HIGMAN IDEAL, STABLE HOCHSCHILD HOMOLOGY AND AUSLANDER-REITEN CONJECTURE

YUMING LIU, GUODONG ZHOU^{∗} AND ALEXANDER ZIMMERMANN

Abstract. Let A and B be two finite dimensional algebras over an algebraically closed field, related to each other by a stable equivalence of Morita type. We prove that A and B have the same number of isomorphism classes of simple modules if and only if their 0-degree Hochschild Homology groups HH_{0}(A) and HH_{0}(B) have the same dimension. The first of these two equivalent conditions is claimed by the Auslander-Reiten conjecture. For symmetric algebras we will show that the Auslander-Reiten conjecture is equivalent to other dimension equalities, involving the centers and the projective centers of A and B. This motivates our detailed study of the projective center, which now appears to contain the main obstruction to proving the Auslander-Reiten conjecture for symmetric algebras. As a by-product, we get several new invariants of stable equivalences of Morita type.

1. Introduction

Equivalences between stable module categories occur in many places in representation theory and algebra and are closed related to derived equivalences. A result by Rickard ([32]) and Keller-Vossieck ([12]) says that a derived equivalence between self-injective algebras induces an equivalence of their stable categories. This result actually says that a stable equivalence induced from a derived equivalence between self-injective algebras has a special form. More precisely, suppose that A and B are two self-injective algebras which are derived equivalent, then there are two bimodules _{A}M_{B} and _{B}N_{A} which are projective as left modules and as right modules such that we have bimodule isomorphisms:

AM ⊗_{B} N_{A} ≅ _{A}A_{A} ⊕_{A}P_{A}, _{B}N ⊗_{A} M_{B} ≅ _{B}B_{B} ⊕_{B}Q_{B}

where _{A}P_{A} and _{B}Q_{B} are projective bimodules. Tensoring with M or N provides stable equivalences, which Broué ([4]) called “stable equivalences of Morita type”. Many stable equivalences constructed in modular representation theory happen to be of Morita type (see, for example, [4], [19], [36]).

The Auslander-Reiten conjecture asserts that if the stable categories of two Artin algebras are equivalent, then the algebras have the same number of isomorphism classes of non-projective simple modules (cf. [1, Page 409]). This conjecture has been studied by many authors (see, for instance, [26], [27], [30], [37]) and it has been established for some special classes of algebras. Based on a complete classification of representation-finite self-injective algebras, the conjecture was solved for this class of algebras in [34]. Subsequently, Martinez-Villa ([26]) proved the conjecture for all algebras of finite representation type. Pogorzarły ([30]) gave a proof of this conjecture for self-injective special biserial algebras, and Tang ([37]) proved it for some class of radical cube zero self-injective algebras. All of these proofs rely heavily on the knowledge of the Auslander-Reiten quiver of the algebras in question. Martinez-Villa ([27]) reduced the general problem to the case where both algebras are self-injective.

2 YUMING LIU, GUODONG ZHOU^{∗} AND ALEXANDER ZIMMERMANN

If two Artin algebras are derived equivalent, then they have isomorphic Hochschild (co)homology groups HH_{∗} and HH^{∗} for ∗ ≥ 0. If two Artin algebras are stably equivalent of Morita type, then they have the same global dimension and isomorphic Hochschild (co)homology groups HH_{∗} and HH^{∗} for ∗ ≥ 1 ([31],[22],[39]). In degree zero, it is only known that the stable centers are isomorphic. The stable center is a quotient of the 0-degree Hochschild cohomology, that is, of the center. It is not known whether the 0-degree Hochschild (co)homology groups HH_{0} and HH^{0} are preserved by a stable equivalence of Morita type. In this article, we will show that roughly speaking, the invariance of the dimension of the 0-degree Hochschild (co)homology group under stable equivalences of Morita type is equivalent to the Auslander-Reiten conjecture. More precisely, we will prove the following

∗ Corresponding author.

Mathematics Subject Classification(2010): 16G10, 16E40.

Keywords: Auslander-Reiten conjecture; Higman ideal; Projective center; Stable equivalence of Morita type; Stable Hochschild homology; Transfer map.

Date: version of November 19, 2010.

The first author is supported by Marie Curie Fellowship IIF. The second and the third authors were supported by a German-French grant “PROCOPE” of the DAAD, respectively a French-German grant “partenariat Hubert Curien PROCOPE dossier 14765WB”. The second author benefits also from financial support via postdoctoral fellowship from the network ”Representation theory of algebras and algebraic Lie theory” and the DAAD.

Theorem 1.1. Let k be an algebraically closed field. Let A and B be two finite dimensional kalgebras which are stably equivalent of Morita type. Then the following statements are equivalent.

(1) A and B have the same number of isomorphism classes of simple modules.

(2) The 0-degree Hochschild homology groups HH_{0}(A) and HH_{0}(B) have the same dimension

(over the ground field k). If A and B have no semisimple direct summands, these two conditions are further equivalent to the following

(3) A and B have the same number of non-isomorphic non-projective simple modules.

We remark that there are also many stable equivalences of Morita type between non-self-injective algebras ([23], [22], [24]). We will give an example showing that the assumption of the stable equivalence to be of Morita type cannot be dropped; there are stably equivalent algebras that have non-isomorphic 0-degree Hochschild homology groups.

Denote by Z(A) the center of an algebra A. It can be considered as the set of homomorphisms as A^{e} := A ⊗_{k} A^{op}-modules from A to A, that is, Z(A) = Hom_{A^{e}}(A, A). The projective center Z^{pr}(A) ⊂ Z(A) is, by definition, the subset of homomorphisms which factor through projective A^{e}-modules. The stable center Z^{st}(A) is defined to be the quotient Z(A)/Z^{pr}(A).

For symmetric algebras, we get further equivalent versions of the Auslander-Reiten conjecture, refining the characterisation given in Theorem 1.1.

Corollary 1.2. Suppose that in the setup of Theorem 1.1, one of the algebras A or B is symmetric. Then the following statements are equivalent.

(1) A and B have the same number of isomorphism classes of simple modules.

(2) The 0-degree Hochschild homology groups HH_{0}(A) and HH_{0}(B) have the same dimension.

(3) The centers Z(A) and Z(B) have the same dimension.

(4) The projective centers Z^{pr}(A) and Z^{pr}(B) have the same dimension.

If A and B have no semisimple summands, these conditions are further equivalent to the following (5) A and B have the same number of non-isomorphic non-projective simple modules.

This result indicates that the main obstruction to the Auslander-Reiten conjecture lies in the projective center; this concept will be studied in detail in the first part of this article. The statements of the corollary and further details to be given later, in particular on Cartan matrices, may lead to applications in modular representation theory. In a subsequent paper of the first two authors joint with Steffen König ([15]), we continue the study in this paper and found some more invariants detecting the validity of the Auslander-Reiten conjecture. Some applications of the results of this paper are already achieved: In a recent paper [40] the last two authors proved the Auslander-Reiten conjecture for stable equivalences of Morita type between the algebras of dihedral, semidihedral or quaternion type in the sense of Erdmann ([7]). The main tool there, in particular for semidihedral and quaternion type, are Theorem 1.1 and its Corollary. Moreover, again using at essential points Theorem 1.1 and its corollary, in [41] the last two authors show that the Auslander-Reiten conjecture holds for a stable equivalence of Morita type between two indecomposable tame symmetric algebras with only periodic modules, as well as for indecomposable tame symmetric algebras of polynomial growth.

The key point of the proof of the main theorem is to define the concept of the 0-degree stable Hochschild homology group HH_{0}^{st}(A) which is analogous to its counterpart in Hochschild cohomology, the stable center Z^{st}(A) = Z(A)/Z^{pr}(A). This stable Hochschild homology group HH_{0}^{st}(A) is a subspace of the usual 0-degree Hochschild homology group HH_{0}(A) and its definition uses the Hattori-Stallings trace map. We then prove that HH_{0}^{st}(A) is invariant under stable equivalences of Morita type. Moreover, we deduce the equality

dimHH_{0}^{st}(A) + rank_{p}C_{A} = dimHH_{0}(A),

where C_{A} is the Cartan matrix and where rank_{p}C_{A} is its p-rank, that is, the rank of C_{A} over k. By analyzing the stable Grothendieck group, we are able to prove that the Auslander-Reiten conjecture for stable equivalences of Morita type is equivalent to the invariance of the p-rank of the Cartan matrices. The theorem thus follows.

We will also study the notions of the projective center and the p-rank of the Cartan matrix for Frobenius algebras. More precisely, let A be a Frobenius k-algebra with a non-degenerate associative bilinear form ( , ) : A × A → k. Let {a_{1}, ⋯, a_{n}} and {b_{1}, ⋯, b_{n}} be a pair of dual bases of A, that is, (a_{i}, b_{j}) = δ_{ij}. We define two linear maps τ : A → A by τ(x) = ∑_{i=1}^{n} b_{i}xa_{i} and θ : A → A by θ(x) = ∑ a_{i}xb_{i}, respectively.

Proposition 1.3. Suppose that k is an algebraically closed field and that A is a Frobenius k-algebra. Then we have the following.

(1) The projective center Z^{pr}(A) is equal to the image Im(τ) of the above map τ.

