Finite generation of Hochschild cohomology of Hecke algebras of finite classical type in characteristic zero

Markus Linckelmann

Abstract

We show that the Hochschild cohomology HH^{∗}(H) of a Hecke algebra H of finite classical type over a field k of characteristic zero and a non-zero parameter q in k is finitely generated, unless possibly if q has even order in k^{×} and H is of type B or D.

1 Introduction

The Hochschild cohomology of an algebra A over a field k is the graded k-algebra HH^{∗}(A) = Ext_{H⊗_{k}H^{0}}^{∗}(A, A), where A^{0} is the opposite algebra of A and where the A is viewed as a left A ⊗_{k} A^{0}-module via multiplication in A. By a result of Gerstenhaber in [16], HH^{∗}(A) is graded commutative; in particular, HH^{∗}(A) is commutative if char(k) = 2. If char ≠ 2, the Krull dimension of HH^{∗}(A) is, by definition, that of the even part of HH^{∗}(A). For any A-A-bimodule M, viewed as A⊗_{k}A^{0}-module in the obvious way, the graded k-module Ext_{A⊗_{k}A^{0}}^{∗}(A, M) becomes a right module over HH^{∗}(A) via the Yondeda product.

Theorem 1.1. Let H be a Hecke algebra of a finite Coxeter group (W, S) over a field k of characteristic zero with non-zero parameter q in k. Suppose that all irreducible components of W are of type A, B, D, and suppose in addition that if W involves a component of type B or D then the order of q in k^{×} is not even. Then, for any finitely generated H-H-bimodule M, the HH^{∗}(H)-module Ext_{H⊗_{k}H^{0}}^{∗}(H, M) is Noetherian. In particular, HH^{∗}(H) is finitely generated as a k-algebra.

If q = 1 or if q has infinite order then H ⊗_{k} H^{0} is semi-simple, in which case HH^{∗}(H) = HH^{0}(H) is finite-dimensional, and so we may assume that q is a primitve ℓ-th root of unity for some integer ℓ ≥ 2. We can be more precise regarding the Krull dimension of the Hochschild cohomology:

Theorem 1.2. Let H a Hecke algebra of type A_{n−1} or B_{n} (n ≥ 2), or of type D_{n} (n ≥ 4) over a field k of characteristic zero and a parameter q of finite order ℓ ≥ 2 in k^{×}; if H is of type B_{n} or D_{n} suppose in addition that ℓ has odd order. Let m, a be the non-negative integers satisfying n = ℓm + a and 0 ≤ a ≤ ℓ − 1. The Krull dimension of HH^{∗}(H) is equal to m.

As in [2], we make use of the transfer maps for Hecke algebras from [6], interpreted as special case of the transfer maps for symmetric algebras in [3], [4]. A key result is J. Du’s Theorem 2.7 in [9], which we use to show that the theory of vertices for modules over Hecke algebras in [9] admits a bimodule version, which then allows us to play the problem back to maximal ℓ-parabolic subalgebras of H. Since these are tensor products of Brauer tree algebras and semi-simple algebras, the result follows from well-known properties of the Hochschild cohomology of selfinjective algebras of finite representation type. We refer to [15, §4.4, §8] for general background material and further references on Hecke algebras. The cohomology H^{∗}(H) = Ext_{H}^{∗}(k, k) for Hecke algebras of type A, B , D with the same restrictions on the parameter q as in 1.1 has been calculated in [2] in terms of stable elements in the cohomology of a maximal ℓ-parabolic subalgebra. It would certainly be desirable to describe HH^{∗}(H) explicitly as well (possibly using the stable elements methods in Hochschild cohomology in [20]). The Hochschild cohomology of tame Hecke algebras is described in [13] and [24]. The main obstacle to a generalisation of the above results to Hecke algebras of type B and D with even ℓ, Hecke algebras of exceptional types, Hecke algebras over fields of positive characteristic, or Hecke algebras with unequal parameters, is that we do not have appropriate versions of [9, Theorem 2.7] in these cases.

2 Traces for symmetric algebras

The trace maps used in the context of Hecke algebras in various sources such as [6], [9], as well as Higman’s criterion extended to modules over Hecke algebras in [19], can be interpreted as special cases of the trace maps and Higman’s criterion for symmetric algebras associated with certain bimodules in [3], [4] (the trace maps are in turn the degree zero components of trace maps for the Hochschild cohomology of symmetric algebras in [20], but we will not need this here). We review this connection in the special case of restrictions to subalgebras - detailed proofs can be found in [4]. Let k be a commutative ring. We adopt the usual convention that if A, B are k-algebras, an A-B-bimodule is the same as an A ⊗_{k} B^{0}-module, where B^{0} is the algebra opposite to B; equivalently, we always assume that the left and right k-module structure of an A-B-bimodule coincide. A k-algebra A is called symmetric if A is isomorphic, as an A-A-bimodule, to its k-dual A^{∗} = Hom_{k}(A, k) and if A is finitely generated projective as a k-module. The image s of 1_{A} under a bimodule isomorphism A ≅ A^{∗} is called a symmetrising form for A; it has the property that s(ab) = s(ba) for all a, b ∈ A and that the bimodule isomorphism A ≅ A^{∗} sends a ∈ A to the map s_{a} ∈ A^{∗} defined by s_{a}(b) = s(ab) for all a, b ∈ A. Since the automorphism group of A as an A-A-bimodule is canonically isomorphic to Z(A)^{×}, any other symmetrising form of A is of the form s_{z} for some z ∈ Z(A)^{×}. Given two symmetric algebras A, B and an A-B-bimodule M which is finitely generated projective as a left A-module and as a right B-module, the functors M ⊗_{B} − and M^{∗} ⊗_{A} − are left and right adjoint to each other. More precisely, any choice of symmetrising forms s for A and t for B induces adjunction isomorphisms as follows. Composition with s and t induces B-A-bimodule isomorphisms Hom_{A}(M, A) ≅ M^{∗} ≅ Hom_{B^{0}}(M, B); similarly for M^{∗} instead of M. The counit of M^{∗} ⊗_{A} − as left adjoint to M ⊗_{B} − is given by the composition of B-B-bimodule homomorphisms M^{∗} ⊗_{A} M ≅ Hom_{B^{0}}(M, B) ⊗_{A} M → B, where the first isomorphism is induced by the isomorphism M^{∗} ≅ Hom_{B^{0}}(M, B) and the second map sends μ ⊗ m to μ(m), where μ ∈ Hom_{B^{0}}(M, B) and m ∈ M. The counit of M as left adjoint to M^{∗} is obtained similarly, and the units of the two adjunctions are obtained by dualising the counits, exploiting the symmetry of A, B and the fact that (M ⊗_{B} N)^{∗} ≅ N^{∗} ⊗_{B} M^{∗}, where N is a B-A-bimodule which is finitely generated projective as a left B-module and as a right A-module. If A is free as a k-module and X a k-basis of A, then the symmetrising form s on A determines a dual basis X^{∨} = {x^{∨} | x ∈ X} satisfying s(xx^{∨}) = 1 for x ∈ X and s(xy^{∨}) = 0 for x, y ∈ X, x ≠ y. Let now A be a symmetric k-algebra with symmetrising form s and let B be a unitary symmetric subalgebra of A such that A is finitely generated projective as a left (or equivalently, as a right) B-module and such that the restriction of s to B is a symmetrising form on B (or equivalently, as a B-B-bimodule, B has a complement in A contained in ker(s)). (Such a subalgebra is called in [4, 5.1] a parabolic subalgebra of A; it is well-known that finitedimensional Hecke algebras are symmetric and that a parabolic subalgebra of a Hecke algebra in the sense of [15, 4.4.7] is also parabolic in the sense of [4, 5.1], but the converse need not be true.) Denote by μ : A ⊗_{B} A → A the A-A-bimodule homomorphism induces by multiplication in A. Dualising yields an A-A-bimodule homomorphism μ^{∗} : A^{∗} → (A⊗_{B} A)^{∗}. Denote by τ_{A/B} : A → A ⊗_{B} A the A-A-bimodule homomorphism given by composing the bimodule homomorphisms

