ON THE USE OF KÜLSHAMMER TYPE INVARIANTS IN REPRESENTATION THEORY

ALEXANDER ZIMMERMANN

Abstract. Since 2005 a new powerful invariant of an algebra emerged using earlier work of Horváth, Héthelyi, Külshammer and Murray. The authors studied Morita invariance of a sequence of ideals of the centre of a finite dimensional algebra over a field of finite characteristic. It was shown that the sequence of ideals is actually a derived invariant, and most recently a slightly modified version of it an invariant under stable equivalences of Morita type. The invariant was used in various contexts to distinguish derived and stable equivalence classes of pairs of algebras in very subtle situations. Generalisations to non symmetric algebras and to higher Hochschild (co-)homology was given. This article surveys the results and gives some of the constructions in more detail.

Contents

Introduction 1 1. Historical facts and basic definitions: the origins by Brauer and Reynolds 3 2. Selfinjective and symmetric algebras revisited 5 2.1. Basic definitions and properties 5 2.2. The Nakayama automorphism 6 2.3. The Nakayama twisted centre 7 2.4. How to get the Nakayama automorphism explicitly 8 2.5. Practical questions for algebras given by quivers and relations 9 3. Külshammer, the new idea in the 1980s 11 3.1. Külshammer’s original construction for symmetric algebras 11 3.2. Extending to general algebras: trivial extension algebras and Külshammer

theory 12

4. Morita, derived and stable invariance 13 4.1. Morita invariance 13 4.2. Derived invariance 13 4.3. Stable invariance 14 5. Applications 17 5.1. Algebras of dihedral, semidihedral and quaternion type 17 5.2. Białkowski-Erdmann-Skowroński deformation of preprojective algebras 22 5.3. Algebras of polynomial growth and domestic weakly symmetric algebras 25 6. Hochschild (co-)homology 29 References 31

Introduction

Brauer studied representations of finite groups over fields of characteristic p dividing the order of the group. In 1956 he showed [12] amongst many other things that, if the field is algebraically closed, the number of simple modules is equal to the number of conjugacy classes of elements of G of order prime to p. His method was rather general already and Külshammer used these ideas 25 years later to define very sophisticated invariants for general symmetric algebras. More precisely Külshammer defined and studied in a series of four papers [30] a sequence of ideals of the centre of a symmetric algebra. In case of group rings over a finite group Külshammer studied lower defect groups, proved Brauer’s main theorems and many other results known in modular representation theory of finite groups by these ideals and related invariants. Külshammer’s approach was left untouched until Murray [35, 36] studied Külshammer’s approach in connection with the question of the existence of real characters and of characters of defect 0. This was the starting point of the collaboration between Breuer, Héthelyi, Horváth, Külshammer and Murray [19] and [13] where the authors study the existence of odd diagonal entries in the Cartan matrix of a group algebra. Moreover, in [19] the Morita invariance of the sequence of ideals is shown and the question of derived invariance is posed.

Date: June 14, 2010.

In [52] completely different methods are used to show that indeed at least for perfect base fields the sequence of ideals is invariant under derived equivalence. Still the assumption that the algebras are symmetric was needed.

After a lecture of the author in September 2005 in Oberwolfach Holm became interested in the sequence of ideals and proposed several improvements and applications. First, in joint work of the author with Bessenrodt and Holm [4] using trivial extension algebras the derived invariance, and also the very definition of the Külshammer ideals was extended to not necessarily symmetric algebras. Further, the Külshammer ideals were used in [23] to distinguish the derived equivalence class of two algebras of dihedral type and of two pairs of algebras of semidihedral type which were not seen to be not derived equivalent in the classification of Holm [20].

During the September 2005 Oberwolfach lecture Adem asked for a generalisation of the Külshammer ideal structure to higher Hochschild (co-)homology. The question is far from trivial and was solved in [53] where actually two approaches were taken, both of which are not exactly what was asked for. The first approach uses Hochschild homology instead of cohomology as in the Külshammer ideal structure, the second approach uses the Stasheff approach to the Gerstenhaber structure to get a non linear analogue. The case of non symmetric algebras was answered in [54] again using trivial extension algebras and the Hochschild homology approach.

The derived equivalence classification of tame domestic weakly symmetric algebras was given by Bocian, Holm and Skowroński for domestic algebras [9, 10, 21] using for the last remaining delicate questions Külshammer ideals. Similarly, also using Külshammer ideals in parts, Białkowski, Holm and Skowroński [6, 7, 22] gave a derived equivalence classification of tame algebras of polynomial growth up to some difficult problems concerning scalars in the relations of certain algebras, similar to the problem solved in [23].

Then, most recently Białkowski, Erdmann and Skowroński classified selfinjective algebras with the property that the third syzygy of every simple module S is again isomorphic to S. They obtain that these algebras are all certain deformations of preprojective algebras of a generalised Dynkin type. The deformations involve parameters in the relations of the algebra. As was seen [21, 22, 23] Külshammer ideals are well suited for this kind of questions. Derived equivalence classes of one family called of type L, defined in detail in Example 2.19 and more generally in Section 5.2 below, were largely given in joint work with Holm [24]. Here we developed quite sophisticated methods to determine the Külshammer subspace structure at the beginning which hold for a priori non symmetric algebras as well. We display this method, even though strictly speaking only small parts of it are really necessary in case of algebras of type L. Nevertheless, the method works in general, is potentially very useful and it seems reasonable to present it here.

During a lecture of the author in October 2007 at Beijing Normal University the question of an invariance under stable equivalences was posed. In most recent results with Yuming Liu and Guodong Zhou a generalisation of Külshammer ideal theory was given for stable categories and an invariance was proved for stable equivalences of Morita type ([33] and [29]). Most interestingly the result has strong links to the Auslander-Reiten conjecture [3, Conjecture 5, page 409], which says that a stable equivalence should preserve the number of simple non projective modules. The result [33, 29] was used in joint work with Zhou [48] to give a classification of algebras of dihedral, of semidihedral and of quaternion type as defined by Erdmann [16] up to stable equivalences of Morita type. Moreover, in [49], in joint work with Zhou we prove that the classification of weakly symmetric tame algebras of polynomial growth up to stable equivalence of Morita type coincides with the derived equivalence classification of Białkowski-Holm-Skowroński.

