Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
A sheaf cohomology theory for C*-algebras
Martin Mathieu
(Queen’s University Belfast)
Banach Algebras 2017 at Ouluon 6 July 2017
Partially supported by UK Engineering and Physical Sciences Research CouncilGrant No. EP/M02461X/1.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
(Commutative) Topology
C∗1com
Comp
category of unital commutative C*-algebras with unital*-homomorphisms as morphisms;
category of compact Hausdorff spaces with continuousmappings as morphisms.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
(Commutative) Topology
C∗1com
Comp
category of unital commutative C*-algebras with unital*-homomorphisms as morphisms;
category of compact Hausdorff spaces with continuousmappings as morphisms.
Comp
X ; Xf−→ Y
A; Bπ−→ A
π(ϕ) = ϕ ◦ π, ϕ ∈ B
C−→
ˆ←−
C∗1com
C (X ); C (Y )C(f )−→ C (X )
C (f )(g) = g ◦ f , g ∈ C (Y )
A; Aπ−→ B
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Noncommutative Topology
C∗1 category of unital C*-algebras with unital*-homomorphisms as morphisms;
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Noncommutative Topology
C∗1 category of unital C*-algebras with unital*-homomorphisms as morphisms;
? Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Noncommutative Topology
A C*-algebra
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Noncommutative Topology
A C*-algebra
Prim(A) space of primitive ideals of A with hull-kernel topology
OPrim(A) 3 U A(U) =⋂
t /∈U t;
I � A U(I ) = {t ∈ Prim(A) | I * t} ∈ OPrim(A).
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Noncommutative Topology
A C*-algebra
Prim(A) space of primitive ideals of A with hull-kernel topology
OPrim(A) 3 U A(U) =⋂
t /∈U t;
I � A U(I ) = {t ∈ Prim(A) | I * t} ∈ OPrim(A).
hence V ⊆ U ⇒ A(V ) ⊆ A(U);
ringed space (A,X = Spec(A),OX )
“C* -ringed space” (A,Prim(A),A), A is a sheaf of C*-algebras
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Noncommutative Topology
A C*-algebra
Prim(A) space of primitive ideals of A with hull-kernel topology
OPrim(A) 3 U A(U) =⋂
t /∈U t;
I � A U(I ) = {t ∈ Prim(A) | I * t} ∈ OPrim(A).
hence V ⊆ U ⇒ A(V ) ⊆ A(U);
ringed space (A,X = Spec(A),OX )
“C* -ringed space” (A,Prim(A),A), A is a sheaf of C*-algebras
Note A separable ⇒ closed prime ideals are primitive
P. Ara, M. Mathieu, Sheaves of C*-algebras, Math. Nachrichten 283 (2010), 21–39.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves of C*-algebras
Example 1. The multiplier sheaf
A C*-algebra with primitive ideal space Prim(A);
MA : OPrim(A) → C∗1 , MA(U) = M(A(U)),
where M(A(U)) denotes the multiplier algebra of the closedideal A(U) of A associated to the open subset U ⊆ Prim(A).
M(A(U))→ M(A(V )), V ⊆ U, the restriction homomorphisms.
Proposition
The above functor MA defines a sheaf of C*-algebras.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves of C*-algebras
Example 2. The injective envelope sheaf
let I (B) denote the injective envelope of B;
IA : OPrim(A) → C∗1 , IA(U) = pU I (A) = I (A(U)),
where pU = pA(U) denotes the unique central open projectionin I (A) such that pA(U)I (A) is the injective envelope of A(U).
I (A(U))→ I (A(V )), V ⊆ U, given by multiplication by pV
(as pV ≤ pU).
{pU | U ∈ OPrim(A)} is a complete Boolean algebra isomorphic tothe Boolean algebra of regular open subsets of Prim(A), and it isprecisely the set of projections of the AW*-algebra Z (I (A)).
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
The Aim: Sheaf Cohomology
H i(X ,F), i ∈ N
where X is a topological space, A a sheaf of C*-algebras on X andF ∈ OMod∞A (X ).
