Representations and Cohomology of Categories
Peter Webb
What is a representation of a category?
Representations and Cohomology of Category cohomology and the Schur Categories multiplier
Xu’s counterexample Peter Webb The orbit category and Alperin’s weight conjecture
University of Minnesota Concluding remarks April 10, 2010 Representations Outline and Cohomology of Categories
Peter Webb
What is a representation of a What is a representation of a category? category? Category cohomology and the Schur Category cohomology and the Schur multiplier multiplier Xu’s counterexample
The orbit category Xu’s counterexample and Alperin’s weight conjecture
Concluding The orbit category and Alperin’s weight conjecture remarks
Concluding remarks Representations Theme and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and the Schur multiplier
Xu’s Representations of categories are remarkably like counterexample
representations of groups! The orbit category and Alperin’s weight conjecture
Concluding remarks Representations Categories and Cohomology of Categories
Peter Webb
Let C be a small category. What is a representation of a Examples: category?
Category I a group cohomology and the Schur I a poset multiplier Xu’s I the free category associated to a quiver. The objects counterexample
are the vertices of the quiver, the morphisms are all The orbit category and Alperin’s possible composable strings of the arrows. weight conjecture
Concluding The theory of representations of the above examples is well remarks developed and we do not expect to get more information about them from this general theory. We are more interested in other categories, such as the orbit category associated to a family of subgroups of a group, or the categories which arise with p-local finite groups. Representations Representations and Cohomology of Categories
Peter Webb
What is a representation of a Let R be a commutative ring with 1. A representation of a category? Category category C over R is a functor M : C → R-mod. cohomology and the Schur multiplier
Straightforward example: Xu’s C is the category with five morphisms • ←− • −→ •. counterexample The orbit category A representation is a diagram of modules B ←− A −→ C. and Alperin’s We may be interested in weight conjecture Concluding I the direct limit of this diagram: the pushout; remarks
I is this operation exact?
I Etc. Representations and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and the Schur multiplier A representation of a category is a diagram of modules. Xu’s counterexample
The orbit category and Alperin’s weight conjecture
Concluding remarks Representations Well-studied examples of representations and Cohomology of Categories
Peter Webb
What is a representation of a I When C is a group we get homomorphism category? C → EndR (V ). Category cohomology and I When C is a poset we get a module for the incidence the Schur multiplier algebra. Xu’s counterexample I When C is the free category associated to a quiver we The orbit category get a representation of the quiver. and Alperin’s weight conjecture When C = • ←− • −→ • its path algebra is I Concluding remarks ∗ ∗ ∗ 0 ∗ 0 0 0 ∗ Representations Further examples and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and I C = finite dimensional vector spaces over some field. the Schur multiplier We get generic representation theory. Xu’s counterexample I C = finite sets with bijective morphisms. We get The orbit category species. and Alperin’s weight conjecture Various constructions in topology and the cohomology I Concluding of groups: homotopy colimits, the Quillen category. remarks Representations Category Algebra and Cohomology of Categories
Peter Webb
What is a representation of a category?
The category algebra RC is the free R-module with the Category cohomology and morphisms of C as a basis. We define the product of these the Schur basis elements to be composition if possible, zero otherwise. multiplier Xu’s Examples: counterexample
The orbit category I When C is a group we get the group algebra. and Alperin’s weight conjecture When C is a poset we get the incidence algebra. I Concluding remarks I When C is the free category associated to a quiver we get the path algebra of the quiver. Representations Equivalence of representations and modules and Cohomology of Categories
Peter Webb
Theorem (B. Mitchell) What is a representation of a Representations are ‘the same’ as RC-modules, if C has category? Category finitely many objects. cohomology and the Schur multiplier
Example: Xu’s counterexample I When C is a group, representations are the same as The orbit category and Alperin’s modules for the group algebra. weight conjecture When C is the free category associated to a quiver, Concluding I remarks representations are the same as modules for the path algebra. Under this correspondence a representation M corresponds L to an RC-module x∈ObC M(x). Natural transformations of functors correspond to module homomorphisms. Representations Constant functors and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and For any R-module A we define the constant functor the Schur A : C → R-mod to be A(x) = A on objects x and multiplier Xu’s A(α) = idA on morphisms α. counterexample
Taking A to be R itself we get the constant functor R. The orbit category and Alperin’s Example: weight conjecture
Concluding I When C is a group we get the trivial module R. remarks Representations Theme and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and the Schur multiplier
Xu’s Representations of categories are remarkably like counterexample
representations of groups! The orbit category and Alperin’s weight conjecture
Concluding remarks Representations Category cohomology and Cohomology of Categories
Peter Webb
What is a representation of a Theorem (Roos, Gabriel-Zisman) category? ∗ ∼ ∗ ExtRC(R, R) = H (|C|, R) where |C| is the nerve of C. Category cohomology and the Schur multiplier We define H∗(C, R) to be the cohomology groups in the last Xu’s theorem. This is the cohomology of C. counterexample More generally, for any representation M of C we put The orbit category and Alperin’s ∗ ∗ weight conjecture H (C, M) := ExtRC(R, M). Concluding remarks Example:
I When C is a (discrete) group the nerve is the classifying space BC and the algebraically computed cohomology is isomorphic to the cohomology of BC. Representations Category extensions: Definition 1 and Cohomology of Categories
Peter Webb
What is a representation of a category?