In the present paper we survey these results and give at certain points quite complete proofs for results which seem to be very useful also in further contexts. In Section 1 we trace some steps in the origins, starting from Brauer and Reynolds. Section 2 reviews properties of selfinjective algebras and develops tools to actually compute Külshammer ideals for quite complicated algebras. These tools were developed during the past years for this purpose, but the origins are, of course, very classical. Section 3 presents Külshammer ideals as they were developed originally by Külshammer and as they were generalised later to non-symmetric algebras. Section 4 displays the invariance of the various forms of Külshammer ideals under Morita, derived and stable equivalences. Section 5 is devoted to various applications mentioned above including a detailed outline of the proof for the deformed preprojective algebras of type L. Section 6 gives the above mentioned two approaches to the Hochschild (co-)homology generalisations of Külshammer invariants.

1. Historical facts and basic definitions: the origins by Brauer and Reynolds

Brauer developed in the 1950’s the far reaching representation theory of groups over fields of finite characteristic. In 1956 he showed in particular

Theorem 1. (Brauer [12, Statement 3B]) If K is an algebraically closed field of characteristic p > 0 and if G is a finite group, then the number of simple KG-modules up to isomorphism equals the number of conjugacy classes of G of elements of order prime to p.

Of course, in the spirit of that time, Brauer did speak of irreducible characters rather than of modules, but the result translates into modern terms as it is shown above. The method of proof he used is somewhat indirect. He defines for any K-algebra A the space of commutators [A, A], which is defined as the K-vector space generated as vector space by all possible expressions ab − ba where a, b ∈ A.

Further he defines

T A := {a ∈ A | ∃n ∈ ℕ : a^{p^{n}} ∈ [A, A]}

The first lemma is a little more general than stated in [12], but with identical proof. The generalisation comes from the fact that actually one can consider more generally

T_{n}A := {a ∈ A | a^{p^{n}} ∈ [A, A]}

for all n ∈ ℕ, a fact which is an observation due to Külshammer. We will need and study T_{n}A later in more detail.

Lemma 1.1. (Brauer [12, Statement 3A]) Let K be a field of characteristic p > 0 and let A be a K-algebra. Then T_{n}A is a K-subspace of A satisfying T A = ⋃_{n∈ℕ} T_{n}A. If A is finite dimensional, and K is a splitting field for A, then the number of simple A-modules up to isomorphism equals the dimension of A/T A.

The proof of this lemma is so simple that we may give it here in almost full detail.

Proof (Brauer). Take x, y ∈ T_{n}A. Then develop (x + y)^{p} and get a sum of all possible words in x and y with p factors, each occurring exactly once. If 1 < s < p, then there are n(s) of such words in which x occurs s times and y occurs p − s times. Take N(s) the set of these words. Then the cyclic group of order p acts on this set by cyclic permutation of the word: c · (a_{1}a_{2} … a_{p−1}a_{p}) := (a_{2}a_{3} … a_{p}a_{1}) for a generator c of the cyclic group and a_{i} ∈ {x, y}. Hence N(s) decomposes into orbits of length p, and the difference of two elements in the same orbit is clearly in [A, A]. Hence,

(x + y)^{p} − x^{p} − y^{p} ∈ [A, A].

Moreover

(xy − yx)^{p} + [A, A] = (xy)^{p} − (yx)^{p} + [A, A] = x ((yx)^{p−1}y) − ((yx)^{p−1}y) x + [A, A] = [A, A]

and

(λx)^{p} = λ^{p}x^{p}

for all x, y ∈ A and λ ∈ K show that T_{n}A is a K-subspace of A.

Traces of commutators of matrices are 0. Therefore, if A = Mat_{n}(K), then [A, A] ⊆ {M ∈ Mat_{n}(K) | trace(M) = 0}. On the other hand, using elementary matrices, one sees that the inclusion actually is an equality. The space of matrices with trace 0 is of codimension 1, and Mat_{n}(K) has exactly one simple module up to isomorphism. We denote by rad(A) the Jacobson radical of the algebra A. Hence, putting \overline{A} := A/rad(A), one gets

\overline{A}/(T \overline{A}) = A/(T A)

as vector spaces since rad(A) is nilpotent, and therefore rad(A) ⊆ T (A). This shows the statement by Wedderburn’s theorem.

Brauer’s Theorem 1 follows from the fact that for a group ring KG a basis is formed by the elements of G, and any element g ∈ G admits a unique (so-called the p-primary decomposition) g = g_{p} · g_{p′} where g_{p} is a p-element and g_{p′} is of order prime to p commuting with g_{p}. Hence

(g − g_{p′})^{p^{n}} + [KG, KG] = (g^{p^{n}} − g_{p′}^{p^{n}}) + [KG, KG] = [KG, KG]

for a certain large n. Then,

hgh^{−1} − g = hgh^{−1} − gh^{−1}h = [h, gh^{−1}] ∀g, h ∈ G

and the rest is straightforward.

Remark 1.2. Observe that actually much more is shown: T A = rad(A) + [A, A].

Another concept due to Reynolds [41] is closely linked. Let G be a finite group, let K be an algebraically closed field of characteristic p > 0. For any g ∈ G and h ∈ G consider the p-primary decomposition g = g_{p} · g_{p′} and h = h_{p} · h_{p′}. Then

S_{h} := {g ∈ G | ∃x ∈ G : x · g_{p′} · x^{−1} = h_{p′}}

be the set of elements in G whose p′-part is conjugate to the p′-part of h. Put

C_{h} := ∑_{g∈S_{h}} g

the sum of all these elements in S_{h}. Recall that the centre of KG has a basis consisting of all conjugacy class sums of elements of G. Therefore C_{h} ∈ Z(KG) for all h ∈ G.

Definition 1.3. (Reynolds [41, Theorem 1]) The Reynolds ideal of KG is the ideal of Z(KG) generated as K-vector space by the elements C_{h}, for h ∈ G.

We get

Proposition 1.4. (Reynolds [41], cf [18, Theorem VI.4.6]) The Reynolds ideal R(KG) of KG is the annihilator of rad(KG) in Z(KG).