A ≅ A^{∗} → (A ⊗_{B} A)^{∗} ≅ A^{∗} ⊗_{B} A^{∗} ≅ A ⊗_{B} A

The image c_{B}^{A} = τ_{A/B}(1_{A}) in A ⊗_{B} A is called the relative Casimir element. Since τ_{A/B} is an A-A-bimodule homomorphism, we have a · c_{B}^{A} = c_{B}^{A} · a for all a ∈ A. Write c_{B}^{A} = ∑_{y∈Y} y ⊗ y′ for some finite subset Y of A and elements y′ ∈ A for each y ∈ Y. For an A-A-bimodule M write M^{A} = {m ∈ M | am = ma (∀a ∈ A)}; similarly for B-B-bimodules. In particular, c_{B}^{A} belongs to (A ⊗_{B} A)^{A}. Note that an A-A-bimodule homomorphism M → N sends M^{A} to N^{A}, and hence any A-A-bimodule homomorphism A ⊗_{B} A → M sends c_{B}^{A} to an element in M^{A}. There is a trace map

tr_{B}^{A} : M^{B} → M^{A}

sending m ∈ M^{B} to ∑_{y∈Y} ymy′. This map depends on the choice of s; since s is unique up to an element in Z(A)^{×}, this is true for tr_{B}^{A} as well. The image of tr_{B}^{A} is a Z(A)-submodule of M^{A}; in particular, if M = A the image of the trace map tr_{B}^{A} : A^{B} → A^{A} = Z(A) is an ideal in Z(A). A quick way to see that tr_{B}^{A} defined in this way sends indeed M^{B} to M^{A} and does not depend on any choice (other than that of the symmetrising form s) is as follows: any element m ∈ M^{B} determines unique B-B-bimodule homomorphism B → M, sending 1_{B} to m. Through a standard adjunction, this corresponds to the A-A-bimodule homomorphism A ⊗_{B} A → M sending a ⊗ a′ to ama′, where a, a′ ∈ A. Thus the image of the relative Casimir element c_{B}^{A} is equal to tr_{B}^{A}(m), hence contained in M^{A} and independent of the choice of the elements y, y′. In particular, if U, V are A-modules then Hom_{A}(U, V) becomes an A-A-module, and the map tr_{B}^{A} applied to this bimodule is a map from Hom_{B}(U, V) to Hom_{A}(U, V) (and [2, Lemma 2.10] is a special case of this observation). We have the following bimodule version of Higman’s criterion in [4, 6.8]:

Lemma 2.1. Let A be a symmetric k-algebra with symmetrising form s and B a unitary symmetric subalgebra of A such that A is finitely generated projective as a left B-module and such that the restriction to B of s is a symmetrising form of B. Then A is isomorphic to a direct summand of A ⊗_{B} A as an A-A-bimodule if and only if the map tr_{B}^{A} : A^{B} → Z(A) is surjective. In particular, if tr_{B}^{A}(1_{A}) is invertible in Z(A) then A is isomorphic to a direct summand of A ⊗_{B} A as an A-A-bimodule.

Proof. By a standard argument for relative projectivity (see e.g. [4, Theorem 6.8]), A is isomorphic to a direct summand of A⊗_{B} A if and only if the map A⊗_{B} A → A induced by multiplication in A splits as an A-A-bimodule homomorphism (because this map represents the counit of the adjunction between A ⊗_{B} − and the restriction from A to B). Thus the dual τ_{A/B} of this map is split injective if and only if A is isomorphic to a direct summand of A ⊗_{B} A. Every A-A-bimodule homomorphism A⊗_{B} A → A is of the form a⊗a ↦ aca′ for some uniquely determined element c ∈ A^{B}, and precomposing such a homomorphism with τ_{A/B} yields an endomorphism of A sending 1_{A} to tr_{B}^{A}(c). Thus A is isomorphic to a direct summand of A ⊗_{B} A if and only if there is an element c ∈ A^{B} such that tr_{B}^{A}(c) is invertible in Z(A). Since the image of tr_{B}^{A} is an ideal in Z(A), this is equivalent to tr_{B}^{A} being surjective.

If A is free over k and X a k-basis for A with dual basis denoted as above by X^{∨}, the relative Casimir element c_{k}^{A} is equal to the sum ∑_{x∈X} x ⊗ x^{∨} in A ⊗_{k} A. Slightly more generally we have the following:

Lemma 2.2. Let A be a k-free symmetric k-algebra with symmetrising form s and let B be a k-free unitary symmetric subalgebra of A such that A is free as a right B-module and such that the restriction to B of s is a symmetrising form of B. Let X be a k-basis of A, with dual basis X^{∨} = {x^{∨} | x ∈ X}. Suppose that X contains a basis Y of A as a left B-module. Then Y^{∨} = {y^{∨} | y ∈ Y } is a basis of A as a left B-module, we have c_{B}^{A} = ∑_{y∈Y} y ⊗ y^{∨} and tr_{B}^{A}(a) = ∑_{y∈Y} yay^{∨} for all a ∈ A^{B}.