(2) The p-rank of the Cartan matrix C_{A} is equal to the dimension of the image Im(θ) of the above map θ.

(3) ^{⊥}Im(θ)/K(A) = HH_{0}^{st}(A), where ^{⊥}Im(θ) is the left orthogonal of Im(θ) in A and K(A) is the commutator subspace of A.

This article is organized as follows. In the second section, we study the notion of the projective center, or equivalently, the Higman ideal. We define the stable Hochschild homology group in the third section. The fourth section contains an analysis of the stable Grothendieck groups. The main theorem and its corollary are proved in the fifth section. The last section contains an alternative proof of the main result in case that the ground field has positive characteristic.

2. Higman ideal and projective center

Throughout this paper, we denote by k a commutative ring. In certain cases we shall need to restrict the nature of k further. All algebras considered are Noetherian k-algebras with identity which are k-free of finite rank as k-module. When k is a field, they are just finite dimensional k-algebras over k. By a module over a k-algebra, we always mean a finitely generated unitary left module, unless stated otherwise. Given a k-algebra A, we give the list of some usual notations.

• A-mod the category of left A-modules

• A-\underline{mod} the stable category of A-mod

• l(A) the number of isomorphism classes of simple A-modules

• Z(A) the center of A

• J(A) the Jacobson radical of A

• Soc(A) the socle of A

• R(A) := Soc(A) ∩ Z(A) the Reynolds ideal of A

• A^{e} := A ⊗_{k} A^{op} the enveloping algebra of A Recall that the stable category A-\underline{mod} is defined as follows: The objects of A-\underline{mod} are the same as those of A-mod, and the morphisms between two objects X and Y are given by the quotient kmodule \underline{Hom}_{A}(X, Y) = Hom_{A}(X, Y)/P(X, Y), where P(X, Y) is the k-submodule of Hom_{A}(X, Y) consisting of those homomorphisms from X to Y which factor through a projective A-module. Two k-algebras A and B are said to be stably equivalent if there is an equivalence F : A-\underline{mod} →B-\underline{mod} between the stable categories.

Note that we can identify any A-A-bimodule with an A^{e}-module. As we have a canonical k-algebra isomorphism End_{A^{e}}(A, A) ≃ Z(A) (f ↦ f(1)), for each A^{e}-module V, the group Ext_{A^{e}}^{1}(A, V) has a natural right Z(A)-module structure defined as follows. We interpret elements of Ext_{A^{e}}^{1}(A, V) as equivalence classes of extensions of V by A. Let ξ ∈ Ext_{A^{e}}^{1}(A, V) correspond to the class of a short exact sequence of A^{e}-modules

ξ : 0 ⟶ V ⟶ X ⟶^{φ} A ⟶ 0.

Given any a ∈ End_{A^{e}}(A, A), the class ξa corresponds to the upper short exact sequence of the pullback diagram:

ξa : 0 ⟶ V ⟶ X_{1} ⟶ A ⟶ 0
1↓  ↓  a↓
ξ : 0 ⟶ V ⟶ X ⟶^{φ} A ⟶ 0.

In analogy to the Higman ideal in the study of orders (see [10] and [35]), we give the following

Definition 2.1. Let k be a commutative ring and A be a k-algebra. Define the Higman ideal to be

H(A) := {a ∈ Z(A)| Ext_{A^{e}}^{1}(A, V) · a = 0, ∀ V ∈ A^{e}-mod}.

Remark 2.2. Clearly H(A) is an ideal of Z(A). Denote by D_{univ} the following universal exact sequence

0 ⟶ Ω_{A^{e}}(A) ⟶ A^{e} ⟶ A ⟶ 0,

where A^{e} ⟶ A is given by the multiplication map. Then it is easy to show that the definition of the Higman ideal can be simplified as follows

H(A) := {a ∈ Z(A)| D_{univ} · a = 0 ∈ Ext_{A^{e}}^{1}(A, Ω_{A^{e}}(A))}.

Set Z^{pr}(A) to be the subset of Z(A) consisting of A^{e}-homomorphisms from A to A which factor through a projective A^{e}-module. Clearly Z^{pr}(A) is an ideal of Z(A) and we call it the projective center of A. It is now easily observed that the Higman ideal is nothing else but the projective center Z^{pr}(A).

Proposition 2.3. Let k be a commutative ring and A a k-algebra. Then we have H(A) = Z^{pr}(A).

Proof In fact, define the k-linear homomorphism

μ : Hom_{A^{e}}(A, A^{e}) ⊗_{A^{e}} A ⟶ End_{A^{e}}(A), μ(f ⊗ a) : a′ ↦ f(a′)a

where a′ ∈ A, f ∈ Hom_{A^{e}}(A, A^{e}), a ∈ A. Using [6, Proposition 29.15] by taking M = A and Λ = A^{e}, we see that the image of μ coincides with H(A).

Since we have a canonical k-isomorphism

Hom_{A^{e}}(A, A^{e}) ⊗_{A^{e}} Hom_{A^{e}}(A^{e}, A) ≃ Hom_{A^{e}}(A, A^{e}) ⊗_{A^{e}} A,

then

H(A) = Im(μ) = {g ∘ f ∈ End_{A^{e}}(A) | f ∈ Hom_{A^{e}}(A, A^{e}), g ∈ Hom_{A^{e}}(A^{e}, A)}.

Since any f ∈ Hom_{A^{e}}(A, A) factors through a projective A^{e}-module if and only if f factors through the regular A^{e}-module A^{e}, we know that H(A) is just another formulation of the projective center Z^{pr}(A).

□

For a symmetric k-algebra A over a field k with symmetrizing bilinear form ( , ), Héthelyi et al. (see [9]) have defined the Higman ideal H(A) as the image of the following map

τ : A → A, x ↦ ∑_{i=1}^{n} b_{i}xa_{i},

where a_{1}, ⋯, a_{n} and b_{1}, ⋯, b_{n} are a pair of dual bases of A. This definition can be generalized to the case of a Frobenius k-algebra over a field k. Recall that a Frobenius k-algebra A is endowed with a non-degenerate associative bilinear form ( , ): A × A → k. Let a_{1}, ⋯, a_{n} and b_{1}, ⋯, b_{n} be a pair of dual bases of A defined by the relations (a_{i}, b_{j}) = δ_{ij}, where δ_{ij} is the Kronecker symbol. Then it is easy to see that the image of the above map τ is independent of the choice of dual bases. So we can define the Higman ideal for a Frobenius k-algebra A as in [9]. Note that for non-symmetric Frobenius k-algebras, we can not reverse the order of the dual bases in the definition of τ : A → A in general (see Example 2.9).

Next we show that the above definition for Frobenius k-algebras coincides with our Definition 2.1. We are grateful to the referee who suggested the following proof which is much simpler than the one proposed in the original version.

Proposition 2.4. Suppose that k is a field and that A is a Frobenius k-algebra. Then the Higman ideal H(A) in Definition 2.1 is equal to the image Im(τ) of the above map τ.

Proof A key observation is that the map f : A → A ⊗_{k} A sending 1 to ∑_{i} b_{i} ⊗ a_{i} is an injective homomorphism of bimodules. In fact, the injectivity is easy as the elements a_{i} form a basis and the bilinear form is non degenerate. Suppose that for a ∈ A, ab_{i} = ∑_{j} λ_{ij}b_{j}, then it is easy to see that a_{i}a = ∑_{j} λ_{ij}a_{j}. So

∑_{i} ab_{i} ⊗ a_{i} = ∑_{ij} λ_{ij}b_{j} ⊗ a_{i} = ∑_{ij} λ_{ji}b_{i} ⊗ a_{j} = ∑_{i} b_{i} ⊗ a_{i}a.

As A is self-injective, then each homomorphism of bimodules from A to itself factoring through a projective bimodule actually factor through f. Notice furthermore that any bimodules homomorphism A ⊗_{k} A → A is of the form g_{x} : b ⊗ a ↦ bxa for some x ∈ A. Therefore, the image of 1 under the composition of f with g_{x} is just τ(x). This proves that Im(τ) = Z^{pr}(A).

□

Remark 2.5. In the case of a symmetric algebra A over a field k, Broué has proven that Z^{pr}(A) = Im(τ), where τ : A → A is defined as above ([5, 3.13 Proposition]).

For the completeness of our discussion, we should mention that Héthelyi et al. (see [9]) obtained another description of the Higman ideal for a symmetric k-algebra over a field k. To state it one needs some notations. Let A be a symmetric k-algebra over a field k. Then A is endowed with a non-degenerate associative symmetric bilinear form ( , ). With respect to this bilinear form, one can define the orthogonal V^{⊥} of a subspace V ⊆ A by V^{⊥} := {a ∈ A | (a, b) = 0, ∀ b ∈ V }. One proves easily ([17]) that J(A)^{⊥} = Soc(A) and K(A)^{⊥} = Z(A), where K(A) is the k-subspace of A spanned by all commutators ab − ba (a, b ∈ A). Moreover for the map τ : A → A, we have Im(τ) ⊆ R(A) and J(A) + K(A) ⊆ Ker(τ) ([9, Lemma 4.1]).

