ON SINGULAR EQUIVALENCES OF MORITA TYPE

GUODONG ZHOU AND ALEXANDER ZIMMERMANN

Abstract. Stable equivalences of Morita type preserve many interesting properties and is proved to be the appropriate concept to study for equivalences between stable categories. Recently the singularity category attained much attraction and Xiao-Wu Chen and Long-Gang Sun gave an appropriate definition of singular equivalence of Morita type. We shall show that under some conditions singular equivalences of Morita type have some biadjoint functor properties and preserve positive degree Hochschild homology.

Introduction

For a Noetherian algebra A over a commutative ring its singularity category D_{sg}(A) is defined to be the Verdier quotient of the bounded derived category of finitely generated modules over A by the full subcategory of perfect complexes. This notion was introduced in an unpublished manuscript [5] by Ragnar-Olaf Buchweitz under the name of stable derived category. He related this category to maximal Cohen-Macaulay modules. Later Dmitri Orlov [22] rediscovered this notion independently in the context of algebraic geometry and mathematical physics, under the name of singularity category. The derived category of an algebra is replaced there by the derived category of coherent sheaves over a scheme. Orlov’s notation for this object seems now to become the standard one, also in the case of the derived category of an algebra, and we shall concentrate here on this case.

If A is a selfinjective algebra, then D_{sg}(A) is equivalent to the stable category of A (cf [15, 25]). By definition D_{sg}(A) is always triangulated and it is easy to see that D_{sg}(A) is trivial if and only if A has finite global dimension. From this point of view D_{sg}(A) seems to have advantages with respect to the stable category of an algebra, in case the algebra is not selfinjective, and may be an appropriate replacement. Recently much work was undertaken to understand the structure of D_{sg}(A) under various conditions on A. We mention in particular Xiao-Wu Chen’s work here [7, 8, 9, 10], but also Bernhard Keller, Daniel Murfet and Michel Van den Bergh [16] as well as Osamu Iyama, Kiriko Kato and Jun-Ichi Miyachi [14].

Abstract equivalences between stable categories of algebras are very ill-behaved, even in case the algebras are selfinjective. Very few properties of the algebras are preserved. However, if the equivalence is induced by an exact functor of the module categories, much more can be said and a rich structure is available. The concept developed for this purpose is Broué’s concept of stable equivalence of Morita type [4]. Since the singularity category generalises the stable category, we cannot expect better properties in the singularity case than we have in the stable case.

Very recently analogous to the notion of stable equivalences of Morita type, Xiao-Wu Chen and Long-Gang Sun defined in [11] the concept of singular equivalences of Morita type. The purpose of the present note is to study this new concept of singular equivalences of Morita type. We obtain two main results. First, we shall prove in Theorem 3.1 that under mild conditions a singular equivalence of Morita type gives rise to a bi-adjoint pair. This section is inspired by an analogous approach by Alex Dugas and Roberto Martinez-Villa [12]. Then we shall investigate Hochschild homology and show in Theorem 4.1 that Hochschild homology of a finite dimensional algebra over a field and in strictly positive degrees is invariant under a singular equivalence of Morita type.

Date: September 27, 2012.

2010 Mathematics Subject Classification. Primary 16G35; Secondary 18E30, 18E35, 16E30, 13D03 .

Key words and phrases. Singularity categories; Singular equivalences of Morita type; Hochschild homology.

The main tool here is Serge Bouc’s generalisation [3] of the Hattori-Stallings trace to Hochschild homology.

The paper is organised as follows. We recall the notion and some properties of singularity categories in Section 1. Section 2 is devoted to the definition and some of the results of Chen and Sun on singular equivalences of Morita type. We prove the biadjoint property in Section 3 and we study Hochschild homology in Section 4.

Acknowledgement: We are very grateful to Xiao-Wu Chen and Long-Gang Sun for sending us their preprint [11].

We thank the two referees for their careful reading of this paper, and in particular for mentioning to us an error in the proof of Theorem 3.1 which lead to a modification of the notion of being strongly right nonsingular.

1. Singularity categories and singularly stable categories

Let A be a right Noetherian ring. We denote by mod(A) the category of finitely generated right A-modules, by D^{b}(mod(A)) the bounded derived category of mod(A), by P^{<∞}(A) the full subcategory of mod(A) consisting of modules of finite projective dimension, and by K^{b}(proj(A)) the homotopy category of bounded complexes of finitely generated projective A-modules.

Definition 1.1 ([5]). Let A be a right Noetherian ring. Then the Verdier quotient category

D_{sg}(A) := D^{b}(mod(A))/K^{b}(proj(A))

is called the singularity category of A.

It is well-known that K^{b}(proj(A)) is a full triangulated subcategory of D^{b}(A). We briefly recall the construction of the Verdier quotient. We refer to Gabriel and Zisman’s book [13, Chapter 1] for more ample details, and give only the basic construction here for the convenience of the reader.

The objects of D_{sg}(A) are the same as those of D^{b}(A). Let X and Y be objects of D_{sg}(A). Then a morphism in Hom_{D_{sg}(A)}(X, Y) is represented by triples (ν, Z, α) where Z is an object in D^{b}(A), where α ∈ Hom_{D^{b}(A)}(Z, Y) and where ν ∈ Hom_{D^{b}(A)}(Z, X) so that the mapping cone of ν is isomorphic to an object in K^{b}(proj(A)). A triple (ν, Z, α) is covered by a triple (ν′, Z′, α′) if there is a morphism ψ ∈ Hom_{D^{b}(A)}(Z′, Z) so that ν′ = ν ∘ ψ and α′ = α ∘ ψ. Two triples (ν, Z, α) and (ν′′, Z′′, α′′) are equivalent if both are covered by some triple (ν′, Z′, α′). This way the category of triples is directed, and the morphisms from X to Y is the limit of this category.

The construction of the singularity category as Verdier quotient implies that D_{sg}(A) is always triangulated.

Let A be any right Noetherian ring. Denote by \underline{mod}(A) the stable category of (finitely generated right) A-modules, with objects being the same as mod(A) and morphisms \underline{Hom}_{A}(M, N) being the equivalence classes of morphisms of A-modules modulo those factoring through a projective module. Recall that the category \underline{mod}(A) admits an endo-functor Ω, the syzygy functor, defined as ker(π_{X}), where for every object X in \underline{mod}(A) we choose a projective object P_{X} in mod(A) and an epimorphism P_{X} ⟶^{π_{X}} X in mod(A).

By the very construction there are natural functors

mod(A) ⟶^{F} \underline{mod}(A)
D^{b}(A) ⟶^{G} D_{sg}(A)
\underline{mod}(A) ⟶^{H} D_{sg}(A)

so that the diagram

mod(A) ⟶ D^{b}(A)
↓ F  ↓ G
\underline{mod}(A) ⟶^{H} D_{sg}(A)

commutes. Moreover, H(M) = 0 if and only if M is of finite projective dimension. Finally H commutes with syzygies in the sense that

H ∘ Ω ≃ [−1] ∘ H.