Proof. Since A is free as right B-module, A^{∗} is free as left B-module, with basis dual to Y, and thus, since A is symmetric, A is free as left B-module with basis Y^{∨}. The isomorphism A ≅ A^{∗} sends 1_{A} to the symmetrising form s. The map μ^{∗} sends s ∈ A^{∗} to s ∘ μ, where μ : A ⊗_{B} A → A is induced by multiplication in A, as above. For any y, z ∈ Y such that y ≠ z we have (s ∘ μ)(y ⊗ y^{∨}) = (yy^{∨}) = 1 and (s ∘ μ)(y ⊗ y^{∨}) = s(yz^{∨}) = 0. Note that the canonical isomorphism (A ⊗_{B} A)^{∗} ≅ A^{∗} ⊗_{B} A^{∗} exchanges the order of the two copies of A. By elementary linear algebra, the image of s∘μ in A^{∗} ⊗_{B} A^{∗} is equal to ∑_{y∈Y} s_{y} ⊗s_{y^{∨}}. Since the isomorphism A^{∗} ≅ A sends s_{y} to y, the image in A ⊗_{B} A of the above element yields the formula for c_{B}^{A} as claimed. The formula for tr_{B}^{A} follows immediately.

This lemma shows that the trace maps considered in the context of parabolic subalgebras of Hecke algebras are indeed special cases of the general construction of trace maps for symmetric algebras. In particular, as a consequence of the quoted result [9, Theorem 2.7] of Du, 2.1 can be applied to maximal ℓ-local parabolic subalgebras of Hecke algebra of type A over a field of characteristic zero.

3 Separably equivalent algebras

Definition 3.1. Two algebras A and B over a commutative ring k are called separably equivalent if there is an A-B-bimodule M which is finitely generated projective as a left A-module and as a right B-module and a a B-A-bimodule N which is finitely generated projective as a left B-module and as a right A-module, such that A is isomorphic to a direct summand of M ⊗_{B} N as an A-A-bimodule and such that B is isomorphic to a direct summand of N ⊗_{B} M as a B-B-bimodule.

The terminology is motivated by the fact that a finite-dimensional algebra A over a field k is separable (that is, projective as an A ⊗_{k} A^{0}-module) if and only if it is separably equivalent to k. Morita equivalent algebras are trivially separably equivalent. If A, B are symmetric algebras and there is a derived equivalence or a stable equivalence of Morita type between them then A and B are separably equivalent. A finite group algebra kG over a field of positive characteristic p is separably equivalent to the group algebra kP of a Sylow-p-subgroup of G. Any block algebra A of kG is separably equivalent to the group algebra kD of a defect group D of A. If A and B are indecomposable algebras over a complete local commutative Noetherian ring k, then as a consequence of the Krull-Schmidt theorem, the bimodules M and N in 3.1 can be chosen to be indecomposable. If A and B are symmetric k-algebras one can always choose M and N such that N ≅ M^{∗}, simply by replacing M by M ⊕ N^{∗}, but then M is no longer indecomposable. Here is how one reunites both properties:

Proposition 3.2. Let A, B be indecomposable symmetric separably equivalent algebras over a complete local commutative Noetherian ring k. There is an indecomposable A-B-bimodule M which is finitely generated projective as a left A-module and as a right B-module such that A is isomorphic to a direct summand of M ⊗_{B} M^{∗} and B is isomorphic to a direct summand of M^{∗} ⊗_{A} M.

Proof. This is again a special case of a standard argument for relative projectivity. Since A, B are indecomposable there are indecomposable bimodules M, N satisfying the properties in the definition 3.1 of separably equivalent algebras. View M ⊗_{B} − as a functor from Mod(B ⊗_{k} A^{0}) to Mod(A ⊗_{k} A^{0}). This functor has M^{∗} ⊗_{A} − as a left and right adjoint. Since A is isomorphic to a direct summand of M ⊗_{B} N, it follows from the implication (ii) ⇒ (v) in [4, Theorem 6.8] that M is isomorphic to a direct summand of M ⊗_{B} M^{∗} ⊗_{A} A ≅ M ⊗_{B} M^{∗}. A similar argument applied to the functor − ⊗_{A} M and its left and right adjoint − ⊗_{B} M^{∗} concludes the proof.

Proposition 3.3. Let A be a symmetric algebra over a commutative ring k with symmetrising form s and B a unitary symmetric subalgebra of A such that A is finitely generated projective as a left B-module and such that the restriction to B of s is a symmetrising form of B. If tr_{B}^{A}(1_{A}) is invertible in Z(A) then A and B are separably equivalent.

Proof. Set M = A, viewed as an A-B-bimodule. Then M^{∗} ⊗_{A} M ≅ A, viewed as a B-B-bimodule, and M^{∗} ⊗_{B} M ≅ A ⊗_{B} A, since A is symmetric. By [4, 5.2], B is isomorphic to a direct summand of A as a B-B-bimodule, and by 2.1, A is isomorphic to a direct summand of A ⊗_{B} A.

Let G be a finite group and B a G-algebra over a commutative ring k. The action of G on B induces an action on Z(B), hence on Z(B)^{×}. Let α ∈ Z^{2}(G; Z(B)^{×}), a 2-cocycle of G with coefficients in Z(B)^{×}, with respect to the induced action of G on Z(B)^{×}. That is, α is a map from G × G to Z(B)^{×} satisfying the 2-cocycle identity α(x, y)α(x, yz) = (^{x}α(y, z))α(x, yz) for x, y, z in G. Set A = B_{α}G; that is, A is the crossed product equal to the free B-module ⊕_{x∈G}Bx̂ with a B-basis {x̂ | x ∈ G} indexed by the elements of G and multiplication induced by (bx̂)(cŷ) = α(x, y)b(^{x}c)x̂y, for x, y ∈ G and b, c ∈ B. The 2-cocycle identity ensures that this multiplication is associative. Up to an isomorphism preserving the image of B, the algebra A depends only on the image of α in H^{2}(G; Z(B)^{×}). If we choose ˆ1 = 1_{B} then α is normalised; that is, α(x, 1) = 1 = α(1, x) for all x ∈ G, and then α(x, x^{−1}) = α(x^{−1}, x) for all x ∈ G. It is well-known (and easy to check) that if B is symmetric with a G-invariant symmetrising form t then A is symmetric with symmetrising form s extending t to A by zero on the subspaces Bx̂ for x ∈ G − {1}.

Proposition 3.4. Let G be a finite group, B a symmetric G-algebra over a commutative ring k having a G-invariant symmetrising form, and let α ∈ Z^{2}(G; Z(B)^{×}). Set A = B_{α}G. If |G| · 1_{k} is invertible in k then A and B are separably equivalent.