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
The Aim: Sheaf Cohomology
H i(X ,F), i ∈ N
where X is a topological space, A a sheaf of C*-algebras on X andF ∈ OMod∞A (X ).
joint work in progress with Pere Ara
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
A key property in abelian categories
in an abelian category,
• the morphism set between any pair of objects is an abeliangroup;
• every morphism has a kernel and a cokernel;
• every morphism can be uniquely factorised as
Eϕ //
π
F
G
µ
>>
where π is an epimorphism and µ is a monomorphism.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
A key property in abelian categories
in an abelian category,
• the morphism set between any pair of objects is an abeliangroup;
• every morphism has a kernel and a cokernel;
• every morphism can be uniquely factorised as
Eϕ //
π
F
G
µ
>>
where π is an epimorphism and µ is a monomorphism.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
A key property in abelian categories
in an abelian category,
• the morphism set between any pair of objects is an abeliangroup;
• every morphism has a kernel and a cokernel;
• every morphism can be uniquely factorised as
Eϕ //
π
F
G
µ
>>
where π is an epimorphism and µ is a monomorphism.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Operator modules
Definition
A right A-module E which at the same time is an operator space isa right operator A-module if it satisfies either:
(a) There exist a complete isometry Φ: E −→ B(H,K ), for someHilbert spaces H, K , and a *-homomorphism π : A −→ B(H)such that Φ(x · a) = Φ(x)π(a) for all x ∈ E , a ∈ A.
(b) The bilinear mapping E × A −→ E , (x , a) 7→ x · a extends to acomplete contraction E ⊗h A −→ E .
(c) For each n ∈ N, Mn(E ) is a right Banach Mn(A)-module inthe canonical way.
E is nondegenerate if the linear span of {x · a | x ∈ E , a ∈ A} isdense in E , and it is unital if A is unital and x · 1 = x for all x ∈ E .
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Our kind of categories
Categories we need to consider (for a C*-algebra A):
OMod∞A
OMod1A
the category with objects the nondegenerate rightoperator A-modules and morphisms the completelybounded A-module maps;
the subcategory of OMod∞A with morphisms thecompletely contractive A-module maps.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Our kind of categories
Categories we need to consider (for a C*-algebra A):
OMod∞A
OMod1A
the category with objects the nondegenerate rightoperator A-modules and morphisms the completelybounded A-module maps;
additive, finitely bicomplete
the subcategory of OMod∞A with morphisms thecompletely contractive A-module maps.
non-additive, bicomplete
None of these categories is abelian!
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
A kernel
Let f ∈ MorC(A,B) for some A,B ∈ C.
A morphism i : K → A is a kernel of f if fi = 0 and for eachD ∈ C and g ∈ MorC(D,A) with fg = 0 there is a uniqueh ∈ MorC(D,K ) making the diagram below commutative
D
g
��
0
��
h
��K
0
66i // A
f // B
Any kernel is a monomorphism and is, up to isomorphism, unique.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
A cokernel
Let f ∈ MorC(A,B) for some A,B ∈ C.
A morphism p : B → C is a cokernel of f if pf = 0 and for eachD ∈ C and g ∈ MorC(B,D) with gf = 0 there is a uniqueh ∈ MorC(C ,D) making the diagram below commutative
A
0
((
0&&
f// B p
//
g
��
C
h��
D
Any cokernel is an epimorphism and is, up to isomorphism, unique.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Our kind of categories
the situation in our categories:
monomorphisms are injective A-module maps
epimorphisms are A-module maps with dense range
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Our kind of categories
the situation in our categories:
monomorphisms are injective A-module maps
epimorphisms are A-module maps with dense range
OMod1A
OMod∞A
T : E → F kernel iff is it a complete isometry;
T : E → F cokernel iff it is a complete quotient map.