A consequence of this relation is an important observation, namely that the singularity category is in general not Hom-finite [9] and is in general not a Krull-Schmidt category. An example is given by the 3-dimensional local algebra A = K[X, Y]/(X^{2}, Y^{2}, XY) over a field K. Indeed, for the simple A-module S one gets Ω(S) ≃ S ⊕ S and this implies isomorphisms

H(S) ≃ H(Ω(S))[1] ≃ H(S)[1] ⊕ H(S)[1] ≃ (H(S)[2])^{4} ≃ ⋯ ≃ (H(S)[n])^{2^{n}}

in D_{sg}(A). Moreover, H(S) ≠ 0 since S is of infinite projective dimension. Therefore,

dim_{K}End_{D_{sg}(A)}(H(S)) = dim_{K}End_{D_{sg}(A)}((H(S)[n])^{2^{n}}) ≥ 2^{n}

for all n ≥ 0 and this implies that dim_{K}End_{D_{sg}(A)}(H(S)) = +∞.

As we have seen, M is an A-module of finite projective dimension if and only if H(M) = 0. Hence, it is natural to consider the following category \underline{mod}_{P^{<∞}}(A).

Definition 1.2. Let A be a finite dimensional algebra. The singularly stable category is by definition the quotient category of mod(A) by P^{<∞}(A), denoted by \underline{mod}_{P^{<∞}}(A) := mod(A)/P^{<∞}(A).

More precisely, the objects of \underline{mod}_{P^{<∞}}(A) are the same objects as those in mod(A) and for two A-modules X and Y define Hom_{\underline{mod}_{P^{<∞}} (A)}(X, Y) to be the equivalence classes of A-module homomorphisms X ⟶ Y modulo those factoring through an object in P^{<∞}(A).

It is clear that H factors through the natural functor

\underline{mod}(A) ⟶^{Π} \underline{mod}_{P^{<∞}}(A)

in the sense that there is a natural functor

\underline{mod}_{P^{<∞}}(A) ⟶^{L} D_{sg}(A)

so that

H = L ∘ Π.

Remark 1.3. Observe that L is not an embedding in general. Let Q be the quiver

[figure: •; •; 1; 2; α; β]

and let A = KQ/⟨α^{2}, βα⟩. Let S_{1} and S_{2} be the two simple A-modules. Then H(S_{1}) ≃ H(S_{2}) since Ω^{2}(S_{2}) ≃ S_{1} ≃ Ω^{2}(S_{1}), but S_{1} ≄ S_{2} in \underline{mod}_{P^{<∞}}(A) since there is no non zero homomorphism of A-modules between these objects.

Remark 1.4. We could consider modules of finite projective dimension as “smooth” objects. Then the singularly stable category measures the singularity of A. Clearly the algebra A has finite global dimension if and only if the singularly stable category has only one object with only one endomorphism. However, the singularly stable category is only an additive category, and in general it is not triangulated. If A is selfinjective, H is an equivalence (cf [15, 25]) and an A-module of finite projective dimension is actually projective. Hence also Π is an equivalence in this case.

Remark 1.5. Let A be the algebra introduced in Remark 1.3. Then it is easy to see that D_{sg}(A) ≃ D_{sg}(K[X]/X^{2}). However, these two algebras are not stably equivalent. In fact, if they were stably equivalent, then there would be a one to one correspondence between the isomorphism classes of non-projective indecomposable modules. However, up to isomorphisms, A has more than two non-projective indecomposable modules and K[X]/(X^{2}) has only one such module.

We are grateful to one of the referees who suggested the above proof which is simpler than our original argument.

2. Singular equivalences of Morita type

As mentioned in the introduction, general stable equivalences have very poor properties, even for selfinjective algebras. A richer concept is given by Broué [4]. Broué defined stable equivalences of Morita type as equivalences between stable module categories induced by tensor product with bimodules. This concept was highly successful in the understanding of equivalence between stable categories of self-injective algebras and was a subject of numerous studies.

We consider the question when the singularly stable categories of two algebras are equivalent. Since equivalences between singular categories of selfinjective algebras coincide with stable equivalences, we need a richer concept than just an equivalence between triangulated categories. Recently Xiao-Wu Chen and Long-Gang Sun introduced singular equivalences of Morita type [11] on the model of Broué’s concept of stable equivalences of Morita type.

Let K be a commutative ring. For a K-algebra A, we denote by A^{e} = A^{op} ⊗_{K} A its enveloping algebra.

Definition 2.1. (cf [11]) Let A and B be two K-algebras for a commutative ring K. Let _{A}M_{B} and _{B}N_{A} be two bimodules so that

• M is finitely generated and projective as A^{op}-module and as B-module;

• N is finitely generated and projective as A-module and as B^{op}-module;

• _{A}M ⊗_{B} N_{A} ≃ _{A}A_{A} ⊕_{A}X_{A} for a module X ∈ P^{<∞}(A^{e});

• _{B}N ⊗_{A} M_{B} ≃ _{B}B_{B} ⊕_{B}Y_{B} for a module Y ∈ P^{<∞}(B^{e}). We then say that the pair (_{A}M_{B},_{B}N_{A}) induces a singular equivalence of Morita type.

We say that A and B are singularly equivalent of Morita type if there is a pair of bimodules (_{A}M_{B},_{B}N_{A}) which induces a singular equivalence of Morita type.

Remark 2.2. • It is immediate from the definition that a pair of bimodules inducing a

stable equivalence of Morita type induces a singular equivalence of Morita type as well. However, a singular equivalence of Morita type will not be a stable equivalence of Morita type in general since the property of X to be in P^{<∞}(A^{e}) is in general much weaker than the condition to be projective as bimodule.

Nevertheless, if A is selfinjective (and thus so is any algebra singularly equivalent of Morita type to A, as is remarked in [11]), any module with finite projective resolution is actually projective, and hence a singular equivalence of Morita type is actually a stable equivalence of Morita type. The concept of a singular equivalence of Morita type and of a stable equivalence of Morita type coincide for selfinjective algebras.

• Let (_{A}M_{B},_{B}N_{A}) induce a singular equivalence of Morita type and let M ⊗_{B}N ≃ A⊕X and N ⊗_{A} M ≃ B ⊕ Y. Then X is projective as A-left module and as A-right module. Indeed, M is projective as B-right module, hence a direct factor of some B^{n}. Hence M ⊗_{B} N is a direct factor of B^{n} ⊗_{B} N ≃ N^{n}. Now, X is by definition a direct factor of N^{n} and since N is projective as A-right module, X is projective as A-right module. Similarly X is projective on the left. Likewise Y is projective as B-left module and as B-right module.

From now on to the end of the present section and in Section 3 fix a field K and K-algebras will always be supposed to be finite dimensional and modules will be always finitely generated.

The following result is a direct consequence of Definition 2.1.