Proof. Clearly B is a direct summand of A as a B-B-bimodule. The homomorphism of left A-modules σ from A to A⊗_{B} A sending a ∈ A to a ∑_{x∈G} ax̂⊗(x̂)^{−1} is in fact a homomorphism of A-A-bimodules. If B is free as a k-module one could show this using 2.2, but one can show this also in general by a direct calculation. Clearly σ is a homomorphism of A-B-bimodules, and so we only need to check that for y ∈ G this map commutes with the right action of ŷ. The map σ sends ŷ to ∑_{x∈G} ŷx̂ ⊗ (x̂)^{−1} = ∑_{x∈G} ŷx̂ ⊗ (ŷx̂)^{−1}ŷ = ∑_{x∈G} α(y, x)ŷx ⊗ (ŷx)^{−1}α(y, x)^{−1}ŷ = ∑_{x∈G} ŷx⊗(ŷx)^{−1}ŷ. Since x runs over G, so does yx, and so this sum is equal to ∑_{x∈G} x̂⊗(x̂)^{−1}ŷ as required. Thus σ is indeed a homomorphism of bimodules from A to A⊗_{B} A. Composed with the homomorphism A⊗_{B} A → A induced by multiplication in A this yields the endomorphism of A given by multiplication with |G|. Since the image of |G| is invertible in k, the homomorphism σ is split injective, and hence A is isomorphic to a direct summand of A ⊗_{B} A as required.

For the sake of completeness we include the following consequence of a result of Erdmann and Nakano [12]:

Proposition 3.5. Let A, B be separably equivalent symmetric algebras over a field k. Then A has finite (resp. wild) representation type if and only if B has finite (resp. wild) representation type. In particular, if k is algebraically closed then A and B have the same representation type.

Proof. Let M be an A-B-bimodule such that M is finitely generated projective as a left Amodule, as a right B-module and such that A is isomorphic to a direct summand of M ⊗_{B} M^{∗} and B is isomorphic to a direct summand of M^{∗} ⊗_{A} M. Suppose that A has wild representation type. The functors M ⊗_{B} − and M^{∗} ⊗_{A} − satisfy the hypotheses (hence the conclusion) of [12, §2, Proposition], implying that B has wild representation type. Suppose next that A has finite representation type. Let V be an indecomposable B-module. Then since B is isomorphic to a direct summand of M^{8} ⊗_{A} M, V is isomorphic to a direct summand of M^{∗} ⊗_{A} M ⊗_{B} V. Thus V is isomorphic to a direct summand of M ⊗_{A} U for some indecomposable A-module U. Since A has only finitely many isomorphism classes of indecomposable modules, the same is true for B. Since (by a result of Drozd) a finite-dimensional algebra over an algebraically closed field k has a uniquely determined representation type which is either wild, tame, or finite, the proposition follows.

4 Separability and finite generation of Hochschild cohomology

The purpose of this section is to show that finite generation of Hochschild cohomology carries through separable equivalences. Let k be a commutative ring. If H^{∗} is a graded k-algebra, we denote by Z(H^{∗}) its center in the graded sense; that is, the degree n component of Z(H^{∗}) consists of all a ∈ H^{n} satisfying ab = (−1)^{nm}ba for all m ≥ 0 and all b ∈ H^{m}. Thus Z(H^{∗}) is graded-commutative, hence has a commutative quotient Z(H^{∗})/I modulo an ideal I generated by nilpotent elements. If Z(H^{∗}) is left or right Noetherian then I is finitely generated, hence nilpotent, and the Krull dimension of Z(H^{∗}) is, by definition, that of Z(H^{∗})/I or, equivalently, if 1 ≠ −1 in k, the Krull dimension of Z(H^{∗}) is defined as that of the (necessarily commutative) even part of Z(H^{∗}).

Theorem 4.1. Let A, B be separably equivalent symmetric k-algebras. Then Ext_{A⊗_{k}A^{0}}(A, U) is Noetherian as an HH^{∗}(A)-module for any finitely generated A-A-bimodule U if and only if Ext_{B⊗_{k}B^{0}}(B, V) is Noetherian as an HH^{∗}(B)-module for any finitely generated B-B-bimodule V. In that case, the Krull dimensions of HH^{∗}(A) and of HH^{∗}(B) are equal.

Remark 4.2. Let A be a finite-dimensional algebra over an algebraically closed field k. The property that HH^{∗}(A, U) is Noetherian as an HH^{∗}(A)-module for any finitely generated A ⊗_{k} A^{0}-module U is by [10, 2.4], equivalent to the property that HH^{∗}(A) is Noetherian and that Ext_{A}^{∗}(V, W) is Noetherian as an HH^{∗}(A)-module for all finitely generated A-modules V, W. By [10, Theorem 2.5] this property forces A to be Gorenstein (that is, of finite injective dimension as a left and right A-module).

The proof of 4.1 uses the following formal observations:

Lemma 4.3. Let A be a k-algebra, let U, V be A-modules and let U′ be a nonzero direct summand of U. If Ext_{A}^{∗}(U, V) is Noetherian as a module over the graded k-algebra Ext_{A}^{∗}(U, U) then Ext_{A}^{∗}(U′, V) is Noetherian as a module over the graded k-algebra Ext_{A}^{∗}(U′, U′).

Proof. Let e ∈ End_{A}(U) = Ext_{A}^{0}(U, U) be an idempotent corresponding to a projection of U onto U′. Then Ext_{A}^{∗}(U′, U′) ≅ e · Ext_{A}^{∗}(U, U) · e and Ext_{A}^{∗}(U′, W) ≅ Ext_{A}^{∗}(U, V) · e. The result follows from standard properties of Noetherian modules and rings.

Given two symmetric k-algebras A, B and an A-B-bimodule which is finitely generated projective as a left A-module and as a right B-module, the functor M ⊗_{B} − has as left and right adjoint the functor M^{∗} ⊗_{B} −. Applying this to the algebras A ⊗_{k} A^{0} and B ⊗_{k} B^{0} yields immediately the following statement:

Lemma 4.4. Let A, B be symmetric k-algebras. Let M be an A-B-bimodule which is finitely generated as a left A-module and as a right B-module. The functor M^{∗} ⊗_{A} − ⊗_{A} M from Mod(A ⊗_{k} A^{0}) to Mod(B ⊗_{k} B^{0}) is left adjoint to the functor M ⊗_{B} − ⊗_{B} M^{∗}.

Lemma 4.5. Let A, B be symmetric k-algebras. Let M be an A-B-bimodule which is finitely generated as a left A-module and as a right B-module. Let V be a B-B-bimodule. Consider Ext_{B⊗_{k}B^{0}}^{∗}(M^{∗} ⊗_{A} M, V) as a right HH^{∗}(A)-module by resticting its right Ext_{B⊗_{k}B^{0}}(M^{∗} ⊗_{A} M, M^{∗} ⊗_{A} M)-module structure via the algebra homomorphism HH^{∗}(A) → Ext_{B⊗_{k}B^{0}}(M^{∗} ⊗_{A} M, M^{∗} ⊗_{A} M) induced by the functor M^{∗} ⊗_{A} − ⊗_{A} M. The canonical adjunction isomorphism

Ext_{B⊗_{k}B^{0}}^{∗}(M^{∗} ⊗_{A} M, V) ≅ Ext_{A⊗_{k}A^{0}}^{∗}(A, M ⊗_{B} V ⊗_{B} M^{∗})

is an isomorphism of right HH^{∗}(A)-modules.