T : E → F kernel iff it is a completely boundedisomorphism onto its image;
T : E → F cokernel iff it is surjective and completelyopen.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Homological Algebra
fundamental concept “short exact sequence”
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Homological Algebra
fundamental concept “short exact sequence”
ModA
OModA
0 // Eα // F
β // G // 0
where E ,F ,G are modules over a ring A andkerα = 0, imβ = G and imα = ker β.
E // M // FP // // G
where E ,F ,G are right operator A-modules over theC*-algebra A and M = Ker P and P = Coker M.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Homological Algebra
fundamental concept “short exact sequence”
ModA
OModA
0 // Eα // F
β // G // 0
where E ,F ,G are modules over a ring A andkerα = 0, imβ = G and imα = ker β.
E // M // FP // // G “kernel–cokernel pair”
where E ,F ,G are right operator A-modules over theC*-algebra A and M = Ker P and P = Coker M.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Exact categories
let C be an additive category;
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Exact categories
let C be an additive category;
if C is abelian
every monomorphism is a kernel and every epimorphism is acokernel;
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Exact categories
let C be an additive category;
A kernel–cokernel pair (M,P) consists of two composablemorphisms in C such that M = Ker P and P = Coker M,
E1// M // E2
P // // E3
where Ei ∈ C.A monomorphism M arising in such a pair is called admissible
E // //F
and an epimorphism arising in such a pair is called admissible
E // //F
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Exact categories (a la Quillen)
An exact structure on an additive category C is a class of kernel–cokernelpairs, closed under isomorphisms, satisfying the following axioms.
[E0] ∀ E ∈ C : 1E is an admissible monomorphism;
[E0op ] ∀ E ∈ C : 1E is an admissible epimorphism;
[E1] the class of admissible monomorphisms is closed undercomposition;
[E1op ] the class of admissible epimorphisms is closed under composition;
[E2] the push-out of an admissible monomorphism along an arbitrarymorphism exists and yields an admissible monomorphism;
[E2op ] the pull-back of an admissible epimorphism along an arbitrarymorphism exists and yields an admissible epimorphism.
Th. Buhler, Exact categories, Expo. Math. 28 (2010), 1–69.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
An exact structure
Definition
Let A be a C*-algebra. We endow the additive category OMod∞Awith the exact structure of all kernel–cokernel pairs
E1// M // E2
P // // E3
where Ei ∈ obj(OMod∞A ), 1 ≤ i ≤ 3, M is a monomorphism inOMod∞A with closed range and completely bounded inverse, P is acompletely open mapping in OMod∞A (in particular, surjective)and ker P = im M.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
An exact structure
Theorem
Let A be a C*-algebra. The class of all kernel–cokernel pairs inOMod∞A is an exact structure on OMod∞A .
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
what is next?
. . . now do the same for sheaves of operator modules
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves in categories
X topological space;OX category of open subsets (with open subsets U as objectsand V → U if and only if V ⊆ U).
C bicomplete category with equalisers and coequalisers.
Definition
A presheaf on X in C is a contravariant functor F : OX → C.
A sheaf with values in C is a presheaf F such that F(∅) = 0 and,for every open subset U of X and every open cover U =
⋃i Ui ,
the maps F(U)→ F(Ui ) are the limit of the diagramsF(Ui )→ F(Ui ∩ Uj) for all i , j .
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Limits in categories
Let I −→ C be a small diagram; we write Ei ∈ C for the imageof an object i ∈ I and, if ϕij : i → j is a morphism, we denote itsimage by Tϕij : Ei → Ej . An object L ∈ C together with morphismsπi : L→ Ei , i ∈ I is a limit of the diagram if they make thediagram below commutative and is final with this property.
