Saltman: ”Triality, cocycles, crossed products,

involutions, Clifford algebras and invariants”).

I A natural construction generalizing the Clifford algebra of a

quadratic form.

For simplicity we shall restrict to fields of characteristic not 2

and consider only situations of type An−1 and Dn.

Our contribution

I Profinite group action, usually an absolute Galois group.

=⇒Algebraic structures, in parallel to the geometric structures.

I F -structures lying over corresponding structures over F1

(D. Saltman: ”Triality, cocycles, crossed products,

involutions, Clifford algebras and invariants”).

I A natural construction generalizing the Clifford algebra of a

quadratic form.

For simplicity we shall restrict to fields of characteristic not 2

and consider only situations of type An−1 and Dn.

Γ-projective spaces

F field with separable closure Fs; Γ = Gal(Fs/F ).

Γ-Projective space P over F : Projective space P over Fs

endowed with the discrete topology and a continuous (left)

action of Γ by collineations.

Γ-Projective space X over F1: (finite) Γ-set X .

Γ acts on the Grassmannians Grk−1(P), resp. Grk−1(X ).

Γ-projective spaces

F field with separable closure Fs; Γ = Gal(Fs/F ).

Γ-Projective space P over F : Projective space P over Fs

endowed with the discrete topology and a continuous (left)

action of Γ by collineations.

Γ-Projective space X over F1: (finite) Γ-set X .

Γ acts on the Grassmannians Grk−1(P), resp. Grk−1(X ).

Γ-projective spaces

F field with separable closure Fs; Γ = Gal(Fs/F ).

Γ-Projective space P over F : Projective space P over Fs

endowed with the discrete topology and a continuous (left)

action of Γ by collineations.

Γ-Projective space X over F1: (finite) Γ-set X .

Γ acts on the Grassmannians Grk−1(P), resp. Grk−1(X ).

Γ-projective spaces

F field with separable closure Fs; Γ = Gal(Fs/F ).

Γ-Projective space P over F : Projective space P over Fs

endowed with the discrete topology and a continuous (left)

action of Γ by collineations.

Γ-Projective space X over F1: (finite) Γ-set X .

Γ acts on the Grassmannians Grk−1(P), resp. Grk−1(X ).

Γ-projective spaces and algebraic structures.

I Γ-projective spaces over F1 are related with étale algebras

over F .

I Γ-projective spaces over F are related with central simple

algebras over F .

Γ-projective spaces and algebraic structures.

I Γ-projective spaces over F1 are related with étale algebras

over F .

I Γ-projective spaces over F are related with central simple

algebras over F .

Étale algebras

L étale over F , dimF L = n ⇐⇒ L⊗F Fs∼→ F n

s

Γ-projective spaces over F1

of dimension n-1

m

étale algebras over F of dim. n

( L 7→ X(L) = HomF -alg(L, Fs))

Γ-spaces with n elements, resp. étale algebras over F of

dimension n are classified by the Galois cohomology set

H1(Γ, Sn).

Γ-projective spaces and central simple algebras

A; central simple F -algebra of degree n, As = A⊗F Fs.

The Severi-Brauer variety SB(A) of A is the Γ-projective space

over Fs whose (k-1)-dimensional linear varieties are the

nk -dimensional right ideals of As.

Remark. L étale of dimension L. Since

X (L) = HomF -alg(L, Fs) ' {Minimal ideals of Ls}

we may say that X (L) is the Severi-Brauer variety of the étale

algebra L.

Embeddings

X : Γ-projective space of dimension n − 1 over F1

P: Γ-projective space of dimension n − 1.

We define a general embedding

ε : X ↪→ P

to be a Γ-equivariant injective map X → Gr0(P), whose image

consists of n points in general position.

There is map induced a map

Grk−1ε : Grk−1(X ) ↪→ Grk−1(P) for 1 ≤ k ≤ n,

which carries each k -element subset of X to the linear variety

spanned by its image.

L: étale F -algebra of dimension n

A: central simple F -algebra of degree n

ε : L ↪→ A

induces a general embedding

ε∗ : X (L) ↪→ SB(A),

ξ ∈ X (L) = HomF -alg(L, Fs) 7→ε∗(ξ) =

{x ∈ As |

(1⊗ ξ(`)

)· x =

(ε(`)⊗ 1

)· x for all ` ∈ L

}

Γ-quadrics and central simple algebras withorthogonal involution

P : Γ-projective space of odd dimension 2n-1.