Proposition 2.3. Let (_{A}M_{B},_{B}N_{A}) be a pair of bimodules inducing a singular equivalence of Morita type between two K-algebras A and B. Then

− ⊗_{A} M_{B} : D_{sg}(A) ⟶ D_{sg}(B)

is an equivalence of triangulated categories with quasi-inverse

− ⊗_{B} N_{A} : D_{sg}(B) ⟶ D_{sg}(A).

Moreover, the same functors establish an equivalence of additive categories between \underline{mod}_{P^{<∞}}(A) and \underline{mod}_{P^{<∞}}(B).

The following result is an adaptation to the singular situation of a proof of Yu-Ming Liu for stable equivalences of Morita type (cf [18, Lemma 2.2]). The proof carries over verbatim.

Proposition 2.4. (cf [11]) Let A and B be K-algebras. Suppose (_{A}M_{B},_{B}N_{A}) induces a singular equivalence of Morita type. Then _{A}M is a progenerator in mod(A^{op}), and likewise for M_{B}, _{B}N and N_{A}.

The following fact is proved in [11] analogous to [19, Proposition 2.1 and Theorem 2.2].

Proposition 2.5. (cf [11]) Let A and B be two K-algebras without direct summands which have finite projective dimension as bimodules. Assume that two bimodules _{A}M_{B} and _{B}N_{A} induce a singular equivalence of Morita type between A and B.

(1) Then A and B have the same number of indecomposable summands. In particular, A is indecomposable if and only if B is indecomposable.

(2) Suppose that A = A_{1} × A_{2} × ⋯ × A_{s} and B = B_{1} × B_{2} × ⋯ × B_{s}, where all A_{i} and all B_{i} are indecomposable algebras. Then, there is a permutation σ of the set {1, …, s} so that A_{i} and B_{σ(i)} are singularly equivalent of Morita type for all i ∈ {1, …, s}.

In analogy of what is known to hold for stable equivalences of Morita type, Chen and Sun also show the following lemma.

Lemma 2.6. (cf [11]) Let K be a field and let A and B be finite dimensional K-algebras. Assume that bimodules _{A}M_{B} and _{B}N_{A} define a singular equivalence of Morita type between A and B, and suppose that A or B is indecomposable as an algebra. Then M and N each have a unique indecomposable bimodule summand of infinite projective dimension. If we denote these summands as M_{1} and N_{1} respectively, then (M_{1}, N_{1}) also induces a singular equivalence of Morita type between A and B.

Let K be a field, and let A and B be finite dimensional K-algebras without direct summands having finite projective dimension as bimodules. Proposition 2.5 and Lemma 2.6 imply that for a singular equivalence of Morita type induced by (_{A}M_{B},_{B}N_{A}) we can always suppose that A and B are indecomposable algebras and that _{A}M_{B} and _{B}N_{A} are indecomposable bimodules.

Remark 2.7. During the ICRA 2012 in Bielefeld, Chang-Chang Xi raised the question whether there are algebras which are singularly equivalent of Morita type, but which are not stably equivalent of Morita type. This remark answers this question.

For any algebra A denote by T_{2}(A) := ( \begin{matrix}A & A \\ 0 & A \end{matrix} ) the algebra of upper 2×2 triangular matrices

over A. In a forthcoming paper, Yu-Ming Liu and the first author give two indecomposable Kalgebras A and B which are stably equivalent but not Morita equivalent, but for which T_{2}(A) is not stably equivalent to T_{2}(B). However, Chen and Sun ([11]) show that if A and B are singularly equivalent of Morita type, then also T_{2}(A) and T_{2}(B) are singularly equivalent of Morita type.

3. Singular equivalences of Morita type give adjoint pairs

Our aim is to prove analogous result of Dugas and Martinez-Villa [12, Theorem 2.7] for singular equivalences of Morita type. For a K-algebra A, denote by J(A) its Jacobson radical.

Theorem 3.1. Let K be a field and let A and B be finite dimensional indecomposable K-algebras. Suppose A and B are not of finite projective dimension as bimodules and suppose that A/J(A) and B/J(B) are separable over K. Let (_{A}M_{B},_{B}N_{A}) be a pair of bimodules inducing a singular equivalence of Morita type between A and B. Suppose that _{A}M_{B} is indecomposable as a bimodule, and suppose that Hom_{A^{op}}(_{A}M_{B},_{A}A_{A}) is projective as a B^{op}-module.

Then

BN_{A} ≃ Hom_{A^{op}}(_{A}M_{B},_{A}A_{A})

as B^{op} ⊗_{K} A-modules, and (− ⊗_{B} N, − ⊗_{A} M) is a pair of adjoint functors between the module categories mod(B) and mod(A).

Remark 3.2. Since a singular equivalence of Morita type induces an equivalence D_{sg}(A) ≃ D_{sg}(B) and \underline{mod}_{P^{<∞}}(A) ≃ \underline{mod}_{P^{<∞}}(B) it is clear that (− ⊗_{A} M, − ⊗_{B} N) is a pair of adjoint functors between D_{sg}(A) and D_{sg}(B), as well as between \underline{mod}_{P^{<∞}}(A) and \underline{mod}_{P^{<∞}}(B). Theorem 3.1 states that the functors form an adjoint pair between the module categories.

In order to prove Theorem 3.1, we shall use the following technical notion, motivated by Dugas and Martinez-Villa [12].

Definition 3.3. An A^{op} ⊗_{K} B-module _{A}U_{B} is called strongly right nonsingular, if for each Amodule T_{A}, the B-module Ω_{A}^{n}(T) ⊗_{A} U_{B} is projective for n >> 0.

Lemma 3.4. Let K be a field and let A and B be finite dimensional K-algebras.

(i) Let _{A}U_{B} be a bimodule which is projective as a left and as a right module. Then _{A}U_{B} is strongly right nonsingular if and only if for each A-module T_{A}, T ⊗_{A} U_{B} has finite projective dimension.

(ii) Objects in P^{<∞}(A^{op} ⊗_{K} B) which are projective as a left and as a right modules are strongly right nonsingular. In particular, for a singular equivalence of Morita type induced by the pair of bimodules (_{A}M_{B},_{B}N_{A}), so that M ⊗_{B} N ≃ A ⊕ X in mod(A ⊗_{K} A^{op}) and N ⊗_{A} M ≃ B ⊕ Y in mod(B ⊗_{K} B^{op}), the two bimodules _{A}X_{A} and _{B}Y_{B} are strongly right nonsingular bimodules.

(iii) Let A be an algebra such that A/J(A) is separable over K. If the A^{e}-module A is not in

P^{<∞}(A^{e}), then the bimodule _{A}A_{A} is not strongly right nonsingular. (iv) Let A and B be finite dimensional indecomposable K-algebras which are not of finite projec-

tive dimension as bimodules and such that A/J(A) and B/J(B) are separable over K. Let (_{A}M_{B},_{B}N_{A}) be a pair of bimodules inducing a singular equivalence of Morita type between A and B. Then the two bimodules _{A}M_{B} and _{B}N_{A} are not strongly right nonsingular.