2. Selfinjective and symmetric algebras revisited

In order to be able to explain more deeply the relations between Reynolds ideals, T (KG) and related objects we need to explain the structure of selfinjective and of symmetric algebras. The theory is classical and originates in Nakayama’s work [37] in the late 1930’s.

Various approaches can be found in the literature, but as far as we know the approach using Picard groups, which we will explain in Section 2.2 did not appear elsewhere, though Yamagata [47] gives some related thoughts.

Throughout this section we suppose for simplicity that K is a field. However, many results stay true under weaker assumptions on K, sometimes K being a commutative ring would be sufficient.

2.1. Basic definitions and properties. Recall that for a K-algebra A the space of linear forms Hom_{K}(A, K) is an A − A-bimodule by

(afb)(x) := f(bxa) ∀a, b, x ∈ A∀f ∈ Hom_{K}(A, K).

A group algebra is a symmetric algebra in the following sense.

Definition 2.1. Let K be a field and let A be a K-algebra. Then A is • symmetric if A ≃ Hom_{K}(A, K) as A − A-bimodules.

• selfinjective if A ≃ Hom_{K}(A, K) as A left-modules.

We shall derive some consequences.

Suppose A is selfinjective and let φ : A ⟶ Hom_{K}(A, K) be an isomorphism of A leftmodules. Then we may define a K-bilinear form

⟨ , ⟩ : A × A ⟶ K

by

⟨a, b⟩ := (φ(b))(a).

The fact that φ is an isomorphism of vector spaces is equivalent to the fact that ⟨ , ⟩ is non degenerate.

The fact that φ is A-linear is equivalent to

⟨a, bc⟩ = (φ(bc))(a) = (bφ(c))(a) = φ(c)(ab) = ⟨ab, c⟩

for all a, b, c, where the linearity is used in the second equality. A bilinear form on an algebra A is called associative if

⟨a, bc⟩ = ⟨ab, c⟩ for all a, b, c ∈ A.

Now φ is an A − A-bimodule homomorphism if and only if ⟨ , ⟩ is associative (i.e. φ is left A-linear) and moreover

⟨a, b⟩ = (φ(b))(a) = (φ(1)b)(a) = φ(1)(ba) = ⟨ba, 1⟩ = ⟨b, a⟩

and so A is symmetric if and only if the associative non degenerate form ⟨ , ⟩ may be chosen symmetric.

We summarise the statements in a (well known) proposition which gives an alternative definition of selfinjective and symmetric algebras.

Proposition 2.2. Let K be a field and let A be a finite dimensional K-algebra. Then we have the following statements.

• The algebra A is selfinjective if and only if there is a non degenerate associative bilinear form on A.

• The algebra A is symmetric if and only if there is a non degenerate associative and symmetric bilinear form on A.

We should mention that the existence statement in Proposition 2.2 is constructive: The bilinear form is as explicit as is the isomorphism. That is, if one knows the explicit isomorphism of A to its dual by an explicit formula, then one knows the bilinear form by an explicit formula, and vice versa.

The non degenerate associative symmetric bilinear form is called symmetrising form for a symmetric algebra. In the remaining parts of Section 2 these ideas are developed further in particular with emphasis on the question how to actually determine the bilinear form and associated questions.

For the moment we shall continue with Reynolds ideals and give the promised link.

In the following we frequently use for a symmetric algebra A and subsets S of A the symbol S^{⊥} to designate the orthogonal space with respect to the symmetrising form of the algebra.

Proposition 2.3. (Külshammer [31]; [30, Part I Lemma A; Satz C; Satz D]) Let K be a field and let A be a finite dimensional symmetric K-algebra. Then [A, A]^{⊥} = Z(A) and soc(A) = rad(A)^{⊥} = Ann_{A}(rad(A)). In particular R(KG) = Z(KG)∩soc(KG) for a finite group G.

Proof.

⟨ab − ba, c⟩ = ⟨ab, c⟩ − ⟨ba, c⟩ = ⟨a, bc⟩ − ⟨c, ba⟩ = ⟨bc, a⟩ − ⟨cb, a⟩ = ⟨bc − cb, a⟩

and hence c ∈ [A, A]^{⊥} if and only if ⟨ab − ba, c⟩ = 0 for all a, b. Therefore c ∈ [A, A]^{⊥} if and only if ⟨bc − cb, a⟩ = 0 for all a, b. In particular c ∈ [A, A]^{⊥} if and only if bc − cb ∈ A^{⊥}. But A^{⊥} = 0 since the form is non degenerate. Hence c ∈ [A, A]^{⊥} if and only if bc = cb for all b ∈ A. This shows [A, A]^{⊥} = Z(A).

⟨I, rad(A)⟩ = ⟨1, I · rad(A)⟩

and hence

I ⊆ rad(A)^{⊥} ⇔ I · rad(A) = 0 ⇔ I ⊆ soc(A).

This finishes the proof.

2.2. The Nakayama automorphism. Selfinjective algebras come along with an automorphism, called the Nakayama automorphism which will be explained now.

If A is a selfinjective K-algebra, then A ≃ Hom_{K}(A, K) as an A left-module. Hence, Hom_{K}(A, K) is a free left A-module of rank 1. Moreover,

End_{A}( _{A}Hom_{K}(A, K)) ≃ End_{A}( _{A}A) ≃ A

and so Hom_{K}(A, K) is a progenerator over A with endomorphism ring isomorphic to A, whence inducing a Morita self-equivalence of A. This implies that the isomorphism class of Hom_{K}(A, K) is in the Picard group P ic_{K}(A) (cf e.g. [40, Section 37]). As is shown there there is a group homomorphism

ω_{0} : Aut_{K}(A) ⟶ P ic_{K}(A)

given by ω_{0}(α) = _{1}A_{α}. Here for any two automorphisms α and β of A the A − A-bimodule αA_{β} denotes A as vector space, on which a ∈ A acts by multiplication by β(a) on the right and by α(a) on the left. To shorten the notation we abbreviate in this context the identity on A by 1. One gets ker(ω_{0}) = Inn(A) the inner automorphisms of A and hence

Out_{K}(A) := Aut_{K}(A)/Inn(A)

is a subgroup of P ic_{K}(A). (Observe that the group of inner automorphisms does not depend on K.) The image of ω_{0} consists of those isomorphism classes of invertible A − A-bimodules which are free on the left (cf. [40, (37.16) Theorem]). Observe that

α^{−1}A_{1} ⟶ _{1}A_{α}
a ↦ α(a)

is an A − A-bimodule homomorphism. Hence ω_{0} may also be defined by twisting the action on the left.