Ei
Tϕij
��
L′
ρi
55
ρj))
// Lπi
::
πj
$$Ej
Also called an inverse limits or projective limit.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves in categories
Let U ∈ OX . Let I be a set and U =⋃
i∈I Ui with Ui ∈ OX . Let
the index category I (2) consist of unordered pairs of elements in I(we allow the case of singleton sets in I (2))where {i , j} → {k, `} if {i , j} ⊆ {k , `}. Consider the diagram
I (2) → {Uij = Ui ∩ Uj | {i , j} ∈ I (2)} ⊆ OU , {i , j} 7→ Uij
composed with the functor F : OX → C. Then F(U) is the limit ofthe diagram
F(Ui )
))F(U)
77
''
F(Ui ∩ Uj)
F(Uj)
55
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves in categories
F(Ui0) // F(Ui0 ∩ Ui1)
F(U)
::
$$
ρ //∏i∈I
F(Ui )
OO
��
µi0 i1
77
νi0 i1''
µ //ν
//∏
(i0,i1)∈I×IF(Ui0 ∩ Ui1)
OO
��F(Ui1) // F(Ui0 ∩ Ui1)
The sheaf property is the requirement that the morphism ρ is theequaliser of the pair (µ, ν).
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves in concrete categories
Notation and Terminology:
suppose C is a concrete category
the elements of F(U) are called sections over U ∈ OX ;
by s|V , V ⊆ U open, we mean the “restriction” of s ∈ F(U) to V ;i.e., the image of s in F(V ) under ρVU : F(U)→ F(V );
the unique gluing property of a sheaf can be expressed as follows:
for each compatible family of sections si ∈ F(Ui ), i.e.,si |Ui∩Uj
= sj |Ui∩Ujfor all i , j , there is a unique section s ∈ F(U)
such that s|Ui= si for all i .
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves in categories
F, G (pre)sheaves on X
morphism ϕ : F→ G is a natural transformation, i.e.,
F(U)
ρVU��
ϕU // G(U)
ρ′VU��
F(V ) ϕV
// G(V )
is commutative for all U,V ∈ OX , V ⊆ U;
hence we have the categories of sheaves on X , Sh(X ,C),and of presheaves, PSh(X ,C).
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves in categories
Stalks
F presheaf on X , t ∈ X ;
stalk of F at t is defined as Ft = lim−→UtF(U),
where Ut denotes the downward directed family of openneighbourhoods of t and lim−→ denotes the (directed) colimit in C.
With stalks we can build bundles and with bundles we can getsheaves again.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves in categories
An Example
Let F be a presheaf on X . For each U ∈ OX , putFp(U) =
∏t∈U Ft and, for V ⊆ U, set ρVU the canonical
morphism from∏
t∈U Ft →∏
t∈V Ft .
In this way we obtain the product sheaf associated with F.
We shall assume that the canonical morphism σU : F(U)→ Fp(U)is a monomorphism whenever F is a sheaf.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves in categories
The stalk functor
Suppose F and G are presheaves on X and ϕ : F→ G is amorphism of presheaves.For each t ∈ X there is a unique morphism ϕt : Ft → Gt .In this way, we obtain the stalk functor at t : PSh(X ,C)→ C.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves in categories
The stalk functor
Suppose F and G are presheaves on X and ϕ : F→ G is amorphism of presheaves.For each t ∈ X there is a unique morphism ϕt : Ft → Gt .In this way, we obtain the stalk functor at t : PSh(X ,C)→ C.
Properties:
Let ϕ(1), ϕ(2) : F→ G be morphisms of sheaves. Then
ϕ(1) = ϕ(2) if and only if ϕ(1)t = ϕ
(2)t for all t ∈ X .
Let ϕ : F→ G be a morphism of sheaves. Then
(i) ϕ is a monomorphism if ϕt is a monomorphism for all t ∈ X ;(ii) ϕ is an epimorphism if ϕt is an epimorphism for all t ∈ X ;(iii) ϕ is an isomorphism only if ϕt is an isomorphism for all t ∈ X .
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Sheaves in categories
The stalk functor
Suppose F and G are presheaves on X and ϕ : F→ G is amorphism of presheaves.For each t ∈ X there is a unique morphism ϕt : Ft → Gt .In this way, we obtain the stalk functor at t : PSh(X ,C)→ C.