(v) A direct summand of a strongly right nonsingular bimodule is also strongly right nonsingular. The direct sum of two right strongly non singular bimodules is also strongly right nonsingular.

Proof (i). Suppose that for each A-module T_{A}, T ⊗_{A} U_{B} has finite projective dimension. Then for an A-module T_{A}, take a minimal projective resolution

⋯ → P_{n} → P_{n−1} → ⋯ → P_{1} → P_{0} → T_{A} → 0

and apply − ⊗_{A} U_{B}. The result is a complex of B-modules

⋯ → P_{n} ⊗_{A} U_{B} → P_{n−1} ⊗_{A} U_{B} → ⋯ → P_{1} ⊗_{A} U_{B} → P_{0} ⊗_{A} U_{B} → T ⊗_{A} U_{B} → 0.

This complex is actually exact, as _{A}U is projective. For n ≥ 1, we have an exact sequence

0 → Ω_{A}^{n}(T_{A}) ⊗_{A} U_{B} → P_{n−1} ⊗_{A} U_{B} → ⋯ → P_{1} ⊗_{A} U_{B} → P_{0} ⊗_{A} U_{B} → T ⊗_{A} U_{B} → 0.

Note that for 0 ≤ i ≤ n − 1, P_{i} ⊗_{A} U_{B} is projective, as U_{B} is projective. Since T ⊗_{A} U_{B} has finite projective dimension, by Schanuel’s Lemma, for n >> 0 we get that Ω_{A}^{n}(T) ⊗_{A} U_{B} is projective as a B-module. This proves that _{A}U_{B} is strongly right nonsingular.

Conversely, suppose that_{A}U_{B} is strongly right nonsingular. Take a minimal projective resolution

⋯ → P_{n} → P_{n−1} → ⋯ → P_{1} → P_{0} → T_{A} → 0

and apply − ⊗_{A} U_{B} to get a complex

0 → Ω_{A}^{n}(T) ⊗_{A} U_{B} → P_{n−1} ⊗_{A} U_{B} → ⋯ → P_{1} ⊗_{A} U_{B} → P_{0} ⊗_{A} U_{B} → T ⊗_{A} U_{B} → 0

of B-modules. This complex is exact, as_{A}U is projective. As for n >> 0, we have that Ω_{A}^{n}(T)⊗_{A}U_{B} is projective, T ⊗_{A} U_{B} has finite projective dimension.

We shall use (i) in the proof of (ii)-(iv).

(ii). Let _{A}U_{B} be a bimodule of finite projective dimension which is projective as left and as right module. Then there exists an exact sequence of A^{op} ⊗_{K} B-modules

0 → P_{n} → P_{n−1} → ⋯ → P_{0} → U → 0,

where for any 0 ≤ i ≤ n, P_{i} is a projective A^{op} ⊗_{K} B-module. As _{A}U is projective, the above sequence splits as exact sequence of left modules. So if we apply T_{A} ⊗_{A} −, it remains exact. Observe that all the B-modules (T ⊗_{A} P_{i})_{B} are projective and thus the B-module (T ⊗_{A} U)_{B} has finite projective dimension. We have proved that _{A}U_{B} is strongly right nonsingular. The second statement follows from the first one by observing that the two bimodules _{A}X_{A} and _{B}Y_{B} are projective as left and as right modules.

(iii). Suppose _{A}A_{A} is strongly right nonsingular. Then by (i) for each right A-module T_{A}, the module T_{A} ≃ T ⊗_{A} A_{A} is of finite projective dimension. Therefore A has finite global dimension and by [6, Section 1], we have A ∈ P^{<∞}(A^{e}). This proves the statement. Note that the relevant conclusion from [6, Section 1] is shown only under the hypothesis that A/J(A) is separable.

(iv). For each right B-module T_{B} we get isomorphisms of B-modules

T ⊗_{B} N ⊗_{A} M_{B} ≃ T ⊗_{B} (_{B}B_{B} ⊕_{B}Y_{B}) ≃ T_{B} ⊕ (T ⊗_{B} Y_{B}).

If _{A}M_{B} is strongly right nonsingular, by (i) T ⊗_{B} N ⊗_{A} M_{B} has finite projective dimension as a right B-module, and thus T_{B} has finite projective dimension. As in (iii), this implies that the B^{e}-module B is an object of P^{<∞}(B^{e}), which is a contradiction to the hypothesis on B.

The case of _{B}N_{A} is similar.

(v) is trivial.

□

Remark 3.5. In [6, Section 1] an example is given showing that we do need the hypothesis in (ii) and (iii) that A/J(A) and B/J(B) are separable over K.

Proof of Theorem 3.1 Denote _{B}M̌_{A} := Hom_{A^{op}}(_{A}M_{B},_{A}A_{A}) to simplify the notation. Then (− ⊗_{B} M̌_{A}, − ⊗_{A} M_{B}) is an adjoint pair of functors between mod(B) and mod(A), because

Hom_{A}(_{B}M̌_{A},_{A}A_{A}) = Hom_{A}(Hom_{A}(_{A}M_{B},_{A}A_{A}),_{A}A_{A}) ≃ _{A}M_{B},

as _{A}M is finitely generated projective. This pair of adjoint functors can be defined on D^{b}(mod(B)) and D^{b}(mod(A)), as _{B}M̌ and _{A}M are finitely generated projective modules. They induce functors between D_{sg}(A) and D_{sg}(B) since −⊗_{A}M_{B} maps K^{b}(proj(A)) to K^{b}(proj(B)), and since −⊗_{B} M̌_{A} maps K^{b}(proj(B)) to K^{b}(proj(A)).

Moreover, −⊗_{A} M_{B} and −⊗_{B} M̌_{A} induce functors between \underline{mod}_{P^{<∞}}(A) and \underline{mod}_{P^{<∞}}(B) since P^{<∞}(A) ⊗_{A} M_{B} belongs to P^{<∞}(B) and likewise for − ⊗_{B} M̌_{A}.

Let η : id_{mod(B)} ⟶ −⊗_{B} M̌ ⊗_{A}M_{B} be the unit of the adjoint pair (−⊗_{B} M̌_{A}, −⊗_{A}M_{B}) between mod(B) and mod(A) and let η_{B} : B → _{B}M̌ ⊗_{A} M_{B} be its evaluation on B. As _{B}M̌ ⊗_{A} M_{B} ≃ End_{A}(_{A}M_{B}) as B^{e}-modules, η_{B} identifies with the structure map of the right B-module structure of M. By Lemma 2.4, η_{B} is injective and we can form a short exact sequence as follows:

0 → _{B}B_{B} →^{η_{B}} _{B}M̌ ⊗_{A} M_{B} → _{B}U_{B} → 0 (∗).

Applying _{A}M_{B} ⊗_{B} − to the exact sequence (*) gives the exact sequence

0 → _{A}M_{B} →^{Id_{M} ⊗η_{B}} _{A}M ⊗_{B} M̌ ⊗_{A} M_{B} → _{A}M ⊗_{B} U_{B} → 0.