Proof. The adjunction in 4.4 extends to an adjunction between the bounded derived categories D^{b}(A ⊗_{k} A^{0}) and D^{b}(B ⊗_{k} B^{0}) of finitely generated A ⊗_{k} A^{0}-modules and B ⊗_{k} B^{0}-modules, respectively. For any integer n ≥ 0, the elements in HH^{n}(A) are morphisms A → A[n] in D^{b}(A ⊗_{k} A^{0}). The naturality of the adjunction isomorphism in the first argument yields the compatibility with the HH^{∗}(A)-module structure as stated.

Proof of 4.1. By 3.2, there is an A-B-bimodule M which is finitely generated projective as a left A-module and as a right B-module such that A is isomorphic to a direct summand of M ⊗_{B} M^{∗} and such that B is isomorphic to a direct summand of M^{∗} ⊗_{A} M, as bimodules. Suppose that Ext_{A⊗_{k}A^{0}}(A, U) is Noetherian as a right HH^{∗}(A)-module for any finitely generated A-A-bimodule U. Let V be a finitely generated B-B-bimodule. It follows from 4.5 that Ext_{B⊗_{k}B^{0}}(M^{∗} ⊗_{A} M, V) is Noetherian as a right HH^{∗}(A)-module. Thus Ext_{B⊗_{k}B^{0}}(M^{∗} ⊗_{A} M, V) is Noetherian as a right Ext_{B⊗_{k}B^{0}}(M^{∗} ⊗_{A} M, M^{∗} ⊗_{A} M)-module. Since B is isomorphic to a direct summand of M^{∗} ⊗_{A} M, it follows from 4.3 that Ext_{B⊗_{k}B^{0}}(B, V) is Noetherian as a right HH^{∗}(B)-module. Exchanging the roles of A and B shows the equivalence in the statement. Suppose now that the two equivalent statements hold. In order to prove the equality of the Krull dimensions we consider the adjunction isomorphisms Ext_{A⊗_{k}A^{0}}^{∗}(A, M ⊗_{B} M^{∗}) ≅ Ext_{A⊗_{k}B^{0}}^{∗}(M, M) ≅ Ext_{B⊗_{k}B^{0}}^{∗}(B, M^{∗} ⊗_{A} M) (cf. [21]). The functors − ⊗_{A} M and M ⊗_{B} − induce algebra homomorphisms from HH^{∗}(A) and HH^{∗}(B) to Ext_{A⊗_{k}B^{0}}^{∗}(M, M). These homomorphisms are injective since A is a summand of M ⊗_{B}M^{∗} and B is a summand of M^{∗}⊗_{A}M. By a result of Snashall and Solberg [25, Theorem 1.1], the images of these algebra homomorphisms are contained in the center Z(Ext_{A⊗_{k}B^{0}}^{∗}(M, M)) as graded algebra, and by the assumptions, this center is finitely generated as a module over both HH^{∗}(A) and HH^{∗}(B). Thus the Krull dimensions of HH^{∗}(A) and HH^{∗}(B) are both equal to that of Z(Ext_{A⊗_{k}B^{0}}^{∗}(M, M)).

If G is a finite group and k a Noetherian ring it is well-known (as a consequence of a theorem of Evens and Venkov) that Ext_{k(G×G)}(kG, U) is Noetherian as an HH^{∗}(kG)-module, and hence 4.1 has the following immediate consequence:

Corollary 4.6. Let G be a finite group, k a commutative Noetherian ring and A a symmetric k-algebra. If A and kG and separably equivalent then Ext_{A⊗A^{0}}^{∗}(A, U) is Noetherian as a right HH^{∗}(A)-module for any finitely generated A-A-bimodule U; in particular, HH^{∗}(A) is finitely generated as a k-algebra.

In order to check the equivalent properties in 4.1 it suffices to verify them for simple modules:

Proposition 4.7. Let A be a finite-dimensional algebra over a field k and U a finitely generated A-module. The following are equivalent:

(i) For any finitely generated A-module V the Ext_{A}^{∗}(U, U)-module Ext_{A}^{∗}(U, V) is Noetherian. (ii) For any simple A-module S the Ext_{A}^{∗}(U, U)-module Ext_{A}^{∗}(U, S) is Noetherian.

Proof. Suppose that (ii) holds. We show (i) by induction over the composition length of V. If V is simple there is nothing to prove. Otherwise there is a short exact sequence of A-modules of the form

\xymatrix{0 \ar[r] & W \ar[r] & V \ar[r] & S \ar[r] & 0}

for some simple A-module S. This induces a long exact sequence of the form

\xymatrix{⋯ \ar[r] & Ext_{A}^{n}(U, W) \ar[r]^{α^{n}} & Ext_{A}^{n}(U, V) \ar[r]^{β^{n}} & Ext_{A}^{n}(U, S) \ar[r]^{δ^{n}} & Ext_{A}^{n+1}(U, W) \ar[r] & ⋯}

The direct sum α = (⊕_{n≥0} α^{n}) is a homomorphism of Ext_{A}^{∗}(U, U)-modules from Ext_{A}^{∗}(U, W) to Ext_{A}^{∗}(U, V), and similarly for the direct sum β = ⊕_{n≥0}β^{n}. By the assumptions and induction, Ext_{A}^{∗}(U, S) and Ext_{A}^{∗}(U, W) are Noetherian as Ext_{A}^{∗}(U, U)-modules. Thus Ext_{A}^{∗}(U, V) is filtered by a submodule of Ext_{A}^{∗}(U, S) and a quotient of Ext_{A}^{∗}(U, W), both of which are again Noetherian, and hence so is Ext_{A}^{∗}(U, V). This shows that (ii) implies (i); the converse is trivial.