Now, as Hom_{K}(A, K) is free of rank 1 as left-module one gets that Hom_{K}(A, K) is in the image of Out_{K}(A) in P ic_{K}(A) and therefore there is an automorphism ν ∈ Aut_{K}(A) so that

Hom_{K}(A, K) ≃ _{1}A_{ν}

as A-A-bimodules. The automorphism ν is unique up to an inner automorphism.

Definition 2.4. Let A be a selfinjective K-algebra. Then there is an automorphism ν of A so that Hom_{K}(A, K) ≃ _{1}A_{ν} as A-A-bimodules. This automorphism is unique up to inner automorphisms and is called the Nakayama automorphism.

Remark 2.5. In Nakayama’s original approach _{ν}A_{1} is used instead of _{1}A_{ν} and so the Nakayama automorphism in Nakayama’s work corresponds to the inverse of what we define here.

In principle, the definition of selfinjectiveness uses an isomorphism of the A-left-module of linear forms on A with the regular A-module. One could use the right module structure as well. We get the well-known result that left selfinjective is equivalent to right selfinjective.

Corollary 2.6. _{A}A ≃ _{A}Hom_{K}(A, K) ⇔ A_{A} ≃ Hom_{K}(A, K)_{A}.

To prove the Corollary one just needs to see that the isomorphism as left-modules implies the following bimodule isomorphisms.

Hom_{K}(A, K) ≃ _{1}A_{ν} ≃ _{ν^{−1}}A_{1}

and so Hom_{K}(A, K) ≃ A as A right-modules.

Now, for a selfinjective K-algebra A, given a simple A-module S, then _{1}A_{ν} ⊗_{A} S ≃ Hom_{K}(A, K) ⊗_{A} S is again a simple A-module.

Definition 2.7. Let K be a field and let A be a finite dimensional K-algebra. Then A is weakly symmetric if A is selfinjective and Hom_{K}(A, K) ⊗_{A} S ≃ S for all simple A-modules S.

This definition will be important in Section 5.3.

2.3. The Nakayama twisted centre. Let K be a field and let A be a finite dimensional K-algebra. For an explicitly given algebra, say as quiver with relations, it is not very hard to write down many commutators. This gives an upper bound for the dimension of A/[A, A]. However, to prove that the commutators found really generate [A, A] is quite difficult in general. The method is to interpret A/[A, A] as different space, in which it is easier to find many linearly independent elements. This then gives a lower bound for the dimension of A/[A, A]. If the lower and the upper bound coincide, then one has proved that the commutators found actually generate the whole space [A, A].

Let A be a selfinjective K-algebra. Given a ring R and a right R-module M and a left R-module N, the very definition of the tensor product M ⊗_{R} N as free abelian on symbols m ⊗ n with relations m ⊗ rn = mr ⊗ n and additivity in each variable gives an isomorphism A/[A, A] ≃ A⊗_{A⊗_{K}A^{op}}A. An alternative way to see this is by the bar resolution of Hochschild homology.

Now

Hom_{K}(A/[A, A], K) ≃ Hom_{K}(A ⊗_{A⊗_{K}A^{op}} A, K)
≃ Hom_{A⊗_{K}A^{op}}(A, Hom_{K}(A, K))
≃ Hom_{A⊗_{K}A^{op}}(A, _{1}A_{ν})

which gives

Hom_{K}(A/[A, A], K) ≃ {a ∈ A | b · a = a · ν(b) ∀b ∈ A}

where the isomorphism is given by sending a homomorphism to the image of 1 ∈ A which will satisfy the equation by the property of the homomorphism being A ⊗_{K} A^{op}-linear.

Definition 2.8. (Holm and Zimmermann [24, Definition 2.2]) Let K be a field and let A be a selfinjective K-algebra with Nakayama automorphism ν. Then the Nakayama twisted centre is defined to be

Z_{ν}(A) := {a ∈ A | b · a = a · ν(b) ∀b ∈ A}.

Remark 2.9. The definition works for K a commutative ring as well.

The automorphism ν is unique only up to an inner automorphism. If ν differs from ν′ by an inner automorphism, let ν(b) = u · ν′(b) · u^{−1} for all b ∈ A and some unit u of A. Then

Z_{ν}(A) = {a ∈ A | b · a = a · ν(b) ∀b ∈ A}
= {a ∈ A | b · a = a · u · ν′(b) · u^{−1} ∀b ∈ A}
= {a ∈ A | b · (a · u) = (a · u) · ν′(b) ∀b ∈ A}
= {a ∈ A |a · u ∈ Z_{ν′}(A)}
= Z_{ν′}(A) · u^{−1}

Remark 2.10. In general the Nakayama twisted centre will not be a ring: if a, b ∈ Z_{ν}(A), then

b(a_{1}a_{2}) = (ba_{1})a_{2} = (a_{1}ν(b))a_{2} = a_{1}(ν(b)a_{2}) = a_{1}a_{2}ν^{2}(b)

and ν^{2} = ν is equivalent to ν = id. Nevertheless, if z ∈ Z(A) and a ∈ Z_{ν}(A) then

b · za = zba = za · ν(b)

and za ∈ Z_{ν}(A). Hence Z_{ν}(A) is a Z(A)-submodule of A. The module structure does not depend on the chosen Nakayama automorphism, up to isomorphism of Z(A)-modules.

Remark 2.11. In case one follows Nakayama’s original definition of a Nakayama automorphism we need to replace ν by ν^{−1} and hence there the Nakayama twisted centre would consist of elements a satisfying ν(b) · a = a · b.

We summarise our results in the following

Proposition 2.12. (Holm and Zimmermann [24, Lemma 2.4]) If A is a selfinjective Kalgebra over a field K with Nakayama automorphism ν. Then Hom_{K}(A/[A, A], K) ≃ Z_{ν}(A) as Z(A)-modules.

Again, the proposition holds as well for K being a commutative ring.

2.4. How to get the Nakayama automorphism explicitly. Let K be a field and let A be a finite dimensional selfinjective K-algebra.