Extension definition EZ: Category cohomology and An extension of a category C is a diagram of categories and the Schur functors multiplier Xu’s K → E → C counterexample
The orbit category which behaves like a group extension and Alperin’s weight conjecture
Concluding 1 → K → E → G → 1 remarks
(i.e. a short exact sequence of groups). Representations Category extensions: Definition 2 and Cohomology of Categories
Peter Webb
An extension of a category C (in the sense of Hoff) is a What is a representation of a diagram of categories and functors category? Category i p cohomology and K −→E −→C the Schur multiplier
Xu’s satisfying counterexample The orbit category 1. K, E and C all have the same objects, i and p are the and Alperin’s identity on objects, i is injective on morphisms, and p is weight conjecture Concluding surjective on morphisms; remarks 2. whenever f and g are morphisms in E then p(f ) = p(g) if and only if there exists a morphism m ∈ K for which f = i(m)g. In that case, the morphism m is required to be unique. Representations Extension properties and Cohomology of Categories
Peter Webb i p Given an extension K −→E −→C it follows (not obviously) What is a representation of a that category?
Category I all morphisms in K are endomorphisms, and are cohomology and the Schur invertible, multiplier I we get a functor E → Groups, x 7→ EndK(x). Xu’s counterexample
The orbit category If all the groups End (x) are abelian and Alperin’s K weight conjecture I we get a functor C → AbelianGroups Concluding remarks i.e. a representation of C, which we denote K.
Compare: for a group extension 1 → K → E → G → 1 there is a conjugation action of E on the normal subgroup K. When K is abelian it becomes a representation of G. Representations Second cohomology parametrizes extensions and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and the Schur multiplier Theorem Xu’s When all the groups EndK(x) are abelian, equivalence counterexample classes of extensions K → E → C biject with elements of The orbit category and Alperin’s H2(C, K). weight conjecture Concluding remarks Representations Other interpretations of cohomology and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and the Schur 1 0 multiplier There are known interpretations of H , H , H0, H1 which Xu’s generalize to categories the familiar results for groups. counterexample A generalization to categories of the group-theoretic The orbit category and Alperin’s interpretation of H2 has not previously been observed. weight conjecture Concluding remarks Representations Schur multiplier basics and Cohomology of Categories
Peter Webb
What is a The Schur multiplier of a category C is defined to be representation of a category? H (C, ) = TorRC( , ). This generalizes the definition for 2 Z 2 Z Z Category groups. cohomology and the Schur Theorem multiplier 0 Xu’s Let G be a group for which G/G is free abelian. There is counterexample universal central extension 1 → K → E → G → 1 with The orbit category 0 and Alperin’s K ⊆ E , unique up to isomorphism. For that extension, weight conjecture K =∼ H (G). Concluding 2 remarks
central: K ⊆ Z(E) universal: every such extension is a homomorphic image of this one. Representations Central extension of categories and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category Questions: cohomology and the Schur 1. What is a central extension K → E → C of categories? multiplier 2. What is the generalization of K ⊆ E 0 to categories? Xu’s counterexample Answers: The orbit category and Alperin’s 1. K is a constant functor. Better: a locally constant weight conjecture Concluding functor (=constant on connected components). remarks 2. H1(E, Z) → H1(C, Z) should be an isomorphism. Representations Universal central extension and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category Theorem (Webb) cohomology and the Schur Let C be a connected category for which H1(C) is free multiplier abelian and H2(C) is finitely generated. Among extensions Xu’s counterexample K → E → C where K is constant and H (E) → H (C) is an 1 1 The orbit category isomorphism, there is up to isomorphism a unique one with and Alperin’s weight conjecture
the property that it has every such extension as a Concluding homomorphic image. In this extension K has the form remarks H2(C). Representations Methods of proof and Cohomology of Categories
Peter Webb I Five-term exact sequences What is a Theorem (Webb) representation of a category? Let K → E → C be an extension of categories, let B be a Category right C-module and let A a left C-module. There are cohomology and Z Z the Schur exact sequences multiplier Xu’s counterexample H2(E, B) → H2(C, B) → The orbit category B ⊗ H (K) → H (E, B) → H (C, B) → 0 and Alperin’s ZC 1 1 1 weight conjecture
Concluding and remarks H2(E, A) ← H2(C, A) ← 1 1 HomZC(H1(K), A) ← H (E, A) ← H (C, A) ← 0.