Now it is easy to see that the monomorphism Id_{M} ⊗ η_{B} is split by the bimodule map

AM ⊗_{B} M̌ ⊗_{A} M ≃ _{A}M ⊗_{B} End_{A}(_{A}M_{B}) → _{A}M_{B}

where the second map is the evaluation map. Hence

AM ⊗_{B} M̌ ⊗_{A} M_{B} ≃ _{A}M_{B} ⊕ (_{A}M ⊗_{B} U_{B}).

Claim 1: U_{B} is projective and _{B}U_{B} is strongly right nonsingular.

We shall use this claim for the moment and and give the proof of Claim 1 just after having finished the proof of Theorem 3.1.

Applying − ⊗_{B} N to the isomorphism

AM ⊗_{B} M̌ ⊗_{A} M_{B} ≃ _{A}M_{B} ⊕ (_{A}M ⊗_{B} U_{B})

gives

AM ⊗_{B} M̌ ⊗_{A} M ⊗_{B} N_{A} ≃ (_{A}M_{B} ⊕ (_{A}M ⊗_{B} U_{B})) ⊗_{B} N_{A}
≃ (_{A}M ⊗_{B} N_{A}) ⊕ (_{A}M ⊗_{B} U ⊗_{B} N_{A})
≃ _{A}A_{A} ⊕_{A}X_{A} ⊕ (_{A}M ⊗_{B} U ⊗_{B} N_{A}).

But we also get

AM ⊗_{B} M̌ ⊗_{A} M ⊗_{B} N_{A} ≃ _{A}M ⊗_{B} M̌ ⊗_{A} (M ⊗_{B} N_{A})
≃ _{A}M ⊗_{B} M̌ ⊗_{A} (_{A}A_{A} ⊕_{A}X_{A})
≃ (_{A}M ⊗_{B} M̌_{A}) ⊕ (_{A}M ⊗_{B} M̌ ⊗_{A} X_{A}).

Claim 2: _{A}M ⊗_{B} U ⊗_{B} N_{A} and _{A}M ⊗_{B} M̌ ⊗_{A} X_{A} are strongly right nonsingular.

Again we shall use this claim for the moment and give the proof of Claim 2 just after having finished the proof of Theorem 3.1.

The indecomposable A^{e}-module A is not strongly right nonsingular by Lemma 3.4 part (iii). The Krull-Schmidt theorem shows that the A^{e}-module A is a direct factor of M ⊗_{B} M̌ or of AM ⊗_{B} M̌ ⊗_{A} X_{A}. Claim 2 shows that _{A}M ⊗_{B} M̌ ⊗_{A} X_{A} is strongly right nonsingular, and hence all of its direct factors. Hence the A^{e}-module A is a direct factor of M ⊗_{B} M̌. This shows that there is an A^{e}-module X̃ such that

AM ⊗_{B} M̌_{A} ≃ _{A}A_{A} ⊕_{A}X̃_{A} (∗∗).

The bimodule _{A}X̃_{A} is strongly right nonsingular by Lemma 3.4 (ii) and (v), as _{A}X̃_{A} is a direct summand of _{A}X_{A} ⊕ (_{A}M ⊗_{B} U ⊗_{B} N_{A}).

Now we apply N ⊗_{A} − to (**) and get

BN ⊗_{A} M ⊗_{B} M̌_{A} ≃ _{B}N_{A} ⊕ (_{B}N ⊗_{A} X̃_{A}),

but

BN ⊗_{A} M ⊗_{B} M̌_{A} ≃ (_{B}B_{B} ⊕_{B}Y_{B}) ⊗_{B} M̌_{A} ≃ _{B}M̌_{A} ⊕ (_{B}Y ⊗_{B} M̌_{A}).

So

BN_{A} ⊕ (_{B}N ⊗_{A} X̃_{A}) ≃ _{B}M̌_{A} ⊕ (_{B}Y ⊗_{B} M̌_{A}).

Claim 3: _{B}N ⊗_{A} X̃_{A} and _{B}Y ⊗_{B} M̌_{A} are strongly right nonsingular; the B^{op} ⊗_{K} A-module _{B}M̌_{A} is indecomposable.

Again we shall use this claim for the moment and give the proof of Claim 3 just after having finished the proof of Theorem 3.1.

As in Lemma 3.4 (iv) the module _{B}N_{A} is not strongly right nonsingular. We hence obtain that the two indecomposable bimodules _{B}N_{A} and _{B}M̌_{A} are isomorphic.

□

Proof of Claim 1 As in the paragraph preceding the statement of Claim 1, we have an isomorphism of bimodules

AM ⊗_{B} M̌ ⊗_{A} M_{B} ≃ _{A}M_{B} ⊕ (_{A}M ⊗_{B} U_{B}).

Since M_{B} and M̌_{A} are projective, M ⊗_{B} U_{B} is projective as a right B-module and since M_{B} is a progenerator by Proposition 2.4, we see that U_{B} is projective.

Given a right B-module T_{B}, we apply T ⊗_{B} − to (*) and we get an exact sequence

T_{B} →^{η_{T}} T ⊗_{B} M̌ ⊗_{A} M_{B} → T ⊗_{B} U_{B} → 0,

where η_{T} = id_{T} ⊗_{B} η_{B}.

As η_{T} is an isomorphism in D_{sg}(B), there exists n >> 0 such that Ω^{n}(η_{T}) is an isomorphism in \underline{mod}(B). In fact, by [15, Example 2.3] or [2, Corollary 3.9(1)], given two B-modules V and W, we have

Hom_{D_{sg}(B)}(V, W) = \underrightarrow{lim} \underline{Hom}_{B}(Ω^{i}V, Ω^{i}W).

Suppose that a module homomorphism f : V → W is invertible in the singularity category. Then its inverse is induced from a module homomorphism g : Ω^{i}(W) → Ω^{i}(V). We see that Ω^{i}(f) ∘ g coincides with Id_{W} (resp. g ∘ Ω^{i}(f) coincides with Id_{V}) in the singularity category, so Ω^{n−i}(Ω^{i}(f)∘g) = Ω^{n}(f)∘Ω^{n−i}(g) coincides with Id_{Ω^{n}(N)} in \underline{Hom}_{B}(Ω^{n}(V), Ω^{n}(W)) for big enough n.