In order to compute the Nakayama automorphism ν we need to find an explicit isomorphism A ⟶ Hom_{K}(A, K) as A-modules.

Proposition 2.13. (Holm and Zimmermann [24, Lemma 2.7]) Let K be a field and let A be a finite dimensional selfinjective K-algebra with associated bilinear form ⟨ , ⟩. Then the Nakayama automorphism ν of A, satisfies ⟨a, b⟩ = ⟨b, ν(a)⟩ for all a, b ∈ A, and any automorphism satisfying this formula is a Nakayama automorphism.

Remark 2.14. If one would use Nakayama’s original definition the form would satisfy ⟨ν(a), b⟩ = ⟨b, a⟩ for all a, b ∈ A.

Proof of Proposition 2.13. There is a non-degenerate associative bilinear form on A, which induces an isomorphism between A and the linear forms on A as A-modules by Proposition 2.2. The isomorphism gives an isomorphism of A-A-bimodules of _{1}A_{ν} and Hom_{K}(A, K) by

1A_{ν} ⟶^{φ} Hom_{K}(A, K)
a ↦ ⟨−, a⟩ = φ(a).

Therefore, φ(a) = φ(1 · ν^{−1}(a)) = φ(1) · ν^{−1}(a) and φ(a) = φ(a · 1) = a · φ(1). Since for f ∈ Hom_{K}(A, K) one has (fa)(b) = f(ab) and (af)(b) = f(ba) for all a, b ∈ A, one gets

⟨b, a⟩ = (φ(a))(b) = (φ(1) · ν^{−1}(a))(b) = φ(1)(ν^{−1}(a)b)
= (b · φ(1))(ν^{−1}(a)) = φ(b)(ν^{−1}(a)) = ⟨ν^{−1}(a), b⟩.

Now, putting a := ν(a′) one gets

⟨b, ν(a′)⟩ = ⟨a′, b⟩

for all a′, b ∈ A. Hence, the Nakayama automorphism has the above property. Conversely, if an automorphism ν satisfies ⟨a, b⟩ = ⟨b, ν(a)⟩ for all a, b ∈ A, then the mapping A ⟶ Hom_{K}(A, K) given by a ↦ ⟨−, a⟩ gives an isomorphism of A and Hom_{K}(A, K) as Amodules, inducing the element _{1}A_{ν} in the Picard group of A.

2.5. Practical questions for algebras given by quivers and relations. In [23] the following very useful result was proved for weakly symmetric algebras. However the statement holds in a more general form.

Proposition 2.15. Let K be a field and let A = KQ/I be a selfinjective algebra given by the quiver Q and ideal of relations I, and fix a k-basis B of A consisting of pairwise distinct non-zero paths of the quiver Q. Assume that B contains a basis of the socle soc(A) of A. Define a K-linear mapping ψ on the basis elements by

ψ(b) = \{ \begin{matrix}1 & if b ∈ soc(A) \\ 0 & otherwise \end{matrix}

for b ∈ B. Then an associative non-degenerate K-bilinear form ⟨−, −⟩ for A is given by ⟨x, y⟩ := ψ(xy).

Remark 2.16. In case A is weakly symmetric Proposition 2.15 was proved in [23]. The assumption that A is weakly symmetric was used in [23] only to prove the non degeneracy of the form. For the reader’s convenience we include a complete proof.

Proof. By definition, since A is an associative algebra, ψ is associative on basis elements, hence is associative on all of A.

Let ν be a Nakayama automorphism of A. We observe now that ψ(x · ν(e)) = ψ(e · x) for all x ∈ A and all primitive idempotents e ∈ A. Indeed, since ψ is linear, we need to show this only on the elements in B. Let b ∈ B. If b is a path not in the socle of A, then bν(e) and eb are either zero or not contained in the socle either, and hence 0 = ψ(b) = ψ(bν(e)) = ψ(eb). If b ∈ B is in the socle of A, then b = e_{b}b = bν(e_{b}) for exactly one primitive idempotent e_{b} and e′b = bν(e′) = 0 for each primitive idempotent e′ ≠ e_{b}. Therefore, ψ(e′b) = ψ(bν(e′)) = 0 and ψ(e_{b}b) = ψ(b) = ψ(bν(e_{b})).

It remains to show that the map (x, y) ↦ ψ(xy) is non-degenerate. Suppose we had x ∈ A \ {0} so that ψ(xy) = 0 for all y ∈ A. In particular for each primitive idempotent e_{i} of A we get ψ(e_{i}xy) = ψ(xyν(e_{i})) = 0 for all y ∈ A. Hence we may suppose that x ∈ e_{i}A for some primitive idempotent e_{i} ∈ A.

Now, xA is a right A-module. Choose a simple submodule S of xA and s ∈ S \{0}. Then, since s ∈ S ≤ xA there is a y ∈ A so that s = xy. Since S ≤ xA ≤ A, and since S is simple, s ∈ soc(A) \ {0}. Moreover, since x ∈ e_{i}A, also s = e_{i}s, i.e. s is in the (1-dimensional) socle of the projective indecomposable module e_{i}A. So, up to a non zero scalar factor, s is a path contained in the basis B (recall that by assumption B contains a basis of the socle). This implies that

ψ(xy) = ψ(s) = ψ(e_{i}s) ≠ 0,

contradicting the choice of x, and hence proving non-degeneracy.

Remark 2.17. It should be noted that the form depends on the chosen basis of the algebra. Indeed, take A = K[X]/X^{2}. The socle is one-dimensional, and take a basis {X}. Then one may complete with an element 1 + μX for μ ∈ K to a basis of A. Hence 1 + X = (1 + μX) + (1 − μ)X and we get ⟨1 + X, 1⟩ = 1 − μ which depends heavily on μ.

Example 2.18. This example was communicated to me by Guodong Zhou in January 2010 during a visit in Paderborn. Let K be a field and let A_{q} = K⟨X, Y ⟩/(X^{2}, Y^{2}, XY − qY X) for q ≠ 0. Then A_{q} is always selfinjective and A_{q} is symmetric if and only if q = 1. Now, ν(X) = qX and ν(Y) = q^{−1}Y defines a Nakayama automorphism. Indeed if we use the K-basis {1, X, Y, XY } for A_{q} we get

1 = ⟨X, Y ⟩ = ⟨Y, ν(X)⟩ = ⟨Y, qX⟩ = 1
1 = ⟨Y, X⟩ = ⟨X, ν(Y)⟩ = ⟨X, q^{−1}Y ⟩ = 1

and likewise for the other basis elements.