A Γ-quadric Q of dimension 2(n-1) is a subset of P defined by a

hyperbolic quadratic form of dimension 2n stable under Γ.

(A, σ) : Central simple algebra with orthogonal involution

of degree 2n.

The set

Q(A, σ) = {Isotropic 2n-dimensional right ideals ⊂ As}

is a Γ-quadric contained in SB(A).

Γ-quadrics over F1

Quadrics Y of dimension 2(n-1) over F1 ⇐⇒Double coverings Y → Y0, |Y | = 2n ⇐⇒Sets with involution (Y , σ), |Y | = 2n.

(Y = {a1, b1, . . . , an, bn}, Y0 = {(a1, b1), . . . , (an, bn)},σ(ai) = bi , σ(bi) = ai )

Γ-quadrics over F1def⇐⇒ Γ-double coveringsdef⇐⇒ Γ-sets with involution.

Γ-quadrics and étale algebras with involution

The Grothendieck correspondence induces:

Γ-Quadrics over F1 ofdimension 2(n-2)

⇐⇒ Étale algebras with involutionof dimension 2n over F

Both types of objects are classified by the Galois cohomology

set

H1(Γ, Sn2 o Sn).

Embeddings of algebras with involutions

(L, σL): étale F -algebra of dimension 2n with involution

(A, σ): central simple F -algebra of degree 2n with orthogonal

involution

ε : (L, σL) ↪→ (A, σ): embedding of algebras with involution.

Proposition. The general embedding of Γ-sets

ε∗ : X (L) ↪→ SB(A) restricts to an embedding

ε∗ : X (L, σL) ↪→ Q(A, σ).

The Clifford variety

Q: Γ-quadric

C(Q) := {(n-1)-dimensional linear varieties ⊂ Q}

is a Γ-set called the Clifford variety of Q.

C(Q) = C+(Q) t C−(Q)

(For v1, v2 ∈ C(Q), v1 ≡ v2 ⇐⇒ dim(v1 ∩ v2) ≡ n − 1 mod 2) is

a decomposition as Γ-sets.

∆(Q) := {C+(Q), C−(Q)},

is the discriminant of Q. We let

δ : C(Q)→ ∆(Q), v ∈ C(Q) 7→ [v ] ∈ ∆(Q).

The Clifford variety of a Γ-quadric over F1

(Y , σ): Γ-set with involution, |Y | =2n

C(Y , σ) = Grn−1(Y ) =

{ n-element subsets of Y not containing any pair (ai , bi)}.

Thus

C(Y , σ) = {Sections of the covering Y → Y0}, |C(Y , σ)| = 2n.

C(Y , σ) = C+(Y , σ) t C−(Y , σ)

∆(Y , σ) := {C+(Y , σ), C−(Y , σ)},

δ : C(Y , σ)→ ∆(Y ), v ∈ C(Y , σ) 7→ [v ] ∈ ∆(Q).

Embeddings of Γ-sets with involution in quadrics

(Y , σ): Γ-quadric over F1 of dimension 2(n-2)

Q: Γ-quadric of dimension 2(n-2).

Proposition. A general embedding

ε : (Y , σ) ↪→ Q

induces a Γ-equivariant map

C(ε) : C(Y , σ) ↪→ C(Q).

The Clifford algebra of an étale algebra

(L, σ) : étale algebra with involution of dim. 2n

X (L, σ) : Γ-quadric corresponding to (L, σ).

C(L, σ) : étale algebra of dimension 2n such that

X C(L, σ) = C X(L, σ).

Example. C(L, σ) = L for n = 1 and

C(L1 × L2, σ1 × σ2) = C(L1, σ1)⊗ C(L2, σ2)

for two étale algebras with involution (L1, σ1) and (L2, σ2).

There is an inclusion ∆(L)→ C(L, σ) corresponding to the

projection ι : C(Q)→ ∆(Q).

(L, σL): étale F -algebra of dimension 2n with involution

(A, σ): central simple F -algebra of degree 2n with orthogonal

involution.

ε : (L, σL) ↪→ (A, σ): embedding of algebras with involution.

Proposition. The map ε induces an embedding

C(ε) : C(L, σL) ↪→ C(A, σ)

Type Dn

Q : Γ-quadric

∆(Q) := {C+(Q), C−(Q)}

To stay in the Dn-setting we assume from now on that Γ acts

trivially on ∆(Q) and we view Γ-quadrics as pairs (Q, ∂), where

∂ is a trivialization of the discriminant, i.e., a fixed isomorphism

of ∆(Q) with the 2-element set {+,−}.