Let a_{1} = e_{1}, a_{2} = e_{2}, ⋯, a_{l} = e_{l} be a set of representatives of conjugacy classes of primitive idempotents in A (this means that Ae_{1}, ⋯, Ae_{l} are representatives for the isomorphism classes of indecomposable projective left A-modules). Let a_{l+1}, ⋯, a_{n} denote a basis of J(A)+K(A). Then we can prove that a_{1}, ⋯, a_{n} is a basis of A. Let b_{1}, ⋯, b_{n} be the dual basis. Then r_{1} = b_{1}, ⋯, r_{l} = b_{l} is a basis of (J(A) + K(A))^{⊥} = Soc(A) ∩ Z(A) = R(A). As J(A) + K(A) ⊂ Ker(τ), Im(τ) is k-spanned by τ(e_{i}) with 1 ≤ i ≤ l. It follows that we only need to know τ(e_{i}) (1 ≤ i ≤ l) for computing the Higman ideal.

Lemma 2.6. [9, Lemma 4.3] With the notations above, for 1 ≤ i ≤ l,

τ(e_{i}) = ∑_{j=1}^{l}(dime_{i}Ae_{j}) · r_{j}.

Before stating some interesting application of this lemma, we first recall the definition of the Cartan matrix. Let A be a finite dimensional k-algebra over a field k. Suppose that P_{1} = Ae_{1}, ⋯, P_{l} = Ae_{l} are representatives for the isomorphism classes of indecomposable projective left A-modules and that S_{1}, ⋯, S_{l} are the corresponding simple modules. By definition, the Cartan matrix C_{A} = (c_{ij})_{i,j=1}^{l} of A is a l × l integer matrix, where c_{ij} is given by the number of composition factors of P_{j} which are isomorphic to S_{i}. It is well known that c_{ij} = dime_{i}Ae_{j}/dimEnd_{A}(S_{i}) and therefore c_{ij} = dime_{i}Ae_{j} over an algebraically closed field k. Let p ≥ 0 be the characteristic of k. If C_{A} is the Cartan matrix of A, then we denote the p-rank of C_{A} by rank_{p}(C_{A}) which, by definition, is the rank of the Cartan matrix over k. Of course, when p = 0 this is just the usual rank of C_{A}.

Corollary 2.7. If A is a symmetric algebra over an algebraically closed field k of characteristic p ≥ 0, then rank_{p}(C_{A}) = dimH(A).

Remark 2.8. If A is a symmetric algebra over an algebraically closed field k of characteristic 0, then clearly rank_{p}(C_{A}) ≠ 0. This implies that the Higman ideal H(A) and therefore the projective center Z^{pr}(A) are always nonzero in this case.

Example 2.9. Let k be an algebraically closed field. Consider the 4-dimensional local Frobenius k-algebra

A = k⟨x, y⟩/(x^{2}, y^{2}, xy − ryx), 0 ≠ r ∈ k.

One can define a non-degenerate associative bilinear form over A by posing

(x, y) = (1, xy) = (xy, 1) = 1, (y, x) = 1/r, (1, 1) = (1, x) = (x, 1) = (1, y) = (y, 1) = 0

(for a construction of such bilinear forms for weakly symmetric algebras, see Proposition 3.1 of [11]). Note that A is symmetric if and only if r = 1. It follows that {a_{1} = 1, a_{2} = x, a_{3} = y, a_{4} = xy} and \{b_{1} = xy, b_{2} = y, b_{3} = rx, b_{4} = 1\} are a pair of dual bases of A. Then Im(τ) = ⟨(2 + r + \frac{1}{r})xy⟩, the subspace generated by (2 + r + \frac{1}{r})xy. However, if we define θ : A → A by x ↦ ∑_{i=1}^{n} a_{i}xb_{i}, then Im(θ) = ⟨4xy⟩. So, if char(k) ≠ 2, the two images are different for r = −1; if char(k) = 2, the two images are different for all r with r ≠ 1. On the other hand, the Cartan matrix of A is a 1 × 1 matrix (4). Notice that if char(k) ≠ 2 and r = −1 or if char(k) = 2 and r ≠ 1, then rank_{p}(C_{A}) ≠ dimH(A). This means that Corollary 2.7 is in general not true for non-symmetric algebras. Clearly in this example rank_{p}(C_{A}) = dimIm(θ). This is not a coincidence. In fact, this is true for arbitrary Frobenius algebras (Proposition 3.16).

Finally, we state some useful properties of the Higman ideal. We thank Shengyong Pan for pointing out an inaccuracy in the original proof of the following proposition.

Proposition 2.10. Suppose that B and C are two k-algebras over a commutative ring k.

(1) Put A = B × C. Then H(A) = H(B) × H(C) and this decomposition is compatible with the decomposition of centers Z(B × C) = Z(B) × Z(C).

(2) If B and C are two derived equivalent algebras which are projective over k as k-modules, then there is an algebra isomorphism of centers of B and C mapping H(B) to H(C).

Proof (1) By Proposition 2.3, the Higman ideal is equal to the projective center. But the latter clearly satisfies the decomposition property.

(2) Since the Higman ideal is equal to the projective center, it suffices to show that the projective center is a derived invariant. The latter has been proven by Broué [4, 4.4 Corollary] for self-injective algebras. In fact, the idea in Broué’s proof can be applied to arbitrary k-algebras which are projective over k. For the convenience of the reader, we shall give the detailed proof here.

By a result of Rickard [33] (see also [16]), there is a complex X^{•} ∈ D^{b}(B ⊗_{k} C^{op}) so that

X^{•} ⊗_{C}^{𝕃} − : D^{b}(C) ⟶ D^{b}(B)

is an equivalence as triangulated categories. The (bounded) complex X^{•} is called a two-sided tilting complex if it satisfies the following equalities in D^{b}(B^{e}) and in D^{b}(C^{e}) respectively,

X^{•} ⊗_{C}^{𝕃} Hom_{B}(X^{•}, B) ≃ B, Hom_{B}(X^{•}, B) ⊗_{B}^{𝕃} X^{•} ≃ C.

Now since B and C are projective over k, we may assume that X^{•} is a complex of bimodules each of which is projective as left-modules and projective as right-modules. Therefore the left derived tensor product can be replaced by the ordinary tensor product, which then is associative. It follows that tensor product with X^{•} from the left and with Hom_{B}(X^{•}, B) from the right is associative and

X^{•} ⊗_{C} − ⊗_{C} Hom_{B}(X^{•}, B) : D^{b}(C^{e}) ⟶ D^{b}(B^{e})

is an equivalence between the derived categories of enveloping algebras over C and B. Under this equivalence, the C^{e}-module C corresponds to B^{e}-module B, and the C^{e}-homomorphisms from C to C correspond to B^{e}-homomorphisms from B to B. Moreover, any C^{e}-homomorphism f from C to C factoring through a projective C^{e}-module corresponds to a B^{e}-homomorphism g from B to B factoring through a complex P^{•} of projective B^{e}-modules in D^{b}(B^{e}). But the homomorphism from P^{•} to B can be seen as a homomorphism in K^{b}(B^{e}) and it factors through the canonical multiplication map B^{e} ⟶ B. It follows that g is a homomorphism which factors through the regular B^{e}-module B^{e} and therefore g lies in Z^{pr}(B). Thus there is an algebra isomorphism of centers of B and C mapping Z^{pr}(B) to Z^{pr}(C).

□

3. The stable 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 M_{B} is isomorphic to a direct summand of some free right B-module B^{s} and there exist elements m_{i} ∈ M and φ_{i} ∈ Hom_{B}(M, B) with 1 ≤ i ≤ s such that for each m ∈ M, m = ∑_{i=1}^{s} m_{i}φ_{i}(m). We define the transfer map as follows.

t_{M} : A/K(A) → (M ⊗_{B} Hom_{B}(M, B))/[A, M ⊗_{B} Hom_{B}(M, B)] → B/K(B)

a ↦ ∑_{i=1}^{s} am_{i} ⊗ φ_{i} ↦ ∑_{i=1}^{s} φ_{i}(am_{i}),

where [A, M ⊗_{B} Hom_{B}(M, B)] = {ax − xa | a ∈ A, x ∈ M ⊗_{B} Hom_{B}(M, B)}. Notice that this map is just the composition of a map induced by the ring homomorphism A → End_{B}(M) ≃ M ⊗_{B} Hom_{B}(M, B) and a map induced by the Hattori-Stallings trace map End_{B}(M) → B/K(B).

We summarize some well known properties of the transfer map in the following

Lemma 3.1. ([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}.

(2) Let

0 → L → M → N → 0

be a short exact sequence of A-B-bimodules which are finitely generated and projective as right Bmodules. Then t_{M} = t_{L} + t_{N}.

(3) Consider A as an A-A-bimodule by left and right multiplications, then t_{A} is the identity map.

Example 3.2. (1) Let k be a field and A be a finite dimensional k-algebra. Let e ∈ A be an idempotent. Considering an indecomposable projective A-module Ae as an A-k-bimodule, we have the transfer map t_{Ae} : A/K(A) → k. Choose a basis {x_{1}, ⋯, x_{s}} of Ae. Then its dual basis {x_{1}^{∗}, ⋯, x_{s}^{∗}} ⊂ (Ae)^{∗} = Hom_{k}(Ae, k). By construction, t_{Ae}(a + K(A)) = ∑_{i=1}^{s} x_{i}^{∗}(ax_{i}). Observe that ∑_{i=1}^{s} x_{i}^{∗}(ax_{i}) is just the trace of the linear map from Ae to Ae induced by the left multiplication by a.