Properties:
Let ϕ(1), ϕ(2) : F→ G be morphisms of sheaves. Then
ϕ(1) = ϕ(2) if and only if ϕ(1)t = ϕ
(2)t for all t ∈ X .
Let ϕ : F→ G be a morphism of sheaves. Then
(i) ϕ is a monomorphism if ϕt is a monomorphism for all t ∈ X ;(ii) ϕ is an epimorphism if ϕt is an epimorphism for all t ∈ X ;(iii) ϕ is an isomorphism only if ϕt is an isomorphism for all t ∈ X .
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
“C*-ringed spaces”
Sheaves of operator modules over sheaves of C*-algebras
X topological space, A sheaf of C*-algebras on X
E : OX −→Ban1 (sheaf in Ban1) right operator A-module on X
if, for each U ∈ OX , E(U) is a (nondegenerate) right operatorA(U)-module and, for U,V ∈ OX with V ⊆ U,
TVU : E(U)→ E(V ) is completely contractive and
TVU(x · a) = TVU(x) · πVU(a) (x ∈ E(U), a ∈ A(U)),
where πVU : A(U)→ A(V ) are the restriction maps in A.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
The categories we are interested in
Categories we need to consider(for a given sheaf A of C*-algebras on X ):
OMod∞A (X )
OMod1A(X )
the category with objects the right operatorA-modules and morphisms (at U) the completelybounded A(U)-module maps;
the subcategory of OMod∞A (X ) with the sameobjects and morphisms (at U) the completelycontractive A(U)-module maps.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
The categories we are interested in
Categories we need to consider(for a given sheaf A of C*-algebras on X ):
OMod∞A (X )
OMod1A(X )
the category with objects the right operatorA-modules and morphisms (at U) the completelybounded A(U)-module maps;
additive, finitely bicomplete
the subcategory of OMod∞A (X ) with the sameobjects and morphisms (at U) the completelycontractive A(U)-module maps.
non-additive, bicomplete
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
The categories we are interested in
Categories we need to consider(for a given sheaf A of C*-algebras on X ):
OMod∞A (X )
OMod1A(X )
the category with objects the right operatorA-modules and morphisms (at U) the completelybounded A(U)-module maps;
we need this one for homological algebra
the subcategory of OMod∞A (X ) with the sameobjects and morphisms (at U) the completelycontractive A(U)-module maps.
we need this one for constructions
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
The categories we are interested in
Categories we need to consider(for a given sheaf A of C*-algebras on X ):
OMod∞A (X )
OMod1A(X )
the category with objects the right operatorA-modules and morphisms (at U) the completelybounded A(U)-module maps;
the subcategory of OMod∞A (X ) with the sameobjects and morphisms (at U) the completelycontractive A(U)-module maps.
None of these categories is abelian!
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
An exact structure on OMod∞A (X )
Definition
Let X be a topological space and let A be a sheaf of C*-algebrason X . Let ExA(X ) denote the collection of all kernel–cokernelspairs in OMod∞A (X )
E1// µ // E2
$ // // E3 .
We call this the canonical exact structure on OMod∞A (X ).
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
An exact structure on OMod∞A (X )
Definition
Let X be a topological space and let A be a sheaf of C*-algebrason X . Let ExA(X ) denote the collection of all kernel–cokernelspairs in OMod∞A (X )
E1// µ // E2
$ // // E3 .
We call this the canonical exact structure on OMod∞A (X ).
Theorem
The class ExA(X ) of all kernel–cokernel pairs defines an exactstructure on OMod∞A (X ).
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Exact sequences
in ExA(X ), every kernel–cokernel pair
E1// µ // E2
$ // // E3
where Ei ∈ OMod∞A (X ) is called a short exact sequence.
Definition
The morphism ϕ ∈ CBA(E,F), E,F ∈ OMod∞A (X ) is calledadmissible if it can be factorised as
Eϕ //
$ �� ��
F
G?? µ
??
for some admissible monomorphism µ and some admissibleepimorphism $ in OMod∞A (X ).