Now, if q ≠ 1, we get

Z(A_{q}) = K · 1 + K · XY

whereas

Z_{ν}(A_{q}) = K · X + K · Y + K · XY.

Hence the Z(A_{q})-module Z_{ν}(A_{q}) is isomorphic to (Z(A_{q})/rad(Z(A_{q})))^{3}.

If A is a symmetric algebra, and ν is chosen to be inner, then Z_{ν}(A) is a rank 1 free Z(A)-module.

Observe that the order of the above automorphism is the multiplicative order of q in K. Hence for big fields K it is possible to create algebras with Nakayama automorphisms of any given order, even infinite order.

Example 2.19. Proposition 2.15 is stated in [21] and in [22] with an additional conclusion. Namely it is stated there that the form ⟨ , ⟩ is symmetric in case the algebra is symmetric. This is not true in general. A counterexample was given during the collaboration on [24]. Namely, the deformed preprojective algebra in the sense of Białkowski, Erdmann and Skowroński [5] of type L_{n} for n ≥ 3 gives an example (see Remark 5.4 below). This algebra is defined by the quiver

[figure: •; 0; •; 1; •; 2; •; 3; •; n − 2; •; n − 1; ⋯; ⋯; ⋯; ϵ; a_{0}; a_{1}; a_{2}; a_{n−2}; ^{\overline{a}^{0}}; ^{\overline{a}^{1}}; ^{\overline{a}^{2}}; ^{\overline{a}^{n−2}}]

subject to the following relations

a_{i}\overline{a}_{i} + \overline{a}_{i−1}a_{i−1} = 0 for all i ∈ \{1, …, n − 2\} ,
\overline{a}_{n−2}a_{n−2} = 0 , ϵ^{2n} = 0 , ϵ^{2} + a_{0}\overline{a}_{0} + ϵ^{3}p(ϵ) = 0

for a polynomial p(X) ∈ K[X]. These algebras are the deformed preprojective algebras of type L_{n}, in the sense of Białkowski, Erdmann and Skowroński [5].

For the special case p(X) = X^{2j} for j ∈ ℕ and for abbreviation we call this algebra by L_{n}^{j} and assume that K is of characteristic 2. Here we just give an example where the bilinear form Proposition 2.15 does not yield a symmetric bilinear form. We will deal with the general case in a later Section 5.2.

In order to be able to apply Proposition 2.15 we need to fix a basis of the socle of L_{n}^{j}. The fact that the elements below is indeed a basis of the algebra is shown in [24] and the basis displayed in Proposition 5.6 can easily be transformed into the basis below. Most recently in a completely independent approach Andreu [1] shows that the basis below is indeed a basis.

For our purpose it seems to be most natural to take as K-basis of the socle the set

\{ϵ^{2n−1}, \overline{a}_{i−1}\overline{a}_{i−2} … \overline{a}_{0}ϵ^{2n−3−2i}a_{0}a_{1} … a_{i−1} | i ∈ \{1, 2 …, n − 2\}\}.

Complete the elements

\overline{a}_{i−1}\overline{a}_{i−2} … \overline{a}_{0}ϵ^{2n−3−2i}a_{0}a_{1} … a_{i−1}

of e_{i}L_{n}^{j}e_{i}, for i ≥ 1, to a basis of e_{i}L_{n}^{j}e_{i} by the elements

\overline{a}_{i−1}\overline{a}_{i−2} … \overline{a}_{0}ϵ^{ℓ}a_{0}a_{1} … a_{i−1}

for ℓ ≤ 2n − 4 − 3i and

a_{i}a_{i+1}a_{i+2} … a_{j}\overline{a}_{j}\overline{a}_{j−1} … \overline{a}_{i+2}\overline{a}_{i+1}\overline{a}_{i}

for i + 1 ≤ j ≤ n − 2. A basis of e_{0}L_{n}^{j}e_{0} is given by ϵ^{ℓ}, for 0 ≤ ℓ ≤ 2n − 1.

Now we verify

⟨\overline{a}_{0}ϵ^{m}, a_{0}⟩ = \{ \begin{matrix}1 & if m = 2n − 3 \\ 0 & if m ≠ 2n − 3 \end{matrix}

⟨a_{0}, \overline{a}_{0}ϵ^{m}⟩ = \begin{cases}1 & if m = 2n − 3 or m = 2n − 4 − 2j \\ 0 & else \end{cases}

Hence the bilinear form from Proposition 2.15 is not symmetric. However, the algebras L_{n}^{j} are symmetric (cf Proposition 8 below).

3. Külshammer, the new idea in the 1980s

We come back to Brauer’s proof displayed in Section 1, Reynolds discoveries and present the original approach which was introduced by Külshammer to improve and unify these earlier approaches. Moreover we explain the generalisation to non symmetric algebras.

3.1. Külshammer’s original construction for symmetric algebras. Recall that for a symmetric K-algebra A one defined T_{n}(A) := {a ∈ A| a^{p^{n}} ∈ [A, A]} and T_{n}^{⊥}(A) is the orthogonal space with respect to the symmetrising form.

Definition 3.1. (Külshammer [30, Part IV]) Let A be a symmetric K-algebra. Then the ideal T_{n}(A)^{⊥} of Z(A) is the n-th Külshammer ideal.

Remark 3.2. • Remark 1.2 gives that T (A) = [A, A] + rad(A). Hence

T (A)^{⊥} = rad(A) ∩ soc(A) =: R(A).

• Külshammer calls the ideals T_{n}(A)^{⊥} the generalised Reynolds ideals.

• Since T_{n}(A) ⊆ T_{n+1}(A) we get T_{n}(A)^{⊥} ⊇ T_{n+1}(A)^{⊥} and the set of Külshammer ideals is a decreasing sequence of ideals of the centre with first term being the centre and last term being the Reynolds ideal R(A).

