Hopf Algebroids in Homological Algebra Uli Kr¨ ahmer joint work with Niels Kowalzig Shanghai, 16.9.2011 Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 1 / 29
A point of departure The following has featured prominently in many talks this week: Theorem (Van den Bergh) Let A be a unital associative algebra over a field k which has a finitely generated projective resolution of finite length as an A e := A ⊗ k A op -module, and for which there exists d such that H i ( A , A e ) = 0 for i � = d. Then one has for all A e -modules M H i ( A , M ) ≃ H d − i ( A , M ⊗ A ω ) , ω := H d ( A , A e ) . Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 2 / 29
A point of departure The following has featured prominently in many talks this week: Theorem (Van den Bergh) Let A be a unital associative algebra over a field k which has a finitely generated projective resolution of finite length as an A e := A ⊗ k A op -module, and for which there exists d such that H i ( A , A e ) = 0 for i � = d. Then one has for all A e -modules M H i ( A , M ) ≃ H d − i ( A , M ⊗ A ω ) , ω := H d ( A , A e ) . Ginzburg added: Theorem (Ginzburg) If in addition ω ≃ A as right A e -module (i.e. if A is Calabi-Yau), then H • ( A , A ) is a Batalin-Vilkovisky algebra. Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 2 / 29
Some questions I’d like to talk about H • ( A , M ) ≃ H d −• ( A , M ⊗ A ω ) is an isomorphism of ...? Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 3 / 29
Some questions I’d like to talk about H • ( A , M ) ≃ H d −• ( A , M ⊗ A ω ) is an isomorphism of ...? We have H j ( A , N ) ≃ Tor A e ( N , A ) , H i ( A , M ) ≃ Ext A e ( A , M ) , and most other classical (co)homology theories in algebra work similarly, only that ( A e , A ) is replaced by other augmented rings. Do the above theorems generalise to other such theories? Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 3 / 29
Some questions I’d like to talk about H • ( A , M ) ≃ H d −• ( A , M ⊗ A ω ) is an isomorphism of ...? We have H j ( A , N ) ≃ Tor A e ( N , A ) , H i ( A , M ) ≃ Ext A e ( A , M ) , and most other classical (co)homology theories in algebra work similarly, only that ( A e , A ) is replaced by other augmented rings. Do the above theorems generalise to other such theories? What is a Batalin-Vilkovisky algebra anyway? Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 3 / 29
Hopf algebroids Hopf algebroids are generalisations of Hopf algebras in which the base field k is replaced by a possibly noncommutative algebra A . Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 4 / 29
Hopf algebroids Hopf algebroids are generalisations of Hopf algebras in which the base field k is replaced by a possibly noncommutative algebra A . They are in particular k -algebras U with a k -algebra map η : A e → U , so every U -module is canonically an A -bimodule. Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 4 / 29
Hopf algebroids Hopf algebroids are generalisations of Hopf algebras in which the base field k is replaced by a possibly noncommutative algebra A . They are in particular k -algebras U with a k -algebra map η : A e → U , so every U -module is canonically an A -bimodule. The Hopf algebroid structure turns U - Mod into a monoidal category whose unit object is A and for which the forgetful functor to A e - Mod is monoidal, just as for Hopf algebras. Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 4 / 29
Hopf algebroids Hopf algebroids are generalisations of Hopf algebras in which the base field k is replaced by a possibly noncommutative algebra A . They are in particular k -algebras U with a k -algebra map η : A e → U , so every U -module is canonically an A -bimodule. The Hopf algebroid structure turns U - Mod into a monoidal category whose unit object is A and for which the forgetful functor to A e - Mod is monoidal, just as for Hopf algebras. For the experts: When I say Hopf algebroid I mean throughout left Hopf algebroid or equivalently left × A -Hopf algebra. Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 4 / 29
Examples of Hopf algebroids Example 1: For A = k , Hopf algebroids are just Hopf algebras. Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 5 / 29