(2) Let A be a Frobenius k-algebra with a non-degenerate associative bilinear form ( , ) : A×A → k. Let {a_{i}} and {b_{i}} be a pair of dual bases of A, that is, (a_{i}, b_{j}) = δ_{ij}. Note that in this case {( , b_{i}) =: a_{i}^{∗}} ⊂ A^{∗} = Hom_{k}(A, k) gives the usual dual basis of {a_{i}}. We define a linear map θ : A → A by θ(a) = ∑ a_{i}ab_{i} for any a ∈ A (cf. next section). Let Ae ⊆ A be a projective A-module. We can choose {a_{i}} as a union of a basis of Ae and that of A(1−e). By (1), t_{Ae}(a+K(A)) = ∑_{a_{i}∈Ae} a_{i}^{∗}(aa_{i}). On the other hand, (a, θ(e)) = (a, ∑ a_{i}eb_{i}) = ∑(aa_{i}e, b_{i}) = ∑_{a_{i}∈Ae}(aa_{i}, b_{i}) = ∑_{a_{i}∈Ae} a_{i}^{∗}(aa_{i}). It follows that t_{Ae}(a + K(A)) = (a, θ(e)).

Remark 3.3. In the definition of trace map given above, we use an A-B-bimodule M which is finitely generated and projective as a right B-module. We can also define another transfer map t_{M} : A/K(A) → B/K(B) for a bimodule _{B}M_{A} which is finitely generated and projective as a left B-module. The definition is similar and we omit it.

Remark 3.4. The construction of transfer maps was generalized to higher degree Hochschild homology by Bouc ([3]) and Keller ([13]). We shall use and refer to this generalization in a subsequent paper [15]. For transfer maps in higher degree Hochschild homology groups, except the properties in Lemma 3.1, there is an additional one:

(4) Suppose that k is an algebraically closed field and that A and B are finite dimensional kalgebras. 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.

Let k be an algebraically closed field and let A and B be two finite dimensional k-algebras. Suppose that two bimodules M and N define a stable equivalence of Morita type between A and B by M ⊗_{B} N ≃ A ⊕ P, N ⊗_{A} M ≃ B ⊕ Q. Since t_{A} = 1_{HH_{n}(A)} and t_{B} = 1_{HH_{n}(B)}, the transfer maps t_{M} : HH_{n}(A) → HH_{n}(B) and t_{N} : HH_{n}(B) → HH_{n}(A) are mutually inverse group isomorphisms for all n > 0 by the properties (1)-(4). This gives a different approach for the result in [22, Theorem 4.4]. We also notice that Example 3 of [22, Section 5] is in fact not a counterexample for the noninvariance of the 0-degree Hochschild homology group since the commutator subspace K(A) = [A, A] of A is not an ideal in general.

From now on, k denotes a field and all algebras are supposed to be finite dimensional over k. In the following, we define a dual notion of the stable center and prove that it is invariant under

stable equivalences of Morita type. We remind the reader that the invariance of the stable centers under stable equivalences of Morita type has been shown by Broué ([4]).

Given a k-algebra A, recall that the center Z(A) of A is equal to the 0-degree Hochschild cohomology algebra HH^{0}(A) and the stable center Z^{st}(A) is defined to be the quotient algebra Z(A)/Z^{pr}(A) where Z^{pr}(A) is the projective center. Motivated by this fact, we introduce the following

Definition 3.5. Let A be a finite dimensional k-algebra over a field k with the decomposition _{A}A = ⨁_{i=1}^{r} Ae_{i}, where Ae_{i} (1 ≤ i ≤ r) are indecomposable projective A-modules. We define the (left) stable Hochschild homology group HH_{0}^{st}(A) of degree zero to be a subgroup of the 0-degree Hochschild homology group HH_{0}(A) = A/K(A), namely

HH_{0}^{st}(A) = ⋂_{i=1}^{r} Ker{t_{Ae_{i}} : A/K(A) → k},

where t_{Ae_{i}} is the transfer map determined by the projective A-k-bimodule Ae_{i}.

Remark 3.6. By Example 3.2 (1), we have

HH_{0}^{st}(A) = {a ∈ A | the trace of the map Ae_{i} → Ae_{i}(b ↦ ab) vanishes for any 1 ≤ i ≤ r}/K(A).

Remark 3.7. In the above definition, we used the transfer maps defined by the left indecomposable modules Ae_{i} considered as right k-modules. If we use the transfer maps defined by the right indecomposable projective modules e_{i}A as in Remark 3.3, then we have a notion of right stable Hochschild homology of degree zero. In the following, we shall mainly study the left Hochschild homology of degree zero. The corresponding properties of the right version are similar and the necessary modifications are left to the reader.

Theorem 3.8. Let A be a finite dimensional k-algebra over an algebraically closed field k of characteristic p ≥ 0. Then

dimHH_{0}^{st}(A) + rank_{p}(C_{A}) = dimHH_{0}(A).

Proof By Lemma 3.1 (2), if two indecomposable projective modules Ae_{i} and Ae_{j} are isomorphic, then t_{Ae_{i}} = t_{Ae_{j}} : A/K(A) → k. Since HH_{0}(A) is invariant under Morita equivalences, so is HH_{0}^{st}(A). Now all terms in the assertion are Morita invariant. So we can assume that A is basic. Now K(A) ⊆ J(A) and by the Wedderburn-Malcev theorem, A = J(A) ⊕ ⨁_{i} ke_{i}, where {e_{1}, ⋯, e_{l}} is a complete list of orthogonal primitive idempotents. So we can take a basis of A/K(A) consisting all e_{i} and the classes of some elements of J(A) in J(A)/K(A).

Since each element a in J(A) is nilpotent, for each j, the trace of the map Ae_{j} → Ae_{j}, b ↦ ab is zero; now for 1 ≤ i, j ≤ l, the trace of the map Ae_{j} → Ae_{j}, b ↦ e_{i}b is the dimension of the space e_{i}Ae_{j}, which is the Cartan invariant c_{ij}. We have proved that the matrix of the map t_{Ae_{i}} : A/K(A) → k under the chosen basis of A/K(A) is the vector (c_{1i}, c_{2i}, ⋯, c_{li}, 0, ⋯, 0) where c_{ij} is the (i, j)-entry of the Cartan matrix C_{A} and therefore the matrix of (t_{Ae_{1}}, ⋯, t_{Ae_{l}}) : A/K(A) → k^{l} is equal to (C_{A}^{T}, 0). The result thus follows.

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Corollary 3.9. Let A and B be two derived equivalent finite dimensional algebras over an algebraically closed field. Then

dimHH_{0}^{st}(A) = dimHH_{0}^{st}(B).

Proof This follows easily from Theorem 3.8 and the fact that both the p-rank of the Cartan matrix and the 0-degree Hochschild homology group are invariant under a derived equivalence. Indeed, a derived equivalence induces a mapping on the Grothendieck groups of the algebras. It is wellknown that this mapping commutes with the Cartan mapping. Hence, the same holds tensoring the Grothendieck groups with the base field over the integers. This shows that the p-rank of the Cartan matrix is a derived invariant.

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Remark 3.10. (1) The proof of Theorem 3.8 implies we have a short exact sequence

0 → (J(A) + K(A))/K(A) → HH_{0}^{st}(A) → Ker(C_{A}^{T}) → 0,

where Ker(C_{A}^{T}) is the kernel of the linear map from k^{l(A)} to itself defined by the transpose of the Cartan matrix. If we consider the right version of the stable Hochschild homology, then there is a short exact sequence

0 → (J(A) + K(A))/K(A) → HH_{0}^{st}(A) → Ker(C_{A}) → 0.

(2) It is well known that the 0-degree Hochschild homology group HH_{0}(A) has a natural Z(A)- module structure. At least for k an algebraically closed field, we can show that the stable Hochschild homology group HH_{0}^{st}(A) is a Z(A)-submodule of HH_{0}(A). In fact, we can assume that A is indecomposable and thus Z(A) is local. Let x ∈ HH_{0}^{st}(A). Then t_{Ae_{i}}(x) = 0 for each i. Let z ∈ Z(A), write z = λ + u where λ ∈ k and u is nilpotent. Now t_{Ae_{i}}(zx) = t_{Ae_{i}}(λx) + t_{Ae_{i}}(ux). We have t_{Ae_{i}}(λx) = λt_{Ae_{i}}(x) = 0 and since u is nilpotent, the left multiplication by ux on Ae_{i} is a nilpotent linear map, t_{Ae_{i}}(ux) = 0. Therefore t_{Ae_{i}}(zx) = 0 for each i and zx ∈ HH_{0}^{st}(A).

Theorem 3.11. Suppose that there is a stable equivalence of Morita type between two finite dimensional k-algebras A and B over an algebraically closed field k. Then their stable Hochschild homology groups HH_{0}^{st}(A) and HH_{0}^{st}(B) are isomorphic.