In what follows, graded modules and algebras are graded in non-negative degrees. If H^{∗}, K^{∗} are graded k-modules, we consider H^{∗} ⊗_{k} K^{∗} as a graded k-module with degree n component ⊕_{i+j=n} H^{i} ⊗_{k} K^{j}. If H^{∗}, K^{∗} are graded k-algebras, we consider H^{∗} ⊗_{k} K^{∗} as graded k-algebra with the multiplication (a ⊗ b)(c ⊗ d) = (−1)^{deg(b) deg(c)}ac ⊗ bd, for homogeneous elements a, c ∈ H^{∗} and b, d ∈ K^{∗}. (This sign convention implies that the tensor product of two graded commutative k-algebras is again graded commutative.) A graded commutative algebra H^{∗} over a field k with finite-dimensional degree zero component H^{0} is finitely generated as a kalgebra if and only if it is left and right Noetherian. Thus the tensor product of two graded commutative (left or right) Noetherian algebras H^{∗}, K^{∗} over a field with finite-dimensional degree zero components is again (left or right) Noetherian, and the Krull dimension of H^{∗} ⊗_{k} K^{∗} is the sum of the Krull dimensions of H^{∗} and K^{∗}. Note that the Krull dimension of the direct product H^{∗} × K^{∗} is the maximum of the Krull dimensions of H^{∗} and K^{∗}. If A, B are two finite-dimensional algebras over a field k it is well-known that, with the above sign convention, we have HH^{∗}(A ⊗_{k} B) ≅ HH^{∗}(A) ⊗_{k} HH^{∗}(B) and hence, if the Krull dimensions of HH^{∗}(A), HH^{∗}(B) are finite then the Krull dimension of HH^{∗}(A ⊗_{k} B) is finite and equal to the sum of the Krull dimensions of HH^{∗}(A) and HH^{∗}(B); see for instance the proof of [5, Proposition 7.4]. Similarly, HH^{∗}(A × B) ≅ HH^{∗}(A) × HH^{∗}(B), and hence the Krull dimension of HH^{∗}(A × B) is the maximum of the Krull dimensions of HH^{∗}(A), HH^{∗}(B) if these are finite. Thus finite generation of Hochschild cohomology passes on to tensor products and direct products. We will need a slightly more precise version of this fact for modules over Hochschild cohomology algebras.

Proposition 4.8. Let A, B be finite-dimensional algebras over a field k having separable semisimple quotients. Set C = A ⊗_{k} B. Suppose that Ext_{A⊗_{k}A^{0}}^{∗}(A, U) is Noetherian as an HH^{∗}(A)- module for any finitely generated A-A-bimodule U and that Ext_{B⊗_{k}B^{0}}^{∗}(B, V) is Noetherian as an HH^{∗}(B)-module for any finitely generated B-B-bimodule V. Then Ext_{C⊗_{k}C^{0}}^{∗}(C, W) is Noetherian as an HH^{∗}(C)-module for any finitely generated C-C-bimodule W.

Proof. By 4.7 it suffices to show this if W is a simple C ⊗_{k} C^{0}-module. Since A, B have separable semi-simple quotients, every simple C ⊗_{k} C^{0}-module is of the form S ⊗_{k} T for a simple A ⊗_{k} A^{0}- module S and a simple B ⊗_{k} B^{0}-module T. The appropriate versions of Künneth’s theorem imply that

Ext_{C⊗_{k}C^{0}}^{∗}(C, S ⊗_{k} T) ≅ Ext_{A⊗_{k}A^{0}}^{∗}(A, S) ⊗_{k} Ext_{B⊗_{k}B^{0}}^{∗}(B, T)

and through the isomorphism HH^{∗}(C) ≅ HH^{∗}(A) ⊗_{k} HH^{∗}(B) this is an isomorphism of right HH^{∗}(C)-modules. Since HH^{∗}(A), HH^{∗}(B) are graded commutative and Noetherian, the same is true for HH^{∗}(C). Thus finitely generated modules over HH^{∗}(C) are Noetherian. Since the tensor product of two finitely generated modules over HH^{∗}(A) and HH^{∗}(B), respectively, is finitely generated as an HH^{∗}(C)-module, hence Noetherian, the result follows.

Given a finite-dimensioanl self-injective algebra A over a field k and a finitely generated A-A-bimodule U we denote as usual by Ω_{A⊗_{k}A^{0}}(U) the kernel of a projective cover P_{U} → U of U; this is unique up to unique isomorphism in the stable category of A ⊗_{k} A^{0}-modules. The following observation is well-known; we include a proof for the convenience of the reader.

Proposition 4.9. Let A be a finite-dimensional self-injective algebra over a field k such that Ω_{A⊗_{k}A^{0}}^{n}(A) ≅ A for some positive integer n. Then Ext_{A⊗_{k}A^{0}}^{∗}(A, U) is Noetherian as an HH^{∗}(A)-module for any finitely generated A-A-bimodule U; in particular, HH^{∗}(A) is finitely generated.

Proof. A bimodule isomorphism Ω_{A⊗_{k}A^{0}}^{n}(A) ≅ A represents an element in HH^{n}(A). Thus HH^{∗}(A) is generated, as a k-algebra, by a k-basis of ⊕_{i=0}^{n} HH^{i}(A), hence HH^{∗}(A) is Noetherian (as HH^{∗}(A) is graded commutative). Similarly, Ext_{A⊗_{k}A^{0}}^{∗}(A, U) is generated, as an HH^{∗}(A)- module, by a k-basis of the finite-dimensional space ⊕_{i=0}^{n} Ext_{A⊗_{k}A^{0}}^{i}(A, U), hence the HH^{∗}(A)- module Ext_{A⊗_{k}A^{0}}^{∗}(A, U) is Noetherian.

Remark 4.10. A finite-dimensional algebra A over a field k satisfying Ω_{A⊗_{k}A^{0}}^{n}(A) ≅ A for some positive integer n is automatically self-injective, by a result of Butler (see [17, 1.5] for a more general result). Self-injective Nakayama algebras have this property by [11, §4.2, Lemma], and hence so do Brauer tree algebras because a Brauer tree algebra is derived equivalent to a symmetric Nakayama algebra (cf. [23]).