Külshammer obtained in [30] many properties of a group in terms of properties of the sequence of ideals T_{n}(KG)^{⊥}. Already in the first discussions in [30] the setup was completely general and the definitions were given for a symmetric algebra over a field of finite characteristic in general. Nevertheless, the applications in focus in Külshammer’s discussions mainly have been group algebras and representations of finite groups.

The main tool for technical proofs is still the symmetrising form. Since we have by Proposition 2.3 that [A, A]^{⊥} = Z(A), the restriction of the symmetrising form ⟨ , ⟩ to Z(A) on the left argument induces a non degenerate form, also denoted by ⟨ , ⟩,

⟨ , ⟩ : Z(A) × A/[A, A] ⟶ K.

As we have seen in Lemma 1.1 the mapping

A/[A, A] ⟶^{μ} A/[A, A]
a + [A, A] ↦ a^{p} + [A, A]

is additive and semilinear, i.e. linear if one applies in addition a twist with the Frobenius automorphism of the field. If V and W are finite dimensional vector spaces over a field K and if ⟨., .⟩ is a non degenerate bilinear pairing V × W ⟶ K. Then any endomorphism φ of W has a unique left adjoint φ^{∗} ∈ End_{K}(V) satisfying ⟨v, φ(w)⟩ = ⟨φ^{∗}(v), w⟩ for all v ∈ V and w ∈ W.

Now, this fact holds for semi-linear maps as well as for linear maps, and the map μ has a left adjoint

ζ : Z(A) ⟶ Z(A).

Now, a ∈ T_{n}(A) ⇔ a ∈ ker(μ^{n}) gives

Lemma 3.3. z ∈ T_{n}(A)^{⊥} ⇔ z ∈ im(ζ^{n}).

This characterisation will be the main tool for most of the abstract statements later.

3.2. Extending to general algebras: trivial extension algebras and Külshammer theory. Up to now, in order to establish a Külshammer ideal theory it was necessary already for the very definition to work over symmetric algebras. There is a method to circumvent this difficulty.

Let A be any finite dimensional K-algebra. Then, as already mentioned, Hom_{K}(A, K) is an A − A-bimodule by

(afb)(c) := f(bca) ∀a, b, c ∈ A and f ∈ Hom_{K}(A, K).

Recall the construction of the trivial extension algebra, which is well-known and very useful in the representation theory of associative algebras. We may form 𝕋A := Hom_{K}(A, K)× A, which is naturally a K-vector space. We may define an algebra structure on this space.

(f, a) · (g, b) := (ag + fb, ab) ∀ a, b ∈ A, f, g ∈ Hom_{K}(A, K) .

It is a tedious but straightforward computation to verify that 𝕋A is a K-algebra by this multiplication. Moreover the projection to the second component is an algebra homomorphism 𝕋A ⟶ A with kernel Hom_{K}(A, K) being an ideal with square 0.

Definition 3.4. For any finite dimensional K-algebra A the algebra 𝕋A is called the trivial extension algebra.

The property that is most interesting for our purposes is that 𝕋A is a symmetric algebra, whatever may be the structure of the algebra A. Indeed,

⟨(f, a), (g, b)⟩ := g(a) + f(b) ∀(f, a), (g, b) ∈ 𝕋A

is a symmetric associative non degenerate bilinear form on 𝕋A. This fact can be found in e.g. [4, Section 3]. Proposition 2.2 shows then the statement.

In [4] Bessenrodt, Holm and the author compute the Külshammer ideals of 𝕋A. We denote

Ann_{Hom_{K}(A,K)}(I) := {f ∈ Hom_{K}(A, K) | f(I) = {0}}

for any subset I ⊆ A. With this notation we showed

Proposition 3.5. (Bessenrodt, Holm, Zimmermann [4, Theorem 4.1]) Let A be a finitedimensional algebra over a field of characteristic p > 0, and let 𝕋A be its trivial extension.

(1) We have T_{0}(𝕋A)^{⊥} = Z(𝕋A) = Ann_{Hom_{K}(A,K)}([A, A]) × Z(A).

(2) For all n ≥ 1 one has T_{n}(𝕋A)^{⊥} = Ann_{Hom_{K}(A,K)}(T_{n}A) × 0.

This result, though not difficult to prove, is most remarkable, since for symmetric algebras we may use the symmetrising form to transport T_{n}(A) via orthogonality to an ideal T_{n}(A)^{⊥} of the centre of A. The ideal structure allows to consider many invariants from commutative algebra attached to this ideal T_{n}(A)^{⊥}. If A is not symmetric, this is not easily possible. Hence it is surprising that enlarging A to 𝕋A, the space T_{n}(𝕋A)^{⊥} keeps the trace of T_{n}(A)^{⊥} faithfully in the sense that T_{n}(A)^{⊥} can be fully recovered by T_{n}(𝕋A)^{⊥}.

If A is already symmetric, the isomorphism 𝕋A ⟶ Hom_{K}(𝕋A, K) takes T_{n}(A)^{⊥} to Ann_{Hom_{K}(A,K)}(T_{n}(A)). This fact is not hard to see. Indeed, if A is symmetric, Hom_{K}(A, K) is the space of all ⟨a, −⟩ for a ∈ A. Now, a linear form being in Ann_{Hom_{K}(A,K)}(T_{n}(A)) is equivalent to ⟨a, T_{n}(A)⟩ = 0 which in turn is equivalent to a ∈ T_{n}(A)^{⊥}.

4. Morita, derived and stable invariance

4.1. Morita invariance. In [19] Héthélyi, Horváth, Külshammer and Murray studied amongst other questions the invariance of Külshammer ideals T_{n}(A)^{⊥} under Morita equivalence. Recall that if

M ⊗_{A} − : A − mod ⟶ B − mod

is an equivalence, then for any z ∈ Z(A) there is a unique φ_{M}(z) ∈ Z(B) so that m · z = φ_{M}(z) · m for all m ∈ M. Then, φ_{M} : Z(A) ⟶ Z(B) is an isomorphism of algebras.

Proposition 4.1. (Héthelyi, Horváth, Külshammer and Murray [19, Corollary 5.3]) Let K be a perfect field of characteristic p > 0 and let A and B be finite dimensional K-algebras. If

M ⊗_{A} − : A − mod ⟶ B − mod

is a Morita equivalence, then φ_{M}(T_{n}(A)^{⊥}) = T_{n}(B)^{⊥} for all n ∈ ℕ.