Let P_{∗} be the minimal projective resolution of T_{B} and let Q_{∗} be the minimal projective resolution of T ⊗_{B} M̌ ⊗_{A} M_{B}. As P_{∗} ⊗_{B} M̌ ⊗_{A} M_{B} is also a projective resolution of T ⊗_{B} M̌ ⊗_{A} M_{B}, the Comparison Lemma gives a chain map f_{∗} : P_{∗} ⊗_{B} M̌ ⊗_{A} M_{B} → Q_{∗}. Therefore, we have a commutative diagram

\xymatrix{P_{∗} \ar[r] \ar[d]_{η_{P∗}} & T_{B} \ar[d]_{η_{T}} \\ P_{∗} ⊗_{B} M̌ ⊗_{A} M_{B} \ar[r] \ar[d]_{f_{∗}} & T ⊗_{B} M̌ ⊗_{A} M_{B} \ar[d]_{=} \\ Q_{∗} \ar[r] & T ⊗_{B} M̌ ⊗_{A} M_{B}}

Note that the induced map

Ω_{B}^{n}(T_{B}) →^{η_{Ω^{n}_{B} (T_{B})}} Ω_{B}^{n}(T_{B}) ⊗_{B} M̌ ⊗_{A} M_{B} →^{f_{n}} Ω_{B}^{n}(T ⊗_{B} M̌ ⊗_{A} M_{B})

is just Ω^{n}(η_{T}), which is an isomorphism as n is supposed to be large enough, as we have seen.

As f_{n} induces an isomorphism between Ω_{B}^{n}(T ⊗_{B} M̌ ⊗_{A} M_{B}) and Ω_{B}^{n}(T) ⊗_{B} M̌ ⊗_{A} M_{B} in \underline{mod}(B), we obtain that η_{Ω_{B}^{n}(T)} : Ω_{B}^{n}(T_{B}) → Ω_{B}^{n}(T) ⊗_{B} M̌ ⊗_{A} M_{B} is an isomorphism in \underline{mod}(B).

As we have an exact sequence of B-modules

Ω_{B}^{n}(T_{B}) →^{η_{Ω^{n}_{B} (T_{B})}} Ω_{B}^{n}(T_{B}) ⊗_{B} M̌ ⊗_{A} M_{B} → Ω_{B}^{n}(T_{B}) ⊗_{B} U_{B} → 0,

we deduce that η_{Ω_{B}^{n}(T_{B})} has projective cokernel. In fact, let S_{B} be an indecomposable direct summand of Ω_{B}^{n}(T_{B}). Then η_{Ω_{B}^{n}(T_{B})} is the direct sum of such η_{S} and η_{S} is an isomorphism in \underline{mod}(B). If S_{B} is projective, η_{S} is injective and η_{S} has projective cokernel, since η_{B} is injective with projective cokernel U_{B} by Claim 1. If S_{B} is not projective, then the fact that η_{S} is an isomorphism in \underline{mod}(B) implies that η_{S} has projective cokernel.

Since η_{Ω_{B}^{n}(T_{B})} has projective cokernel, Ω_{B}^{n}(T_{B}) ⊗_{B} U_{B} is projective and the module _{B}U_{B} is strongly right nonsingular.

□

Proof of Claim 2 Let T_{A} be an A-module. For n ≥ 1, Ω_{A}^{n}(T_{A}) ⊗_{A} M_{B} ≃ Ω_{B}^{n}(T ⊗_{A} M_{B}) ⊕ P_{B} with P_{B} projective. Then

Ω_{A}^{n}(T) ⊗_{A} M ⊗_{B} U ⊗_{B} N_{A} ≃ (Ω_{B}^{n}(T ⊗_{A} M_{B}) ⊗_{B} U ⊗_{B} N_{A}) ⊕ (P ⊗_{B} U ⊗_{B} N_{A}).

The A-module Ω_{B}^{n}(T ⊗_{A} M_{B}) ⊗_{B} U ⊗_{B} N_{A} is projective for n big enough, as _{B}U_{B} is strongly right nonsingular and that _{B}N_{A} is projective as a left and right module; the module P ⊗_{B} U ⊗_{B} N_{A} is projective since U_{B} is projective. We have proved that Ω_{A}^{n}(T) ⊗_{A} M ⊗_{B} U ⊗_{B} N_{A} is projective for n >> 0 and that _{A}M ⊗_{B} U ⊗_{B} N_{A} is strongly right nonsingular.

The fact that _{A}M ⊗_{B} M̌ ⊗_{A} X_{A} is strongly right nonsingular follows from the fact that _{A}X_{A} is in P^{<∞}(A^{e}) and that _{A}M ⊗_{B} M̌ ⊗_{A} X_{A} is projective as a left and right module.

□

Proof of Claim 3 The fact that _{B}N ⊗_{A} X̃_{A} is strongly right nonsingular follows from that _{B}N_{A} is projective as a left and right module and that _{A}X̃_{A} is strongly right nonsingular.

The fact that _{B}Y ⊗_{B} M̌_{A} is strongly right nonsingular follows from that _{B}Y_{B} is in P^{<∞}(B^{e}) and that _{B}M̌_{A} is projective as a left and right module.

Suppose M̌ = M̌_{1} ⊕ M̌_{2} as B^{op} ⊗_{K} A-modules. Then Hom_{A}(_{B}M̌_{A},_{A}A_{A}) ≃ _{A}M_{B} is indecomposable as B^{op} ⊗_{K} A-module implies that Hom_{A}(M̌_{1},_{A}A_{A}) = 0 or Hom_{A}(M̌_{2},_{A}A_{A}) = 0. But M_{B} is projective, and so this happens only if M_{1} = 0 or M_{2} = 0. Therefore _{B}M̌_{A} is indecomposable.

□

We obtain the analogous result to [12, Corollary 3.1].

Corollary 3.6. Under the same assumption of Theorem 3.1, suppose further that Hom_{B}(_{A}M_{B},_{B}B_{B}) is projective as an A-module, or Hom_{B^{op}}(_{B}N_{A},_{B}B_{B}) is projective as a left A-module. Then

BN_{A} ≃ Hom_{A^{op}}(_{A}M_{B},_{A}A_{A}) ≃ Hom_{B}(_{A}M_{B},_{B}B_{B})

and

AM_{B} ≃ Hom_{A}(_{B}N_{A},_{A}A_{A}) ≃ Hom_{B^{op}}(_{B}N_{A},_{B}B_{B}).

Moreover (M ⊗_{B} −, N ⊗_{A} −) and (N ⊗_{A} −, M ⊗_{B} −) are adjoint functors between mod(A^{op}) and mod(B^{op}), which induce pairs of equivalences of the corresponding singularity categories.

Finally (− ⊗_{A} M_{B}, − ⊗_{B} N_{A}) and (− ⊗_{B} N_{A}, − ⊗_{A} M_{B}) are adjoint functors between mod(A) and mod(B), which induce pairs of equivalences of the corresponding singularity categories.

Proof As a left (resp. right) adjoint to a given functor is unique up to isomorphisms, Theorem 3.1 shows that Hom_{A}(_{B}N_{A},_{A}A_{A}) ≃ _{A}M_{B} and in particular, Hom_{A}(_{B}N_{A},_{A}A_{A}) is projective as a right B-module.