Proof Suppose that the stable equivalence of Morita type between A and B are defined by

M ⊗_{B} N ≃ A ⊕ P, N ⊗_{A} M ≃ B ⊕ Q.

Then we have the transfer maps t_{M} : HH_{0}(A) → HH_{0}(B) and t_{N} : HH_{0}(B) → HH_{0}(A). Moreover,

t_{N} ∘ t_{M} = t_{M⊗_{B}N} = t_{A} + t_{P} = 1_{HH_{0}(A)} + t_{P} : HH_{0}(A) → HH_{0}(A).

First we show that the restriction of t_{P} to the stable Hochschild homology group HH_{0}^{st}(A) is zero, that is, t_{P} |_{HH_{0}^{st}(A)}= 0. Since P is a projective A-A-bimodule and therefore is isomorphic to a direct sum of bimodules of the form Ae_{i} ⊗_{k} e_{j}A, where e_{i} and e_{j} are primitive idempotents in A. By definition of HH_{0}^{st}(A), we know that

t_{Ae_{i}⊗_{k}e_{j}A} |_{HH_{0}^{st}(A)}= t_{e_{j}A} ∘ t_{Ae_{i}} |_{HH_{0}^{st}(A)}= 0

by Lemma 3.1. It follows that t_{P} |_{HH_{0}^{st}(A)}= 0.

Next we show that t_{M}(HH_{0}^{st}(A)) ⊆ HH_{0}^{st}(B). For any indecomposable projective B-module Bf, we have

t_{M⊗_{B}Bf} = t_{Bf} ∘ t_{M} : HH_{0}(A) → HH_{0}(B) → HH_{0}(k) = k.

Since M ⊗_{B} Bf is a projective A-module, we have that t_{M⊗_{B}Bf} |_{HH_{0}^{st}(A)}= 0. This implies that t_{M}(HH_{0}^{st}(A)) ⊆ HH_{0}^{st}(B). Similarly, we can show that

t_{Q} |_{HH_{0}^{st}(B)}= 0 and t_{N}(HH_{0}^{st}(B)) ⊆ HH_{0}^{st}(A).

Combing our discussion above, we have proved that the transfer maps t_{M} and t_{N} induce inverse group isomorphisms between HH_{0}^{st}(A) and HH_{0}^{st}(B).

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Remark 3.12. The algebraically closed field condition is necessary in Theorem 3.11. For example, let k = ℝ. Consider two k-algebras A = ℝ and B = ℂ. Since they are separable algebras, they are stably equivalent of Morita type, but we see easily that dimHH_{0}^{st}(A) = 0 and dimHH_{0}^{st}(B) = 1.

We will establish a realization of the stable Hochschild homology group for Frobenius algebras. Let A be a Frobenius k-algebra with a non-degenerate associative bilinear form ( , ) : A × A → k. Since this bilinear form is not necessarily symmetric, we have the notions of left orthogonal and right orthogonal. Let V ⊆ A be a subspace. Then we define the subspaces

⊥V := {x ∈ A : (x, a) = 0, ∀ a ∈ V }, V^{⊥} := {x ∈ A : (a, x) = 0, ∀ a ∈ V }.

The following lemma is easy and its proof is left to the reader.

Lemma 3.13. (1) If I is a left ideal of A, then II^{⊥} = 0. (2) J(A)^{⊥} = Soc(A).

Let {a_{i}} and {b_{i}} be a pair of dual bases of A, that is, (a_{i}, b_{j}) = δ_{ij}. We define a linear map θ : A → A by θ(x) = ∑ a_{i}xb_{i} for any x ∈ A. It is readily seen that the definition does not depend on the choice of the dual bases. However, this map depends on the choice of the bilinear form, as can be seen in the following example given by Ohtake and Fukushima. We present it in a more general form than the original one.

Example 3.14. ([29]). Let k be an algebraically closed field of characteristic two and A = M_{2}(k) the algebra of 2 × 2 matrices. For 1 ≤ i, j ≤ 2, denote by E_{ij} the matrix whose entry at the position (i, j) is 1 and is zero elsewhere. Let P be an invertible matrix. Define two bilinear form ( , ) and ( , )′ on A by posing (E_{ij}, E_{st}) = δ_{it}δ_{js} and (M, N)′ = (M, NP^{−1}) for M, N ∈ M_{2}(k). It is readily seen that the image of θ corresponding to the first bilinear form is the one-dimensional vector space generated by the identity matrix and the image of θ′ corresponding to the second bilinear form is the one-dimensional space generated by P.

We can nevertheless prove the following

Lemma 3.15. Let A be a Frobenius algebra with two non-degenerate associative bilinear forms ( , ) and ( , )′. Denote by θ, θ′ the maps defined above corresponding to the two bilinear forms. Then dimIm(θ) = dimIm(θ′).

Proof Indeed, there is an invertible element u ∈ A such that (a, b)′ = (a, bu^{−1}) for any a, b ∈ A. So if {x_{i}}, {y_{i}} is a pair of dual bases for ( , ), then {x_{i}}, {y_{i}u} is a pair of dual bases for ( , )′. Computing Im(θ) and Im(θ′) using these bases, one get Im(θ′) = Im(θ)u ⊆ A. So their dimensions are equal.

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The following proposition summarizes some properties of the above map θ. Some idea of our proof comes from [9, Section 4]. Notice that ^{⊥}Im(θ) in the following proposition is independent to the choice of bilinear forms.

Proposition 3.16. Let A be a Frobenius algebra over an algebraically closed field k of characteristic p ≥ 0. Then we have the following.

(1) Im(θ) ⊆ Soc(A) and J(A) ⊆ Ker(θ).

(2) K(A) ⊆ ^{⊥}Im(θ) and ^{⊥}Im(θ)/K(A) = HH_{0}^{st}(A).

(3) dim Im(θ) = rank_{p}C_{A}.

(4) If A is basic, then under suitable bases of A/J(A) and Soc(A), the matrix of the map \overline{θ} : A/J(A) → Soc(A) induced from θ is the transpose of the Cartan matrix C_{A}.

Proof We choose carefully a basis {a_{i}} and its dual basis {b_{i}} in A. Suppose that

A/J(A) ≃ M_{u_{1}}(k) × ⋯ M_{u_{r}}(k).

Write E_{ij}^{t} for the matrix in M_{u_{t}}(k) whose entry at the position (i, j) is 1 and is zero elsewhere. Then take a_{1} = e_{1}, a_{2} = e_{2}, ⋯, a_{m} = e_{m} ∈ A such that their images in A/J(A) correspond to the matrices E_{ii}^{t} for 1 ≤ i ≤ u_{t} and 1 ≤ t ≤ r and take a_{m+1}, ⋯, a_{n} such that their images in A/J(A) correspond to E_{ij}^{t} for 1 ≤ i ≠ j ≤ u_{t} and 1 ≤ t ≤ r. Then {a_{1} ⋯, a_{n}} are linearly independent in A and their images in A/J(A) form a basis of A/J(A) as a vector space. Notice that for m+1 ≤ u ≤ n, a_{u} ∈ e_{i}Ae_{j} for some 1 ≤ i ≠ j ≤ m. Moreover let a_{n+1}, ⋯, a_{s} ∈ J(A) such that their images in J(A)/J^{2}(A) form a basis of J(A)/J^{2}(A) as a vector space, let a_{s+1}, ⋯, a_{t} ∈ J^{2}(A) such that their images in J^{2}(A)/J^{3}(A) form a basis of J^{2}(A)/J^{3}(A) as a vector space, etc. Let b_{1}, b_{2}, ⋯ be the dual basis. Then {r_{1} = b_{1}, ⋯, r_{n} = b_{n}} is a basis of J(A)^{⊥} = Soc(A), {b_{1}, ⋯, b_{s}} is a basis of J^{2}(A)^{⊥}, {b_{1}, ⋯, b_{t}} is a basis of J^{3}(A)^{⊥}, etc.

Now we can prove the first assertion. Since for any x ∈ A,

J(A)x = 0 ⟺ x ∈ Soc(A) ⟺ xJ(A) = 0,

we need to prove that J(A) · Im(θ) = 0. Let y ∈ J(A) and x ∈ A. For 1 ≤ i ≤ n, we get

ya_{i}xb_{i} ∈ J(A) · A · A · Soc(A) = J(A) · Soc(A) = 0;

for n + 1 ≤ i ≤ s, we get

ya_{i}xb_{i} ∈ J(A) · J(A) · A · J^{2}(A)^{⊥} = J^{2}(A) · J^{2}(A)^{⊥} = 0;

for s + 1 ≤ i ≤ t, we get

ya_{i}xb_{i} ∈ J(A) · J^{2}(A) · A · J^{3}(A)^{⊥} = J^{3}(A) · J^{3}(A)^{⊥} = 0;

etc. This proves that Im(θ) ⊆ Soc(A).

Now let x ∈ J(A). Then for 1 ≤ i ≤ n, we get

a_{i}xb_{i} ∈ A · J(A) · Soc(A) = J(A) · Soc(A) = 0;

for n + 1 ≤ i ≤ s, we get

a_{i}xb_{i} ∈ J(A) · J(A) · J^{2}(A)^{⊥} = J^{2}(A) · J^{2}(A)^{⊥} = 0;

for s + 1 ≤ i ≤ t,

a_{i}xb_{i} ∈ J^{2}(A) · J(A) · J^{3}(A)^{⊥} = J^{3}(A) · J^{3}(A)^{⊥} = 0;

etc. This proves that J(A) ⊆ Ker(θ).