5 Proof of Theorem 1.1 and Theorem 1.2

Let k be a field of characteristic zero and q a non-zero element in k. Let H = H(W, q) be a Hecke algebra over k with parameter q of the finite Coxeter group (W, S). That is, H has a k-basis {T_{w} | w ∈ W } indexed by the elements of W such that T_{1} is the unit element of H, with multiplication given by T_{w}T_{w′} = T_{ww′} if w, w′ ∈ W such that the length of ww′ is the sum of the length of w, w′, and the quadratic relations (T_{s})^{2} = qT_{1} + (1 − q)T_{s} for s ∈ S. If S is a disjoint union of two non-empty subsets S_{1}, S_{2} such that any element in S_{1} commutes with any element in S_{2}, then the subgroups W_{i} of W generated by S_{i}, for i = 1, 2, can be identified with the Coxeter groups (W_{i}, S_{i}), and W = W_{1} × W_{2}. If we denote by H_{i} the corresponding Hecke algebra of W_{i}, for i = 1, 2, then H ≅ H_{1} ⊗_{k} H_{2}. Thus, by 4.8, we may assume that (W, S) is irreducible, hence of type A_{n−1} or B_{n} or D_{n} (n ≥ 4). Assume first that H = H(S_{n}, q) is of type A_{n−1}. If q = 1 or if q has infinite order in k^{×} then H is semi-simple (by [8, 4.3]), hence separable as char(k) = 0. Thus, in that case, H⊗_{k} H^{0} is again semi-simple, and hence Ext_{H⊗_{k}H^{0}}^{∗}(H, M) = Ext_{H⊗_{k}H^{0}}^{0}(H, M) is finite-dimensional for any finitely generated H ⊗_{k} H^{0}-module, so 1.1 and 1.2 hold trivially. Assume that q is a primitive ℓ-th root of unity for some integer ℓ ≥ 2. It is well-known (cf. [15, Proposition 8.1.1]) that H is symmetric, with a canonical symmetrising form, such that the restriction of this form to the Hecke algebra H′ of a parabolic subgroup of S_{n} is the canonical symmetrising form of H′; in other words, parabolic subalgebras in the context of Hecke algebras (cf. [15, 4.4.7]) are indeed parabolic subalgebras in the sense of [4, 5.1]. As in [2], denote by B a maximal ℓ-parabolic subalgebra H(λ, q) of H, where λ is the partition (ℓ^{m}, 1^{a}), with n = ℓm + a and 0 ≤ a ≤ ℓ − 1. An immediate consequence of a result of Du [9, Theorem 2.7] in conjunction with 2.1 is the following observation:

Proposition 5.1. With the notation above, the symmetric algebras H and B are separably equivalent. More precisely, B is a direct summand of H as a B-B-bimodule, and H is isomorphic to a direct summand of H ⊗_{B} H as an H-H-bimodule.

Proof. The fact that B is a direct summand of H as a B-B-bimodule is a general fact of parabolic subalgebas (as an immediate consequence of the distinguished double coset representatives [15, 2.1.7], [19, (2.28)]). Du’s result [9, Theorem 2.7] says that tr_{B}^{H}(1) is invertible in Z(H). It follows thus from 2.1 that H is isomorphic to a direct summand of H ⊗_{B} H, completing the proof.

It follows therefore from 4.1 that in order to prove 1.1 and 1.2 for the Hecke algebra H of type A_{n−1} it suffices to prove the conclusion for the maximal ℓ-parabolic subalgebra B instead. The algebra B in turn is the tensor product of m copies of the Hecke algebra H(S_{ℓ}, q), and thus, by 4.8, it suffices to prove 1.1 and 1.2 for H(S_{ℓ}, q). This is a product of a Brauer tree algebra and a semi-simple algebra (cf. [14], [26]), and so both results hold as the Hochschild cohomology of a Brauer tree algebra is periodic (cf. [11] and 4.9, 4.10). This concludes the proof of 1.1 and 1.2 in the case of type A_{n−1}. For the types B and D we proceed as in [2, §6], playing the problem back to type A. Suppose that H is of type B_{n} and that either the order of q is infinite or that its order ℓ is not even. By [7, Theorem 4.17], H is Morita equivalent to the direct product of tensor products of Hecke algebras of type A of the form

∏_{j=0}^{n} H(S_{j}, q) ⊗_{k} H(S_{n−j}, q)

where S_{0} is the trivial group, by convention. As a consequence of 4.8, the theorems 1.1 and 1.2 follow in this case from the fact that they hold in type A. Suppose that H is of type D_{n} for some odd integer n ≥ 5. Then by [22, Theorems 3.6, 3.7] (made explicit in [18]), the algebra H is Morita equivalent to the algebra

∏_{j=(n+1)/2}^{n} H(S_{j}, q) ⊗_{k} H(S_{n−j}, q)

and so the results in type A imply again both 1.1 and 1.2. Suppose finally that H is of type D_{n} for some even integer n ≥ 4. Then, by the main result in [18], H is Morita equivalent to the algebra

A(n/2) × ∏_{j=(n+1)/2}^{n} H(S_{j}, q) ⊗_{k} H(S_{n−j}, q)

where A(n/2) is a subalgebra of H(S_{n}, q) generated by H(S_{n/2}, q)⊗_{k}H(S_{n/2}, q) and an invertible element in H(S_{n}, q) which exchanges the two factors in this tensor product and whose square is in the center of this subalgebra. In other words, the explicit description of A(n/2) in [18, Remark 2.4] shows that A(n/2) is a crossed product of the form (H(S_{n/2}, q)⊗_{k} H(S_{n/2}, q))_{α}C_{2}, where C_{2} is a cyclic group of order 2 and α a 2-cocycle of C_{2} with values in Z(H(S_{n/2}, q) ⊗_{k} H(S_{n/2}, q))^{×}. It follows from 3.4 that A(n/2) and H(S_{n/2}, q) ⊗_{k} H(S_{n/2}, q) are separably equivalent. Thus 1.1 and 1.2 follow yet again from the corresponding results in type A. This concludes the proof of both theorems.

6 Further remarks

The purpose of this section is to sketch some arguments which may be used to show the finite generation of HH^{∗}(A) in some situations in which it is not known that Ext_{A⊗_{k}A^{0}}^{∗}(A, U) is Noetherian for all finitely generated A-A-bimodules U. Let k be a commutative Noetherian ring and let A be a k-algebra. An A-module U is called relatively k-injective if every A-homomorphism from U to another A-module V which is split injective as k-homomorphism, is split injective as A-homomorphism. An injective module is relatively k-injective, and if k is a field the converse holds as well. Dually, U is relatively k-projective if every A-homomorphism V → U which is split surjective as k-homomorphism is split surjective as A-homomorphism. If A is symmetric, the classes of relatively k-projective A-modules and relatively k-injective A-modules coincide (and the content of this section can be generalised to the class of algebras with this property). We denote by stmod(A) the k-stable category of the category mod(A) of finitely generated A-modules; that is, stmod(A) has the same objects as mod(A), and for any two finitely generated A-modules U, V, the homomorphism space in stmod(A) from U to V is the quotient space \underline{Hom}_{A}(U, V) = Hom_{A}(U, V)/Hom_{A}^{pr}(U, V), where Hom_{A}^{pr}(U, V) is the space of all A-homomorphisms from U to V which factor through a relatively k-projective Amodule. If A is symmetric then the category stmod(A) is triangulated, with suspension functor Σ sending a finitely generated A-module U to the cokernel of a k-split embedding U → I_{U} of U into a relatively k-injective A-module I_{U}. Such a module I_{U} always exists; for instance, one could take I_{U} = Hom_{k}(A, U) with the map U → I_{U} sending u ∈ U to the map a ↦ au. For two k-algebras A, B we denote by perf(A, B) the category of A-B-bimodules which are finitely generated projective as left A-modules and as right B-modules. If A and B are finitely generated projective as k-modules then the category perf(A, B) contains the finitely generated projective A-B-bimodules. If A, B are symmetric k-algebras then all relatively k-projective modules in perf(A, B) are actually projective A-B-bimodules and the category stperf(A, B) is a thick subcategory of the k-stable category stmod(A ⊗_{k} B^{0}); in particular, stperf(A, B) is again triangulated, with suspension functor, denoted abusively again by Σ, sending a bimodule M in stperf(A, B) to the cokernel of a relatively k-injective envelope M → I_{M} of M in the category of A-B-bimodules.