I Construction of a resolution (Gruenberg resolution) given a surjection F → C where F is a free category. Representations The Hopf fibration and Cohomology of Categories
Peter Webb
What is a representation of a category?
2 Category Take a category C whose nerve is a 2-sphere S cohomology and 2 the Schur (for example, take a triangulation of S and let C be the multiplier poset of the simplices). Xu’s counterexample We have H1(C) = 0, H2(C) = , so there is a universal Z The orbit category constant extension and Alperin’s weight conjecture → E → C Z Concluding remarks 1 3 2 Then |Z| → |E| → |C| is the Hopf fibration S → S → S . Representations Theme and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and the Schur multiplier
Xu’s Representations of categories are not always like counterexample
representations of groups! The orbit category and Alperin’s weight conjecture
Concluding remarks Representations Finite generation of cohomology and Cohomology of Categories
Peter Webb
Question: When is the cohomology ring What is a ∗ ∗ representation of a H (C, R) = ExtRC(R, R) finitely generated? category? Category cohomology and Presumably we should put some finiteness conditions on C. the Schur Suppose that C is finite. Also suppose C is an EI category: multiplier Xu’s every Endomorphism is an Isomorphism (endomorphism counterexample monoids are groups). The orbit category and Alperin’s weight conjecture Evidence for finite generation: it’s true when C is a finite Concluding group (Evens-Venkov). When C is a free category or a poset remarks the cohomology ring is finite dimensional.
Answer (Xu): For a finite EI category the cohomology ring is very often not finitely generated. Representations Example of non-finite generation of cohomology and Cohomology of Categories
Peter Webb
Let C be the category with two objects x, y and seven What is a representation of a morphisms as pictured below: category? Category {α,β} cohomology and the Schur −−−−−−−−−−−→ multiplier C2 × C2 = G × H • • 1 x −−−−−−−−−−−→ y Xu’s counterexample Here End(x) = G × H, End(y) = 1 and there are two The orbit category and Alperin’s homomorphisms α, β : x → y. Composition is determined by weight conjecture letting G interchange α and β, and letting H fix them. Concluding remarks Proposition (Xu et al) ∗ H (C, F2) is isomorphic to the subring of F2[u, v] spanned by the monomials ur v s where r ≥ 1. This ring is not finitely generated and is a domain. Representations The conjecture of Snashall and Solberg and Cohomology of Categories
Peter Webb
What is a representation of a category?
Conjecture (Snashall and Solberg, Proc. LMS 88 (2004)) Category cohomology and Let A be a finite dimensional algebra over a field. Then the the Schur Hochschild cohomology HH∗(A) is finitely generated modulo multiplier Xu’s nilpotent elements. counterexample
∗ ∗ The orbit category Here HH (A) := Ext op (A, A). A ⊗A and Alperin’s weight conjecture The conjecture was verified by Green, Snashall and Solberg Concluding remarks for self-injective algebras of finite representation type (2003) and ‘monomial’ algebras (2006) (path algebras of quivers with monomial relations of length 2). Representations Xu’s counterexample and Cohomology of Categories
Peter Webb
Theorem (Fei Xu, Adv. Math 219 (2008)) What is a representation of a Let kC be the category algebra of a category C over a field category? ∗ ∗ Category k. The ring homomorphism HH (kC) → H (C, k) induced cohomology and the Schur by the functor − ⊗kC k is a split surjection. multiplier This result was already known for group algebras. For Xu’s counterexample category algebras it required a new idea. The orbit category and Alperin’s Corollary weight conjecture Concluding The Snashall-Solberg conjecture is false in general. remarks For the proof we observe that if HH∗(A) is finitely generated modulo nilpotents, so is every homomorphic image of this ring. Taking A = kC where C is the previously described category, we get an image with no nilpotent elements which is not finitely generated. Representations The use of category representations? and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and the Schur multiplier
Xu’s Why did we need to know about representations of counterexample
categories to do this? The orbit category and Alperin’s weight conjecture
Concluding remarks Representations Simple representations of an EI category and Cohomology of Categories
Peter Webb