Next we come to the second and the third assertions. We prove first that a_{m+1}, ⋯, a_{n} ∈ Ker(θ). We will choose another basis of A and its dual basis. Let X_{ij} ⊂ e_{i}Ae_{j} be a basis of e_{i}Ae_{j} for 1 ≤ i, j ≤ m. Then X = ∪X_{ij} is a basis of A. If x ∈ X_{ij}, denote by x^{∗} the corresponding element in the dual basis and it is easy to see that x^{∗} ∈ e_{j}A. Now for m + 1 ≤ u ≤ n, θ(a_{u}) = ∑_{x∈X} xa_{u}x^{∗}. Note that a_{u} ∈ e_{s}Ae_{t} for some s ≠ t. Since xa_{u}x^{∗} ∈ e_{i}Ae_{j}e_{s}Ae_{t}e_{j}A = 0 for x ∈ X_{ij}, we have θ(a_{u}) = 0. This implies that a_{m+1}, ⋯, a_{n} ∈ Ker(θ) and that θ(e_{1}), ⋯, θ(e_{m}) generate Im(θ) as a vector space.

Since by Example 3.2(2), t_{Ae_{i}}(\overline{x}) = (x, θ(e_{i})), for x ∈ A and 1 ≤ i ≤ m where \overline{x} is the class of x in A/K(A), we have \overline{x} ∈ HH_{0}^{st}(A) ⟺ x ∈ ^{⊥}Im(θ). This proves that K(A) ⊆ ^{⊥}Im(θ) and ⊥Im(θ)/K(A) = HH_{0}^{st}(A). Thus the second assertion holds. For the third assertion, by Theorem 3.8,

dim(A/K(A)) = dim(^{⊥}Im(θ)/K(A)) + rank_{p}C_{A}

and we see that

dim(Im(θ)) = dim(A) − dim(^{⊥}Im(θ)) = dim(A/K(A)) − dim(^{⊥}Im(θ)/K(A)) = rank_{p}C_{A}.

Now suppose that A is basic. As above let e_{1}, ⋯, e_{m} be a complete set of orthogonal primitive idempotents and extend them to a basis of A and take the dual basis {b_{1} = r_{1}, ⋯, b_{m} = r_{m}, ⋯ }. Remark that now the images of e_{1}, ⋯, e_{m} in A/J(A) are a basis of A/J(A) and {r_{1}, ⋯, r_{m}} is a basis of J(A)^{⊥} = Soc(A). By (1), we have an induced map \overline{θ} : A/J(A) → Soc(A). Note that the source and the target of this map are both of dimension l(A). We will prove that its matrix under the basis {e_{1}, ⋯, e_{m}} of A/J(A) and the basis {r_{1}, ⋯, r_{m}} of Soc(A) is just the transpose of the Cartan matrix C_{A}. Suppose that \overline{θ}(e_{i}) = ∑ λ_{ij}r_{j}. Then λ_{ij} = (e_{j}, \overline{θ}(e_{i})) = (1, e_{j}\overline{θ}(e_{i})). To compute the latter, as above, choose a basis X_{ij} of e_{i}Ae_{j} and let X = ∪X_{ij}. Now

e_{j}\overline{θ}(e_{i}) = ∑_{x∈X} e_{j}xe_{i}x^{∗} = ∑_{x∈X_{ji}} xx^{∗}.

We have thus

λ_{ij} = (1, e_{j}\overline{θ}(e_{i})) = (1, ∑_{x∈X_{ji}} xx^{∗}) = ∑_{x∈X_{ji}}(x, x^{∗}) = dim(e_{j}Ae_{i}) · 1.

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For a symmetric k-algebra A, we can give a simple description for the stable Hochschild homology group HH_{0}^{st}(A).

Proposition 3.17. Let A be a finite dimensional symmetric k-algebra over an algebraically closed field k. Then we have that HH_{0}^{st}(A) = Z^{pr}(A)^{⊥}/K(A), where Z^{pr}(A)^{⊥} is the orthogonal space of the projective center Z^{pr}(A).

Proof For symmetric algebras, the map θ is the same as the map τ introduced in Section 2. So by Proposition 3.16 (2) and Proposition 2.3, we have that

HH_{0}^{st}(A) = ^{⊥}Im(θ)/K(A) = ^{⊥}Im(τ)/K(A) = H(A)^{⊥}/K(A) = Z^{pr}(A)^{⊥}/K(A).
□

Remark 3.18. (1) Let k be an algebraically closed field. If A is a finite dimensional k-algebra of finite global dimension, then HH_{0}^{st}(A) = 0. In fact, in this case the determinant of the Cartan matrix is invertible in ℤ, so the p-rank of the Cartan matrix is full, that is, rank_{p}C_{A} = l(A) = dimA/(J(A)+ K(A)) where the last equality is Formula (5) of [17]. So we have the equality HH_{0}^{st}(A) = (J(A) + K(A))/K(A) by Remark 3.10 (1). On the other hand, according to [18], (J(A) + K(A))/K(A) = 0 since A has finite global dimension.

(2) In [8], Eu and Schedler also defined a notion of stable Hochschild (co)-homology groups for any Frobenius k-algebra. In particular, for a Frobenius algebra A over a field k, the 0-degree stable Hochschild homology group of A is defined there as the kernel of the canonical homomorphism A ⊗_{A^{e}} A → A ⊗_{A^{e}} I, where I is the injective envelope of the A^{e}-module A. We point out that even for symmetric k-algebras, the two definitions of 0-degree stable Hochschild homology group may be different. For example, let k be a field of characteristic 0 and let A = k[x]/(x^{2}). Then it is easy to compute that the 0-degree stable Hochschild homology group in our sense is a one dimensional k-space but it is zero in Eu and Schedler’s sense.

Example 3.19. Let A be an indecomposable non-simple self-injective Nakayama algebra over an algebraically closed field k of characteristic p ≥ 0. That is, there is a cyclic oriented quiver Q with s vertices such that A = kQ/kQ^{≥sn+t} with n ≥ 0, 0 < t ≤ s and sn + t ≥ 2 and where kQ^{≥sn+t} is the ideal generated by all paths of length ≥ sn + t. We can prove that

dimHH_{0}^{st}(A) = \{ \begin{matrix}n + u − 1, & if p ∤ \frac{sn+t}{u} \\ n + u, & if p | \frac{sn+t}{u}, \end{matrix}

where u is the greatest common divisor of s and t. Notice that the dimension of the degree zero stable Hochschild homology group depends on p. In fact, since dimA/K(A) = s+n, by Theorem 3.8, one only needs to compute the p-rank of the Cartan matrix C_{A}. This has been done in [28] which gives the following formula

rank_{p}C_{A} = \{ \begin{matrix}s − u + 1, & if p ∤ \frac{sn+t}{u} \\ s − u, & if p | \frac{sn+t}{u}. \end{matrix}

4. Cartan matrices and stable Grothendieck groups

Let k be a field and A be a finite-dimensional k-algebra. The stable Grothendieck group G_{0}^{st}(A) is by definition the cokernel of the Cartan map. In other words, we have the following short exact sequence

K_{0}(A) →^{C_{A}} G_{0}(A) → G_{0}^{st}(A) → 0,

where C_{A} is the Cartan matrix of A and where K_{0}(A) (respectively, G_{0}(A)) is a free abelian group of finite rank generated by isomorphism classes of indecomposable projective modules (respectively, isomorphism classes of simple modules). Suppose that the invariant factors of the Cartan matrix are {0, ⋯, 0, 1, ⋯, 1, δ_{1}, δ_{2}, ⋯, δ_{r}}, where δ_{i} ≥ 2. Denote by m_{i} (i = 0, 1) the number of i in the above sequence. Then the stable Grothendieck group is isomorphic to ℤ^{m_{0}} ⊕ ℤ/δ_{1}ℤ ⊕ ⋯ ⊕ ℤ/δ_{r}ℤ. We remark that the stable Grothendieck group gives the precise values of m_{0} and r but cannot detect m_{1} which is the number of 1 in the sequence of invariant factors of C_{A}.

Now we consider two finite dimensional k-algebras A and B. If they are stably equivalent of Morita type, then it is known that their stable Grothendieck groups are isomorphic (see [39, Section 5]). Note that for a self-injective k-algebra, its stable category is naturally triangulated and the stable Grothendieck group is just the Grothendieck group of the stable category as a triangulated category (see [38, Proposition 1]). Thus if the stable categories of two self-injective k-algebras are triangle equivalent, then their stable Grothendieck groups are isomorphic. Our aim is to prove the following result.

Proposition 4.1. Let A and B be two finite dimensional k-algebras over a field k. Suppose that A and B are stably equivalent of Morita type or that they are self-injective and their stable categories are triangle equivalent. Then the following statements are equivalent.