On the other hand, if we suppose in Theorem 3.1 that Hom_{A}(_{B}N_{A},_{A}A_{A}) is projective as a right B-module instead of being projective for Hom_{A^{op}}(_{A}M_{B},_{A}A_{A}) as a left B-module, a dual proof as that of Theorem 3.1, by considering the functors (Hom_{A}(_{B}N_{A},_{A}A_{A}) ⊗_{B} −, N ⊗_{A} −) between left module categories mod(B^{op}) and mod(A^{op}), gives that

AM_{B} ≃ Hom_{A}(_{B}N_{A},_{A}A_{A})

as A^{op} ⊗_{K} B-modules, and (M ⊗_{B} −, N ⊗_{A} −) is a pair of adjoint functors between the module categories mod(A^{op}) and mod(B^{op}). As in the first paragraph, we see that Hom_{A^{op}}(_{A}M_{B},_{A}A_{A}) ≃ BN_{A} and in particular, Hom_{A^{op}}(_{A}M_{B},_{A}A_{A}) is projective as a left B-module.

This shows that the the condition that Hom_{A^{op}}(_{A}M_{B},_{A}A_{A}) is projective as a left B-module and the condition that Hom_{A}(_{B}N_{A},_{A}A_{A}) is projective as a right B-module are equivalent. Furthermore, under these two equivalent conditions, we know that

(i) _{B}N_{A} ≃ Hom_{A^{op}}(_{A}M_{B},_{A}A_{A}) and _{A}M_{B} ≃ Hom_{A}(_{B}N_{A},_{A}A_{A}).

(ii) (M ⊗_{B} −, N ⊗_{A} −) is a pair of adjoint functors between mod(B^{op}) and mod(A^{op}), which

induce pairs of equivalences of the corresponding singularity categories. (iii) (− ⊗_{B} N_{A}, − ⊗_{A} M_{B}) is a pair of adjoint functors between mod(B) and mod(A), which

induce pairs of equivalences of the corresponding singularity categories. A dual proof of the above argument shows that the condition that Hom_{B}(_{A}M_{B},_{B}B_{B}) is pro-

jective as an A-module, and the condition that Hom_{B^{op}}(_{B}N_{A},_{B}B_{B}) is projective as an A-module, are equivalent; under these two conditions, we have

(i) _{B}N_{A} ≃ Hom_{B}(_{A}M_{B},_{B}B_{B}) and _{A}M_{B} ≃ Hom_{B^{op}}(_{B}N_{A},_{B}B_{B}).

(ii) (_{B}N ⊗_{A} −,_{A}M ⊗_{B} −) is a pair of adjoint functors between mod(A^{op}) and mod(B^{op}), which

induce pairs of equivalences of the corresponding singularity categories. (iii) (− ⊗_{A} M_{B}, − ⊗_{B} N_{A}) is a pair of adjoint functors between mod(A) and mod(B), which

induce pairs of equivalences of the corresponding singularity categories.

□

Let ν_{A} := Hom_{K}(Hom_{A}(−, A), K) be the Nakayama functor on mod(A). If Q_{A} is a projective A-module, then ν(Q_{A}) is an injective A-module, and if I_{A} is an injective A-module, then ν(I_{A}) is a projective A-module.

Lemma 3.7. Under the same assumption of Theorem 3.1, if I is injective as a B-module, then M ⊗_{B} I is injective as an A-module. Moreover (M ⊗_{B} −) ∘ ν_{B} ≃ ν_{A} ∘ (M ⊗_{B} −).

Proof We know that N ≃ Hom_{A}(M, A) and that N ⊗_{A} − is (left and) right adjoint to M ⊗_{B} −. Hence for an injective A-module I we get

Hom_{B}(−, N ⊗_{A} I) ≃ Hom_{A}(M ⊗_{B} −, I)

by Corollary 3.6. Moreover M ⊗_{B} − is exact since M is projective as a B-module. Hom_{A}(−, I) is exact since I is injective as an A-module. Therefore Hom_{B}(−, N ⊗_{A} I) is exact as a functor B − mod ⟶ (A − mod)^{op}, and we get therefore that N ⊗_{A} I is injective.

We have

Hom_{A}(M ⊗_{B} −, A) ≃ Hom_{B}(−, Hom_{A}(M, A))
≃ Hom_{B}(−, N)
≃ Hom_{B}(−, B) ⊗_{B} N

as right A modules, since N is projective as B-module. Hence,

ν_{A}(M ⊗_{B} −) = Hom_{K}(Hom_{A}(M ⊗_{B} −, A), K)
≃ Hom_{K}(Hom_{B}(−, B) ⊗_{B} N, K)
≃ Hom_{B}(N, Hom_{K}(Hom_{B}(−, B), K))
≃ Hom_{B}(N, B) ⊗_{B} Hom_{K}(Hom_{B}(−, B), K)
≃ M ⊗_{B} ν_{B}(−)

This shows the lemma.

□

Corollary 3.8. Under the same assumption of Theorem 3.1, the functor −⊗_{A}M_{B} sends projective injective A-modules to projective injective B-modules.

4. Singular equivalences of Morita type and Hochschild homology

In this section, we consider invariant property of Hochschild homology under singular equivalences of Morita type. For stable equivalences of Morita type, in [20], Yu-Ming Liu and Chang-Chang Xi proved that a stable equivalence of Morita type preserves Hochschild homology groups of positive degrees. Remark that by [21, Theorem 1.1] the invariance of degree zero Hochschild homology group under a stable equivalence of Morita type is equivalent to the famous Auslander-Reiten conjecture on the invariance of the number of non projective simple modules under stable equivalence.

We shall now prove that a singular equivalence of Morita type induces an isomorphism of Hochschild homology in positive degrees.

Theorem 4.1. Let K be a Noetherian commutative ring and let A and B be Noetherian K-algebras which are projective as K-modules. Suppose that (_{A}M_{B},_{B}N_{A}) induce a singular equivalence of Morita type.

(1) Then there is n_{0} ∈ ℕ so that HH_{n}(A) ≃ HH_{n}(B) for each n > n_{0}.

(2) If K is a field, and if A and B are finite dimensional, then HH_{n}(A) ≃ HH_{n}(B) for each n > 0.

Our proof of the first statement, inspired by [27, Section 1.2], is similar to that of [20, Theorem 4.4], which uses a change-of-rings argument. Notice that our argument is simpler than the proof in [20] and in fact works also for stable equivalences of Morita type. Our proof of the second statement makes use of transfer maps and is similar to that of [21, Remark 3.3]. Proof of Theorem 4.1.(1). Let 𝔹A be the bar resolution of A, that is

𝔹A : … ⟶ A^{⊗5} ⟶ A^{⊗4} ⟶ A^{⊗3} ⟶ A^{⊗2}(⟶ A ⟶ 0).

Then, we my apply N ⊗_{A} − ⊗_{A} M and obtain an exact sequence N ⊗_{A} 𝔹A ⊗_{A} M of B^{e}-modules:

… ⟶ N ⊗_{A} A^{⊗4} ⊗_{A} M ⟶ N ⊗_{A} A^{⊗3} ⊗_{A} M ⟶ N ⊗_{A} A^{⊗2} ⊗_{A} M(⟶ N ⊗_{A} M ⟶ 0). of B^{e}-modules, since M and N are projective on the right, resp. on the left.