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Exact sequences
Definition
A sequence of admissible morphisms in OMod∞A (X )
E1ϕ1 //
$1 $$ $$
E2ϕ2 //
$2 $$ $$
E3
G1
:: µ1
::
G2
:: µ2
::
is said to be exact if the short sequence G1// µ1 // E2
$2 // // G2
is exact. An arbitrary sequence of admissible morphisms inOMod∞A (X ) is exact if the sequences given by any twoconsecutive morphisms are exact.
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
what are the next steps in our programme?
♦ to introduce injective sheaves;
♦ to construct injective resolutions;
♦ to define the homology of a complex F• in OMod∞A (X )
. . . // Fi−1δi−1
// Fiδi // Fi+1
// . . .
♦ to use homological algebra (such as the Horseshoe Lemma) inOMod∞A (X ) to ensure that homotopic injective resolutions yieldthe same homology
♦ to introduce the cohomology groups as the right derived functorof the global section functor applied to an injective resolution ofF ∈ OMod∞A (X )
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
what are the next steps in our programme?
♦ to introduce injective sheaves;
♦ to construct injective resolutions;
♦ to define the homology of a complex F• in OMod∞A (X )
. . . // Fi−1δi−1
// Fiδi // Fi+1
// . . .
♦ to use homological algebra (such as the Horseshoe Lemma) inOMod∞A (X ) to ensure that homotopic injective resolutions yieldthe same homology
♦ to introduce the cohomology groups as the right derived functorof the global section functor applied to an injective resolution ofF ∈ OMod∞A (X )
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
what are the next steps in our programme?
♦ to introduce injective sheaves;
♦ to construct injective resolutions;
♦ to define the homology of a complex F• in OMod∞A (X )
. . . // Fi−1δi−1
// Fiδi // Fi+1
// . . .
♦ to use homological algebra (such as the Horseshoe Lemma) inOMod∞A (X ) to ensure that homotopic injective resolutions yieldthe same homology
♦ to introduce the cohomology groups as the right derived functorof the global section functor applied to an injective resolution ofF ∈ OMod∞A (X )
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
what are the next steps in our programme?
♦ to introduce injective sheaves;
♦ to construct injective resolutions;
♦ to define the homology of a complex F• in OMod∞A (X )
. . . // Fi−1δi−1
// Fiδi // Fi+1
// . . .
♦ to use homological algebra (such as the Horseshoe Lemma) inOMod∞A (X ) to ensure that homotopic injective resolutions yieldthe same homology
♦ to introduce the cohomology groups as the right derived functorof the global section functor applied to an injective resolution ofF ∈ OMod∞A (X )
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
what are the next steps in our programme?
♦ to introduce injective sheaves;
♦ to construct injective resolutions;
♦ to define the homology of a complex F• in OMod∞A (X )
. . . // Fi−1δi−1
// Fiδi // Fi+1
// . . .
♦ to use homological algebra (such as the Horseshoe Lemma) inOMod∞A (X ) to ensure that homotopic injective resolutions yieldthe same homology
♦ to introduce the cohomology groups as the right derived functorof the global section functor applied to an injective resolution ofF ∈ OMod∞A (X )
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
what are the next steps in our programme?
♦ to introduce injective sheaves;
♦ to construct injective resolutions;
♦ to define the homology of a complex F• in OMod∞A (X )
. . . // Fi−1δi−1
// Fiδi // Fi+1
// . . .
♦ to use homological algebra (such as the Horseshoe Lemma) inOMod∞A (X ) to ensure that homotopic injective resolutions yieldthe same homology
♦ to introduce the cohomology groups as the right derived functorof the global section functor applied to an injective resolution ofF ∈ OMod∞A (X )
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Progress!
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras
Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps
Martin Mathieu (Queen’s University Belfast)
A sheaf cohomology theory for C*-algebras