(1) A and B have the same number of isomorphism classes of simple modules, that is, l(A) = l(B);

(2) rank_{p}(C_{A}) = rank_{p}(C_{B});

(3) The number of 1 in the set of invariant factors of the Cartan matrix of A and that of B are the same, that is, m_{1}(A) = m_{1}(B).

Proof By the discussion above, the stable Grothendieck groups of A and B are isomorphic, so are the stable Grothendieck groups with coefficients in k, that is, G_{0}^{st}(A) ⊗_{ℤ} k ≃ G_{0}^{st}(B) ⊗_{ℤ} k. Tensoring by k on the short exact sequence defining the stable Grothendieck group, we have

K_{0}(A) ⊗_{ℤ} k ⟶^{C_{A}⊗_{ℤ}k} G_{0}(A) ⊗_{ℤ} k → G_{0}^{st}(A) ⊗_{ℤ} k → 0.

Note that the rank of the map C_{A} ⊗_{ℤ} k is just the p-rank of C_{A} and G_{0}(A) ⊗_{ℤ} k is a vector space over k whose dimension is the number l(A) of isomorphism classes of simple modules. We have

rank_{p}(C_{A}) = l(A) − dim(G_{0}^{st}(A) ⊗_{ℤ} k).

This establishes the equivalence of (1) and (2).

The equivalence of (1) and (3) follows from the facts that m_{0} + m_{1}(A) + r = l(A) and that stable Grothendieck groups can detect m_{0} and r.

□

Remark 4.2. In case of a block A of a group algebra kG over an algebraically closed field k of positive characteristic p > 0, the invariant factors of the Cartan matrix C_{A} are always powers of p. So the number of 1 in the set of invariant factors of C_{A} is just the p-rank of C_{A} and thus is the dimension of the Higman ideal, since A is symmetric.

Corollary 4.3. Let k be an algebraically closed field. Suppose that A and B are two Frobenius kalgebras which are stably equivalent of Morita type. Then they have the same number of isomorphism classes of simple modules if and only if dim(Im(θ_{A})) = dim(Im(θ_{B})) where θ_{A} and θ_{B} are the maps introduced in Section 4.

Proof This follows from Proposition 3.16 (3) and Proposition 4.1 .

□

Corollary 4.4. Let k be an algebraically closed field. Suppose that A and B are two finite dimensional k-algebras which are stably equivalent of Morita type. Then

dim (J(A) + K(A))/K(A) = dim (J(B) + K(B))/K(B).

Proof We have

dim A/K(A) = dim A/(J(A)+K(A))+dim (J(A)+K(A))/K(A) = l(A)+dim (J(A)+K(A))/K(A).

By the proof of Proposition 4.1, we have

l(A) = rank_{p}C_{A} + dim(G_{0}^{st}(A) ⊗_{ℤ} k)

and by Theorem 3.8,

dim HH_{0}^{st}(A) + rank_{p}(C_{A}) = dim HH_{0}(A).

Combining these three equalities, we obtain

dim (J(A) + K(A))/K(A) = dim HH_{0}^{st}(A) − dim(G_{0}^{st}(A) ⊗_{ℤ} k)

and the result follows from the fact that both dimHH_{0}^{st}(A) and dim(G_{0}^{st}(A)⊗_{ℤ} k) are invariant under stable equivalences of Morita type.

□

5. Equivalent conditions of the Auslander-Reiten conjecture

Using our results in previous sections, we can now give some easily expressed equivalent conditions of the Auslander-Reiten conjecture for stable equivalences of Morita type.

Theorem 5.1. Let k be an algebraically closed field. Let A and B be two finite dimensional k-algebras which are stably equivalent of Morita type. Then the following two statements are equivalent.

(1) A and B have the same number of isomorphism classes of simple modules;

(2) dimHH_{0}(A) = dimHH_{0}(B).

Moreover, if A and B have no semisimple summand, then (1) and (2) are again equivalent to the following

(3) A and B have the same number of isomorphism classes of non-projective simple modules.

Proof The equivalence of (1) and (2) follows from Theorem 3.8, Theorem 3.11 and Proposition 4.1. If A and B have no semisimple summand, then by [20], the stable equivalence of Morita type induces a bijection between the isomorphism classes of simple projective modules over A and B, and therefore (1) and (3) are equivalent in this case.

□

Now we specialize to symmetric algebras. Let A be a symmetric k-algebra with a non-degenerate associative symmetric bilinear form ( , ). Since K(A)^{⊥} = Z(A) (cf. Section 2), the form ( , ) induces a well-defined non-degenerate bilinear form

Z(A) × A/K(A) → k, (z, a + K(A)) ↦ (z, a).

It follows that we have a duality between Hochschild homology and cohomology, that is,

HH_{0}(A) = A/K(A) ≃ Hom_{k}(Z(A), k) = Z(A)^{∗} = HH^{0}(A)^{∗}.

In particular, dimHH_{0}(A) = dimHH^{0}(A). We obtain the following

Corollary 5.2. Let k be an algebraically closed field. Suppose that two finite dimensional k-algebras A and B are stably equivalent of Morita type and that A is symmetric. Then they have the same number of isomorphism classes of simple modules if and only if dimH(A) = dimH(B), and if and only if dimZ(A) = dimZ(B).

Proof By [21, Corollary 2.4], a stable equivalence of Morita type preserves the property of being symmetric. Thus B is also symmetric. On the other hand, we know that dimH(A) = dimZ^{pr}(A) = dimZ(A) − dimZ^{st}(A) and that dimZ^{st}(A) is an invariant under a stable equivalence of Morita type. Now the conclusion follows from Theorem 5.1 and the remark before this corollary.

□

Remark 5.3. By Theorem 5.1, for stable equivalences of Morita type, the Auslander-Reiten conjecture is equivalent to the invariance of 0-degree Hochschild homology groups. This is not true, however, for general stable equivalences. For example, let A be the path algebra over a field k given by the quiver ∘^{1} ⟶^{α} ∘^{2} ⟶^{β} ∘^{3}. If we glue the source vertex 1 and the sink vertex 3 and put a zero relation in the above quiver, then we get a subalgebra B of A which is given by the following quiver

[figure: ∘^{1}; ∘^{2}; α; β]

with relations αβ = 0. By [25], A and B are stably equivalent. In [14], it was shown that such stable equivalence is not of Morita type, but still induced by tensoring a pair of bimodules. On the other hand, we have dimHH_{0}(A) = 3 and dimHH_{0}(B) = 2. Thus dimHH_{0}(A) ≠ dimHH_{0}(B).

6. Power-p maps and the Auslander-Reiten conjecture

In this section, we work over an algebraically closed field k of positive characteristic p > 0 and in this special case we will give another proof of the main Theorem 5.1. As an application, we get some new computable invariants of stable equivalences of Morita type.

One of the properties of the 0-degree Hochschild homology group is that it admits a power-p map, that is,

μ_{p}^{A} : A/K(A) → A/K(A), x + K(A) ↦ x^{p} + K(A).

The above map is well-defined and additive (see, for example, [17]). We shall prove that this map restricts to the stable 0-degree Hochschild homology group. We begin with a lemma.

Lemma 6.1. Let B be a finite dimensional k-algebra over a field k of characteristic p > 0. For a finitely generated projective right B-module M, the trace map

tr_{M} : End_{B}(M) ≃ M ⊗_{B} Hom_{B}(M, B) → B/K(B)

satisfies that for any f ∈ End_{B}(M), we have tr(f^{p}) = tr(f)^{p}. Consequently, the same holds for the degree zero transfer map t_{M} : A/K(A) → B/K(B).

Proof The first assertion should be well known, but we could not find a proof in the literature, so we include one. Suppose that M ≃ (e_{1}B ⊕ e_{2}B ⊕ ⋯ ⊕ e_{n}B) for some primitive idempotents e_{1}, ⋯, e_{n} ∈ B. We can write an element f ∈ End_{B}(M) ≃ End_{B}(e_{1}B ⊕ e_{2}B ⊕ ⋯ ⊕ e_{n}B) as a matrix (f_{ij})_{1≤i,j≤n} where f_{ij} ∈ Hom_{B}(e_{i}B, e_{j}B). The trace map gives tr_{M}(f) = ∑_{i} f_{ii} in B/K(B). We want to prove that tr_{M}(f^{p}) = ∑_{i} f_{ii}^{p} ∈ B/K(B), therefore tr_{M}(f^{p}) = (tr_{M}(f))^{p} in B/K(B) and we are done. In fact, the trace of f^{p} is the sum of terms of the form

f_{i_{1}i_{2}}f_{i_{2}i_{3}} ⋯ f_{i_{p−1}i_{p}}f_{i_{p}i_{1}}

for 1 ≤ i_{1}, ⋯, i_{p} ≤ n. If all the indices i_{1}, ⋯, i_{p} are equal to some i for 1 ≤ i ≤ n, then the term is f_{ii}^{p}. We shall prove that the sum of all other terms is zero in B/K(B). Consider the sequence of pairs

((i_{1}, i_{2}), (i_{2}, i_{3}), ⋯, (i_{p−1}, i_{p}), (i_{p}i_{1}))

which corresponds to a term above. The cyclic group of order p acts on such a sequence by permuting cyclically the pairs in this sequence. If not all of the indices are equal, then the action is free and an orbit contains p sequences. Notice that any non-trivial cyclic permutation of this sequence corresponds to a different expressed term, but these terms are all equal in B/K(B). So the sum of all the terms corresponding to the cyclic permutations of a sequence is zero in B/K(B) in case that not all the indices are equal. The fixed points of this action are just the sequences in which all of the indices are equal. We have now tr_{M}(f^{p}) = ∑_{i} f_{ii}^{p} in B/K(B). The proof is complete.