The key observation is the following isomorphism of complexes

\begin{matrix}(_{A}M ⊗_{B} N_{A}) ⊗_{A^{e}} 𝔹A & ≃ & B ⊗_{B^{e}} (_{B}N ⊗_{A} 𝔹A ⊗_{A} M_{B}) \\ (m ⊗ n) ⊗ u & ↦ & 1 ⊗ (n ⊗ u ⊗ m). \end{matrix}

which is easily verified. Taking homology groups gives

HH_{n}(A) ⊕ T or_{n}^{A^{e}}(X, A) ≃ T or_{n}^{A^{e}}(M ⊗_{B} N, A) ≃ T or_{n}^{B^{e}}(B, N ⊗_{A} M) ≃ HH_{n}(B) ⊕ T or_{n}^{B^{e}}(B, Y)

for each n ≥ 0. When n is large, T or_{n}^{A^{e}}(X, A) ≃ 0 ≃ T or_{n}^{B^{e}}(B, Y), as X ∈ P^{<∞}(A^{e}) and Y ∈ P^{<∞}(B^{e}), we obtain that HH_{n}(A) ≃ HH_{n}(B) for n >> 0.

□

For the proof of Theorem 4.1.(2), let us recall some properties of transfer maps in Hochschild homology.

Let A and B be two algebras over a commutative ring k. Let M be an A-B-bimodule such that M_{B} is finitely generated and projective. Then we can define a transfer map t_{M} : HH_{n}(A) → HH_{n}(B) for each n ≥ 0. As we don’t need the construction of this map, we refer the reader to Bouc [3] (see also [21, 17] for a summary of Bouc’s results).

Proposition 4.2. [3, Section 3] Let A, B and C be k-algebras over a commutative ring k.

(1) If M is an A-B-bimodule and N is a B-C-bimodule such that M_{B} and N_{C} are finitely generated and projective, then we have t_{N} ∘ t_{M} = t_{M⊗_{B}N} : HH_{n}(A) → HH_{n}(C), for each n ≥ 0.

(2) Let

0 → L → M → N → 0

be a short exact sequence of A-B-bimodules which are finitely generated and projective as right B-modules. Then t_{M} = t_{L} + t_{N} : HH_{n}(A) → HH_{n}(B), for each n ≥ 0.

(3) Suppose that k is an algebraically closed field and that A and B are finite dimensional k-algebras. Then for a finitely generated projective A-B-bimodule P, the transfer map t_{P} : HH_{n}(A) → HH_{n}(B) is zero for each n > 0.

(4) Consider A as an A-A-bimodule by left and right multiplications, then t_{A} : HH_{n}(A) → HH_{n}(A) is the identity map for any n ≥ 0.

Proof of Theorem 4.1.(2). For n ≥ 0, we have transfer maps t_{M} : HH_{n}(A) → HH_{n}(B) and t_{N} : HH_{n}(B) → HH_{n}(A). By the above result,

t_{N} ∘ t_{M} = t_{M⊗_{B}N} = t_{A} + t_{X} = Id + t_{X}

as maps from HH_{n}(A) to itself.

Let \overline{K} be the algebraic closure of K and write

\overline{A} = A ⊗_{K} \overline{K},
\overline{B} = B ⊗_{K} \overline{K},
\overline{M} = M ⊗_{K} \overline{K},
\overline{N} = N ⊗_{K} \overline{K},
\overline{X} = X ⊗_{K} \overline{K},
\overline{Y} = Y ⊗_{K} \overline{K}.

Then one verifies easily that (\overline{A}\overline{M}\overline{B}, \overline{B}\overline{N}\overline{A}) induces a singular equivalence of Morita type between \overline{A} and \overline{B}, because

\overline{A}\overline{M} ⊗\overline{B} \overline{N}\overline{A} ≃ \overline{A}\overline{A}\overline{A} ⊕ \overline{A}\overline{X}\overline{A}

with \overline{X} ∈ P^{<∞}(\overline{A}^{e});

\overline{B}\overline{N} ⊗\overline{A} \overline{M}\overline{B} ≃ \overline{B}\overline{B}\overline{B} ⊕ \overline{B}\overline{Y} \overline{B}

with \overline{Y} ∈ P^{<∞}(\overline{B}^{e}). We also have t\overline{M} = t_{M} ⊗_{K} id\overline{K}.

Since \overline{X} ∈ P^{<∞}(\overline{A}^{e}), there is an exact sequence of \overline{A}^{e}-modules

0 → \overline{P}_{n} → ⋯ → \overline{P}_{0} → \overline{X} → 0

with \overline{P}_{0}, ⋯, \overline{P}_{n} projective. By the point (2)(3) of Proposition 4.2, for n > 0, we have t\overline{X} = ∑_{i=0}^{n}(−1)^{i}t_{\overline{P}_{i}} = 0 as a homomorphism from HH_{n}(\overline{A}) → HH_{n}(\overline{A}), and thus t_{X} = 0 : HH_{n}(A) → HH_{n}(A) for n > 0. This shows that

t_{N} ∘ t_{M} : HH_{n}(A) → HH_{n}(A)

and

t_{M} ∘ t_{N} : HH_{N}(B) → HH_{n}(B)

are isomorphisms for n > 0. We deduce that

t_{M} : HH_{n}(A) → HH_{n}(B)

is an isomorphism for n > 0.

□

Remark 4.3. Finally we briefly mention what is known in this context about invariance of Hochschild cohomology under stable equivalence of Morita type and under singular equivalence of Morita type.

Chang-Chang Xi proved in [26, Theorem 4.2] that a stable equivalence of Morita type between Artin algebras preserves the Hochschild cohomology groups of positive degrees, generalising a previous result of Zygmunt Pogorzały [24, Theorem 1.1] for selfinjective algebras. Sheng-Yong Pan and the first author further showed in [23] the invariance of stable Hochschild cohomology rings under stable equivalences of Morita type.

Chen and Sun prove in [11] that Tate-Hochschild cohomology rings of Gorenstein algebras are preserved under singular equivalences of Morita type. A careful study of the proof of [26, Theorem 4.2] shows that the proof of [26, Theorem 4.2] works for singular equivalences of Morita type. We obtain from this study that a singular equivalence of Morita type preserves Hochschild cohomology groups of large degrees. However, we don’t know the algebra structure.

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Guodong Zhou Department of Mathematics East China Normal University Dong Chuan Road 500 Min Hang District Shanghai 200241 P.R.China

   E-mail address: gdzhou@math.ecnu.edu.cn

Alexander Zimmermann Université de Picardie, Département de Mathématiques et LAMFA (UMR 7352 du CNRS), 33 rue St Leu, F-80039 Amiens Cedex 1, France

   E-mail address: alexander.zimmermann@u-picardie.fr