Such Γ-quadrics occur from pairs((A, σ), ∂

)where

∂ : ∆(A, σ)∼→ F 2 is a trivialization of the discriminant ∆(A, σ).

Pairs (Q, ∂), resp.((A, σ), ∂

)are classified by the cohomology

set

H1(Γ, PGO+2n).

The Grothendieck correspondence induces:

Γ-Quadrics over F1 ofdimension 2(n-2) and trivial discriminant

⇐⇒ Étale algebras with involution ofdimension 2n over F and triv. disc.

Both types of objects are classified by the Galois cohomology

set

H1(Γ, Sn−12 o Sn).

The C+-variety and the Clifford algebra C+

Choice of a trivialization =⇒ Choice of one component C+ of

the Clifford functor C.

1. C+(Y , σ) is a Γ-set of 2n−1 elements if |Y | = 2n.

2. C+(L, σ) is an étale algebra of dimension 2n−1 if

dim L = 2n.

3. C+(Q) is a Γ-projective space of dimension 2n−1 if Q is a

quadric of dimension 2(n − 1).

4. C+(A, σ) is a central simple algebra of degree 2n−1 if

deg A = 2n.

Everything is compatible

(L, σL): étale F -algebra of dimension 2n with involution

(A, σ): central simple F -algebra of degree 2n with involution

∂ : ∆(A, σ)→ F 2; trivialization such that

∂|L : ∆(L, σL)→ F 2

ε : (L, σL) ↪→ (A, σA)

Proposition. The following diagram is commutative:

C+X (L, σL)C+(ε∗)−−−−→ C+

(Q(A, σ)

)∥∥∥ yψXC+(L, σL)

C+(ε)∗−−−−→ SB(C+(A, σ, f )).

A3 = D3 in Geometry

There is a bijective correspondence

4-element Γ-sets3-dim. Γ-proj. spaces

Gr1−→C+

←−6-element Γ-sets with inv. and triv. disc.5-dim. Γ-quadrics and triv. disc.

The 4-dimensional quadric in P5 whose points are parametrized

by a 3-dimensional projective space is known as Klein’s

Quadric.

A3 = D3 in Algebra

4-dim. étale algebrasc.s. algebras of degree 4

λ2

−→C+

←−6-dim. étale algebras. with inv. and triv. disc.deg. 6 c.s. algebras with orth. inv. and triv. disc.

Remark. The 6-dimensional étale algebra λ2L attached to a

4-dimensional étale algebra L has an involution and the fixed

subalgebra under the involution is the cubic resolvent of L. This

corresponds on the level of Galois cohomology to the

isomorphism H1(Γ, S22 o S3)

∼→ H1(Γ, S4) induced by the

isomorphism

S22 o S3

∼→ S4.

Triality

The Dynkin diagram of D4 admits an automorphism of order 3.

As a ”consequence”, the groups PGO+8 and S3

2 o S4 admit

outer automorphisms of order 3. This is known as triality.

Theorem. Let T be either

1. A 8-dimensional Γ-set with involution and trivial

discriminant.

2. An étale algebra of dimension 8 with involution and trivial

discriminant.

3. A 6-dimensional Γ-quadric with trivial discriminant or

4. A central simple algebra of degree 8 with orthogonal

involution and trivial discriminant.

In all four cases C+(T ) and C−(T ) are objects of the same

nature as T and triality permutes the triples(T , C+(T ), C−(T )

)cyclically.

Example.

An étale algebra L of dimension 8 with involution and trivial

discriminant over a field F can be written as

L = F [x ]/(p(x2)

)where p(x) = x4 + ax3 + bx2 + cx + e2 is a polynomial of

degree 4 with a, b, c and e in F .

We have

C+(L)∼→ F [x ]/

(p+(x2)

)and C−(L)

∼→ F [x ]/(p−(x2)

)where

p+(x) = x4 + ax3 + (38a2 − 1

2b + 3e)x2+

( 116a3 − 1

4ab + c + 12ae)x +

( 116a2 − 1

4b − 12e

)2

p−(x) = x4 + ax3 + (38a2 − 1

2b − 3e)x2+

( 116a3 − 1

4ab + c − 12ae)x +

( 116a2 − 1

4b + 12e

)2

What about G2, F4 ?

END