Proposition 6.1. Let A, B be symmetric algebras over a commutative Noetherian ring k and let M be an A-B-bimodule which is finitely generated projective as a left A-module and as a right B-module. Suppose that B is isomorphic to a direct summand of M^{∗} ⊗_{A} M as a B-B-bimodule and that M ⊗_{B} M^{∗} belongs to the thick subcategory of the k-stable category of A-A-bimodules generated by A. If HH^{∗}(A) is finitely generated as a k-algebra then so is HH^{∗}(B).

The proof of 6.1 uses the following two lemmas.

Lemma 6.2. Let A be a symmetric algebras over a commutative Noetherian ring k and U, V finitely generated A-modules such that Ext_{A}^{∗}(U, V) is Noetherian as an Ext_{A}^{∗}(U, U)-module then Ext_{A}^{∗}(U, W) is Noetherian as an Ext_{A}^{∗}(U, U)-module for any A-module W belonging to the thick subcategory ⟨V ⟩ of the stable module category stmod(A) generated by V.

Proof. Clearly Ext_{A}^{∗}(U, V′) is Noetherian as an Ext_{A}^{∗}(U, U)-module for any direct summand V′ of V. We observe next that Ext_{A}^{∗}(U, Σ^{n}(V)) is Noetherian as an Ext_{A}^{∗}(U, U)-modules, for any integer n. Indeed, we have Ext_{A}^{n}(U, V) = \underline{Hom}_{A}(U, Σ^{n}(V)) if n is a positive integer and Ext_{A}^{n}(U, V) = {0} if n is a negative integer. Therefore, if i and n are integers such that both i and i − n are positive then Ext_{A}^{i}(U, V) ≅ Ext_{A}^{i−n}(U, Σ^{n}(V)). Thus the kernel and cokernel of the obvious graded map of degree −n from Ext_{A}^{∗}(U, V) to Ext_{A}^{∗}(UΣ^{n}(U)) extending the above isomorphisms are finitely generated as k-modules, and hence Ext_{A}^{∗}(U, Σ^{n}(V)) is Noetherian. Let X → Y → Z → Σ(X) be an exact triangle in \underline{mod}(A). Suppose that Ext_{A}^{∗}(U, X) and Ext_{A}^{∗}(U, Y) are Noetherian as Ext_{A}^{∗}(U, U)-modules. The maps in the exact triangle induce a long exact sequence

\xymatrix{⋯ \ar[r] & Ext_{A}^{n}(U, X) \ar[r]^{α^{n}} & Ext_{A}^{n}(U, Y) \ar[r]^{β^{n}} & Ext_{A}^{n}(U, Z) \ar[r]^{δ^{n}} & Ext_{A}^{n+1}(U, X) \ar[r] & ⋯}

The direct sum β = (⊕_{n≥0} β^{n}) is a homomorphism of Ext_{A}^{∗}(U, U)-modules from Ext_{A}^{∗}(U, Y) to Ext_{A}^{∗}(U, Z), and similarly for the direct sum δ = ⊕_{n≥0}δ^{n}. Thus, by the assumptions, both the image and the kernel of δ are Noetherian Ext_{A}^{∗}(U, U)-modules, hence so is Ext_{A}^{∗}(U, Z). The result follows.

Lemma 6.3. Let A be a symmetric algebra over a commutative Noetherian ring k. Denote by T the thick subcategory of stperf(A, A) generated by the A-A-bimodule A. If M, N belong to T then so does M ⊗_{A} N.

Proof. Every module in the subcategory T of stperf(A, A) is obtained from applying a finite number of times the shift functor Σ with respect to A ⊗_{k} A^{0}, taking direct summands, direct sums, and completing triangles. Thus we may filter T by full additive subcategories T_{m}, m ≥ 0, defined inductively as follows: we denote by T_{0} the full additive subcategory of T consisting of all finite direct sums of summands of the A-A-bimodules Σ^{n}(A), where n ∈ ℤ. If T_{m} is defined for some m ≥ 0, we define T_{m+1} as full additive subcategory of finite direct sums of summands of bimodules W for which there exists an exact triangle M → N → W → Σ(M) such that M, N belong to T_{m}. Clearly every module in T belongs to T_{m} for some m ≥ 0. What we will show is that if M belongs to T_{t} and N belongs to T_{s} then M ⊗_{A} N belongs to T_{t+s}. We do this by induction over t + s. For t + s = 0 being clear because T_{0} is closed under taking tensor products over A; indeed, Σ^{m}(A) ⊗_{A} Σ^{n}(A) ≅ Σ^{m+n}(A) in stperf(A, A) for any two integers m, n. Suppose t + s ≥ 0. Let again M → N → W → Σ(M) be an exact triangle in stperf(A, A) such that M, N belong to T_{t}. Let X be a bimodule in T_{s}. Then the triangle M ⊗_{A} X → N ⊗_{A} X → W ⊗_{A} X → Σ(M) ⊗_{A} X is exact, and by the assumptions, M ⊗_{A} X, N ⊗_{A} X are in T_{t+s}. Thus W ⊗_{A} X is in T_{t+s+1}. This holds then clearly for any W in T_{t+1} and any X in T_{s}. By reversing the roles of t and s one concludes the proof.

Proof of 6.1. By the assumptions, M⊗_{B}M^{∗} is in the thick subcategory of stperf(A, A) generated by A. Thus, by 6.3, M ⊗_{B} M^{∗} ⊗_{A} M ⊗_{B} M^{∗} is in the thick subcategory of stperf(A, A). It follows from 6.2 that Ext_{A⊗_{k}A^{0}}(A, M ⊗_{B} M^{∗}⊗_{A}M ⊗_{B} M^{∗}) is Noetherian as an HH^{∗}(A)-module. Therefore, by 4.5, Ext_{B⊗B^{0}}(M^{∗} ⊗_{A} M, M^{∗} ⊗_{A} M) is Noetherian as an HH^{∗}(A)-module, hence Noetherian itself because any left or right ideal in this algebra is in particular an HH^{∗}(A)- submodule. Since B is isomorphic to a direct summand of M^{∗} ⊗_{A} M it follows from 4.3 that HH^{∗}(B) is Noetherian, hence finitely generated as a k-algebra.

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