The second assertion follows from the first and the construction of the transfer map.

□

We now prove that the power-p map can restrict to the stable Hochschild homology group.

Corollary 6.2. μ_{p}^{A}(HH_{0}^{st}(A)) ⊆ HH_{0}^{st}(A).

Proof For x ∈ HH_{0}^{st}(A), t_{Ae_{i}}(x^{p}) = t_{Ae_{i}}(x)^{p} = 0 for each i.

□

Recall that T_{n}(A) = {x ∈ A | x^{p^{n}} ∈ K(A)} is a k-subspace of A and that (see [17, (9)])

⋃_{n=0}^{∞} T_{n}(A) = J(A) + K(A).

Lemma 6.3. For each n ≥ 0, T_{n}(A) ⊆ HH_{0}^{st}(A). As a consequence, (J(A) + K(A))/K(A) ⊆ HH_{0}^{st}(A).

Proof Indeed, T_{n}(A)/K(A) = {x ∈ A/K(A)|x^{p^{n}} = 0}. So for x ∈ T_{n}(A)/K(A) and for any projective A-module Ae, t_{Ae}(x)^{p^{n}} = t_{Ae}(x^{p^{n}}) = 0. It follows that t_{Ae}(x) = 0 for each Ae and x ∈ HH_{0}^{st}(A).

□

Combining Corollary 6.2 and Lemma 6.1, we obtain the main result of this section.

Proposition 6.4. Let k be an algebraically closed field of characteristic p > 0. Given an A-B bimodule _{A}M_{B} which is finitely generated and projective as right B-module, then we have a commutative diagram:

\xymatrix{HH_{0}^{st}(A) \ar[r]^{μ_{p}^{A}} \ar[d]_{t_{M}} & HH_{0}^{st}(A) \ar[d]^{t_{M}} \\ HH_{0}^{st}(B) \ar[r]^{μ_{p}^{B}} & HH_{0}^{st}(B).}

Corollary 6.5. Let k be an algebraically closed field of characteristic p > 0. Let A and B be two finite dimensional k-algebras which are stably equivalent of Morita type. Then dim(T_{n}(A)/K(A)) = dim(T_{n}(B)/K(B)).

Proof This follows from Proposition 6.4, Theorem 3.11 and the following commutative diagram:

\xymatrix{0 \ar[r] & T_{n}(A)/K(A) \ar@{-->}[d]^{≃} \ar[r] & HH_{0}^{st}(A) \ar[r]^{(μ_{p}^{A})^{∘n}} \ar[d]_{t_{M}}^{≃} & HH_{0}^{st}(A) \ar[d]_{t_{M}}^{≃} \\ 0 \ar[r] & T_{n}(B)/K(B) \ar[r] & HH_{0}^{st}(B) \ar[r]^{(μ_{p}^{B})^{∘n}} & HH_{0}^{st}(B),}

where (μ_{p}^{A})^{∘n} denotes the composition of μ_{p}^{A} with itself n times.

□

Remark 6.6. Let A and B be two finite dimensional algebras over an algebraically closed field of positive characteristic. If they are derived equivalent, then dim(T_{n}(A)/K(A)) = dim(T_{n}(B)/K(B)). This fact was proved by Bessenrodt, Holm and the third author ([2]).

Now one can give an alternative proof of Theorem 1.1 in case of positive characteristic.

Corollary 6.7. Let k be an algebraically closed field of characteristic p > 0. Suppose that there is a stable equivalence of Morita type between two finite dimensional k-algebras A and B. Then the following are equivalent.

(1) A and B have the same number of isomorphism classes of simple modules;

(2) dimHH_{0}(A) = dimHH_{0}(B).

Proof Since ⋃_{n=0}^{∞} T_{n}(A)/K(A) = (J(A)+K(A))/K(A), by Corollary 6.5, dim(J(A)+K(A)/K(A)) is invariant under a stable equivalence of Morita type. Since l(A) = dim(A/J(A) + K(A)) (see [17, (5)]), we know that l(A) = dim(A/K(A)) − dim(J(A) + K(A)/K(A)).

□

Now we consider symmetric algebras. Let us first recall some notations. Let k be an algebraically closed field of positive characteristic p > 0 and let A be a finite-dimensional symmetric k-algebra. The n-th Külshammer ideal of A is defined as the orthogonal space (with respect to the symmetrizing form on A)

T_{n}^{⊥}(A) = {x ∈ A | (x, y) = 0 for all y ∈ T_{n}(A)}.

We then have the following fundamental lemma.

Lemma 6.8 ([17], No.(36) and (37)). The subspaces T_{n}^{⊥}(A) form a descending chain of ideals of the center Z(A)

Z(A) = K(A)^{⊥} = T_{0}^{⊥}(A) ⊇ T_{1}^{⊥}(A) ⊇ T_{2}^{⊥}(A) ⊇ · · · .

Moreover, the intersection of Külshammer ideals is the Reynolds ideal:

⋂_{i=0}^{∞} T_{n}^{⊥}(A) = R(A) := Soc(A) ∩ Z(A).

We can now state a theorem of the third author saying that the Külshammer ideals are derived invariants. This theorem motivates the work in this article.

Theorem 6.9 ([42]). Let k be an algebraically closed field of characteristic p > 0. Let A and B be symmetric k-algebras. If A and B are derived equivalent (that is, their derived module categories D^{b}(A) and D^{b}(B) are equivalent as triangulated categories), then there exists an algebra isomorphism φ : Z(A) → Z(B) such that φ(T_{n}^{⊥}(A)) = T_{n}^{⊥}(B) for any n ≥ 0.

We now define a stable version of Külshammer ideals. Recall that Z(A) ≃ Hom_{A^{e}}(A, A) and the projective center Z^{pr}(A) is an ideal of Z(A). The stable center Z^{st}(A) is defined to be Z(A)/Z^{pr}(A). Notice that by [9, Lemma 4.1 (iii)], Z^{pr}(A) = H(A) ⊆ R(A) ⊆ T_{n}^{⊥}(A). We define

T_{n}^{⊥,st}(A) := T_{n}^{⊥}(A)/Z^{pr}(A) ⊆ Z^{st}(A)

and R^{st}(A) := R(A)/Z^{pr}(A) ⊆ Z^{st}(A). We call T_{n}^{⊥,st}(A) the n-th stable Külshammer ideal and R^{st}(A) the stable Reynolds ideal, respectively. Since A is finite dimensional, when n is large, T_{n}^{⊥,st}(A) = R^{st}(A).

Proposition 6.10. Let k be an algebraically closed field of characteristic p > 0. Suppose that two finite dimensional k-algebras A and B are stably equivalent of Morita type and that A is symmetric. Then dimT_{n}^{⊥,st}(A) = dimT_{n}^{⊥,st}(B) for any n ≥ 0. In particular dimR^{st}(A) = dimR^{st}(B).

Proof Since K(A)^{⊥} = Z(A), we have a well-defined non-degenerate bilinear form

Z(A)/T_{n}^{⊥}(A) × T_{n}(A)/K(A) → k, (z̄, ā) ↦ (z, a).

It follows that we have a duality between Z(A)/T_{n}^{⊥}(A) and T_{n}(A)/K(A). In particular, their dimensions are the same. Note that

dim(Z(A)/T_{n}^{⊥}(A)) = dim(Z(A)/Z^{pr}(A)) − dim(T_{n}^{⊥}(A)/Z^{pr}(A)) = dimZ^{st}(A) − dimT_{n}^{⊥,st}(A).

Since dim(T_{n}(A)/K(A)) and Z^{st}(A) are invariant under a stable equivalence of Morita type, so is dimT_{n}^{⊥,st}(A).

□

Remark 6.11. Notice that for symmetric algebras, the dimension of the Reynolds ideal R(A) is just the number of simple modules l(A). Since dimR(A) = dimR^{st}(A) + dimH(A), the above proposition gives another proof of Corollary 5.2 in positive characteristic.

Acknowledgement: The authors thank Serge Bouc for his kind explications on his transfer construction. The authors wish to express their gratitude to Steffen König for his most useful remarks and suggestions on a preliminary version of this paper.

We are grateful very much to the referee whose suggestions simplified several proofs considerably.

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Yuming Liu School of Mathematical Sciences Laboratory of Mathematics and Complex Systems Beijing Normal University Beijing 100875 P.R.China

   E-mail address: ymliu@bun.edu.cn

Guodong Zhou Ecole Polytechnique Fédérale de Lausanne SB/IGAT/CTG, Batîment MA B3 424, CH-1015, Lausanne, Switzerland

   E-mail address: guodong.zhou@epfl.ch

Alexander Zimmermann Université de Picardie, Département de Mathématiques et LAMFA (UMR 6140 du CNRS), 33 rue St Leu, F-80039 Amiens Cedex 1, France

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