Examples of Hopf algebroids Example 1: For A = k , Hopf algebroids are just Hopf algebras. Example 2: For any algebra A , U = A e is a Hopf algebroid over A . Here η : A e → U is the identity. Every Hopf algebroid has a counit ε : U → A , and this is here the multiplication map of A . In particular, its in general not a ring map, but only U -linear. Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 5 / 29
Examples of Hopf algebroids Example 1: For A = k , Hopf algebroids are just Hopf algebras. Example 2: For any algebra A , U = A e is a Hopf algebroid over A . Here η : A e → U is the identity. Every Hopf algebroid has a counit ε : U → A , and this is here the multiplication map of A . In particular, its in general not a ring map, but only U -linear. Example 3: U = U ( A , L ), the universal enveloping algebra of a Lie-Rinehart algebra aka Lie algebroid. Here A is a commutative algebra and L is a Lie algebra and A and L are both acting on each other. Special cases are Weyl algebras and more generally the rings of algebraic differential operators on smooth affine varieties X (here A = k [ X ], L = Der k ( A )). Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 5 / 29
Generalising Van den Bergh’s result The Hopf algebroid structure also deifnes a functor ⊗ : U - Mod × U op - Mod → U op - Mod which for A = k is ⊗ k with right action on M ⊗ N given by ( m ⊗ k n ) u = S ( u (2) ) m ⊗ k nu (1) . Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 6 / 29
Generalising Van den Bergh’s result The Hopf algebroid structure also deifnes a functor ⊗ : U - Mod × U op - Mod → U op - Mod which for A = k is ⊗ k with right action on M ⊗ N given by ( m ⊗ k n ) u = S ( u (2) ) m ⊗ k nu (1) . Using this one obtains: Theorem Let U be an A-biprojective Hopf algebroid for which A has a finitely generated projective resolution of finite length as a U-module, and for which there exists d such that Ext i U ( A , U ) = 0 for i � = d. Then one has for all U-modules M Ext i U ( A , M ) ≃ Tor U ω := Ext d d − i ( M ⊗ ω, A ) , U ( A , U ) . Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 6 / 29
Gerstenhaber algebras Definition A Gerstenhaber algebra is a graded commutative k -algebra ( V , � ) � V i , v � w = ( − 1) ij w � v ∈ V i + j , v ∈ V i , w ∈ V j V = i ∈ Z with a graded Lie bracket {· , ·} : V i +1 ⊗ k V j +1 → V i + j +1 on � V i +1 V [1] := i ∈ Z of V for which all operators { v , ·} satisfy the graded Leibniz rule { u , v � w } = { u , v } � w + ( − 1) ij v � { u , w } , u ∈ V i +1 , v ∈ V j . Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 7 / 29
Examples The classical example is the Hochschild cohomology of any associative algebra. Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 8 / 29
Examples The classical example is the Hochschild cohomology of any associative algebra. In fact, we have: Theorem (Shoikhet) If U is a Hopf algebroid which is right A-projective, then Ext U ( A , A ) is naturally a Gerstenhaber algebra. Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 8 / 29
Examples The classical example is the Hochschild cohomology of any associative algebra. In fact, we have: Theorem (Shoikhet) If U is a Hopf algebroid which is right A-projective, then Ext U ( A , A ) is naturally a Gerstenhaber algebra. Shoikhet’s result deals in fact with only mildly restricted abelian monoidal categories and mimicks Schwede’s elegant treatment of the Hochschild case. In our ofrthcoming paper, we will give explicit formulas for the structure in terms of the canonical cochain complex computing Ext U ( A , A ). Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 8 / 29
Gerstenhaber modules Definition A module over V is a graded ( V , � )-module (Ω , � ), � u ∈ V i , ω ∈ Ω j Ω = Ω j , u � ω ∈ Ω j − i , j ∈ Z with a representation of the graded Lie algebra ( V [1] , {· , ·} ) L : V i +1 ⊗ k Ω j → Ω j − i , u ⊗ k ω �→ L u ( ω ) which satisfies for u ∈ V i +1 , v ∈ V j , ω ∈ Ω the mixed Leibniz rule L u ( v � ω ) = { u , v } � ω + ( − 1) ij v � L u ( ω ) . Uli Kr¨ ahmer (joint work with Niels Kowalzig) Hopf algebroids Shanghai, 16.9.2011 9 / 29
Recommend
More recommend