What is a If C is an EI category, the simple representations have the representation of a category? form Sx,V where x is an object of C and V is a simple Category k EndC(x)-module: cohomology and the Schur ( multiplier V if y = x Xu’s Sx,V (y) = counterexample 0 otherwise The orbit category and Alperin’s This gives a parametrization of the indecomposable weight conjecture Concluding projective modules: Px,V is the projective cover of Sx,V . remarks The relation (x, V ) ≤ (y, W ) if and only if there exists a morphism x → y in C is a preorder. Representations Stratifications of algebras and Cohomology of Categories
Peter Webb
The category algebra kC is standardly stratified What is a representation of a (Cline-Parshall-Scott, Dlab) if there are modules ∆x,V such category? Category that cohomology and the Schur I all composition factors Sy,W of ∆x,V have multiplier (y, W ) ≤ (x, V ), and Xu’s counterexample
I there is a filtration of Py,W with factors ∆x,V where The orbit category and Alperin’s (y, W ) < (x, V ), except for a single copy of ∆y,W . weight conjecture
Concluding Theorem (Webb (J. Algebra 320 (2008)) remarks Let C be a finite EI-category and k a field. Then kC is standardly stratified if and only if for every morphism α : x → y in C the group StabAut(y)(α) has order invertible in k. Representations The p-subgroup orbit category and Cohomology of Categories
Peter Webb
What is a representation of a Let G be a finite group and let O be the category with category? Category objects the transitive G-sets G/H where H is a p-subgroup cohomology and the Schur of G. The morphisms are the equivariant mappings of multiplier
G-sets. Xu’s The morphisms are always surjective, and so the criterion for counterexample The orbit category standard stratification is always satisfied, and O is an EI and Alperin’s category. weight conjecture Concluding Corollary remarks Over any field k the category algebra kO is standardly stratified. Representations Further structure and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and Because kO is standardly stratified it also has modules the Schur multiplier ∇ = largest submodule of the injective I with I x,V x,V Xu’s composition factors smaller than Sx,V , except for a counterexample The orbit category single copy of Sx,V and Alperin’s weight conjecture I (partial) tilting modules Tx,V . They have a filtration Concluding with ∆ factors, and also a filtration with ∇ factors. remarks Representations Structural versions of AWC and Cohomology of Categories
Peter Webb Theorem What is a The following are equivalent. representation of a category? (1)∆ = S is a simple kO -module, x,V x,V S Category cohomology and (2) ∇x,V = Ix,V is injective, the Schur multiplier (3)( x, V ) is a weight: V is a projective simple module. Xu’s counterexample
The orbit category Theorem and Alperin’s The following are equivalent. weight conjecture Concluding (1)∆ x,V = Tx,V , remarks
(2)∆ H,V = IH,V is injective, (3) x = G/1, V is a simple kG-module.
This gives structural reformulations of Alperin’s weight conjecture: the number of weights equals the number of simple kG-modules. Representations References and Cohomology of Categories
Peter Webb
What is a representation of a Available from http://www.math.umn.edu/ webb category?
Category cohomology and An introduction to the representations and cohomology of the Schur categories pp. 149-173 in: M. Geck, D. Testerman and J. multiplier Xu’s Thvenaz (eds.), Group Representation Theory, EPFL Press counterexample
(Lausanne) 2007. The orbit category and Alperin’s weight conjecture
Resolutions, relation modules and Schur multipliers for Concluding categories J. Algebra, to appear. remarks
Standard stratifications of EI categories and Alperin’s weight conjecture Journal of Algebra 320 (2008), 4073-4091. Representations and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category For more in this direction: cohomology and the Schur multiplier Liping Li: Representation types of finite EI categories, 4:30 Xu’s counterexample today in Combinatorial Representation Theory II, Olin-Rice The orbit category 241. and Alperin’s weight conjecture
Concluding remarks Representations and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category cohomology and the Schur multiplier An apology ... Xu’s counterexample
The orbit category and Alperin’s weight conjecture
Concluding remarks Representations and Cohomology of Categories
Peter Webb
What is a representation of a category?
Category Look at cohomology and the Schur www.northfieldartsguild.org multiplier for information. Xu’s counterexample The show is in Northfield, The orbit category about 40 miles to the south of and Alperin’s weight conjecture here. Concluding I play the role of Robert. remarks