The authors show that the mapping ζ of Lemma 3.3 behaves well with respect to multiplication by idempotents. Using this statement it is possible to reduce to basic algebras, and to use then that two Morita equivalent basic algebras are isomorphic.

The existence of basic algebras, i.e. an up to isomorphism unique minimal algebra which is Morita equivalent to the given algebra, is very specific for Morita equivalences. Such a concept does not exist for weaker equivalences such as derived equivalences or stable equivalences of Morita type.

4.2. Derived invariance. [19, Question 5.4] asked if Külshammer ideals are also invariant under derived equivalences. The method used for the Morita invariance does not apply since as said before a concept of ”derived basic” algebras do not exist. Nevertheless, an equivalence between derived categories of finite dimensional algebras imply the existence of an isomorphism of the centres.

More precisely, denote by D^{b}(A) the derived category of bounded complexes of finitely generated A-modules with non zero homology in only finitely many degrees.

The main tool is the following result.

Theorem 2. (Rickard [43, Theorem 3.3], Keller [26, Section 8]; cf also e.g. [28, Theorem 6.2.8]) Let K be a field, let A and B be finite dimensional K-algebras and suppose that D^{b}(A) ≃ D^{b}(B) is an equivalence of triangulated categories. Then there is a complex X ∈ D^{b}(B ⊗_{K} A^{op}) which is formed by modules which are projective as B-modules and projective as A-modules, so that

X ⊗_{A}^{𝕃} − : D^{b}(A) ⟶ D^{b}(B)

is an equivalence of triangulated categories.

A complex X as in the theorem is called two-sided tilting complex. It is unknown if every equivalence between derived categories is of the form

X ⊗_{A}^{𝕃} − : D^{b}(A) ⟶ D^{b}(B)

This is shown to hold on the level of objects, but it is not known if there may be an exotic equivalence behaving differently on morphisms (cf Rickard [43, Corollary 3.5]).

The result has many consequences. In particular

X ⊗_{A}^{𝕃} (− ⊗_{A}^{𝕃} Hom_{A}(X, A)) : D^{b}(A ⊗_{K} A^{op}) ⟶ D^{b}(B ⊗_{K} B^{op})

is an equivalence. Therefore, if D^{b}(A) ≃ D^{b}(B) there exists a two-sided tilting complex realising an equivalence. The given equivalence one started with may be different. Then, this two-sided tilting complex induces the equivalence of the derived categories of bimodules, and this then has the property that

End_{D^{b}(A⊗A^{op})}(A) ⟶^{≃} End_{D^{b}(B⊗B^{op})}(B)

is an equivalence.

It is known that the for any algebra C the module category is a full subcategory of the derived category by identifying a module with the complex concentrated in a single degree 0 (cf e.g. Verdier [45, Chapitre III Section 1.2.9]). Therefore for a C-module M one gets End_{C}(M) ≃ End_{D^{b}(C)}(M). Hence End_{D^{b}(A⊗A^{op})}(A) ≃ End_{A⊗A^{op}}(A) ≃ Z(A) and End_{D^{b}(B⊗B^{op})}(B) ≃ End_{B⊗B^{op}}(B) ≃ Z(B).

Corollary 4.2. (Rickard [43, Proposition 2.5]) Let K be a field and let A and B be finite dimensional K-algebras. If D^{b}(A) ≃ D^{b}(B), then a choice of a two-sided tilting complex X realising this equivalence as tensor product induces an isomorphism φ_{X} as algebras between the centre of A and the centre of B.

We see that still an equivalence between the derived categories of finite dimensional algebras yield an isomorphism between the centres of the algebras. The isomorphism however is far less explicit and somewhat complicate.

Despite these difficulties we obtain the following result.

Theorem 3. [52, Theorem 1] Let A and B be finite dimensional symmetric K-algebras over a perfect field K of characteristic p > 0. Suppose that D^{b}(A) ≃ D^{b}(B) as triangulated categories. Then any choice of a two-sided tilting complex X yields an isomorphism φ_{X} : Z(A) ⟶ Z(B) satisfying φ_{X}(T_{n}(A)^{⊥}) = T_{n}(B)^{⊥} for all n ∈ ℕ.

It is worth writing that we use here the mapping that is induced by a functor on the morphisms. We recall that possibly non standard derived equivalences exist. Non standard derived equivalences are not standard only on morphisms, but we only use morphisms here. Hence non standard derived equivalences would possibly induce an isomorphism between the centres which does not preserve the Külshammer ideal structure.

The proof is much more involved than the proof for Morita invariance in the sense that one needs to reformulate the construction of Külshammer ideals in a ”derived category readable form”. Instead of explicit constructions of particular sets one needs to argue via homological properties of morphism spaces.

One should mention that being symmetric is an invariant under derived equivalences.

Proposition 4.3. (Rickard [43, Corollary 5.3] for fields R, [51] for more general rings) Let R be a Dedekind domain and let A and B be R-algebras of finite rank over R so that D^{b}(A) ≃ D^{b}(B). Then if A is symmetric, B is symmetric as well.

Applications of Theorem 3 will be given in Section 5. Actually the invariance of Külshammer ideals proved to be a rather powerful tool in particular in order to distinguish algebras given by quivers and relations where the relations depend on certain parameters. The structure of the quotient Z(A)/T_{n}(A)^{⊥} tends to depend on the parameters in several cases.

4.3. Stable invariance. As usual the stable category A−\underline{mod} of a module category A−mod is the category with objects being A-modules and morphisms between two A-modules in the stable category are equivalence classes of morphisms between these A-modules modulo those which factor through a projective A-module. We denote by \underline{Hom}_{A}(M, N) the morphisms in A − \underline{mod} from M to N.

Equivalences between stable categories can behave badly in general. An example was given by Auslander and Reiten in 1973 [2, Example 3.5].

Example 4.4. Let K be a field. Then the algebras

A := ( \begin{matrix}K & K \\ 0 & K \end{matrix} ) × ( \begin{matrix}K & K \\ 0 & K \end{matrix} )

and

B := ( \begin{matrix}K & K & K \\ 0 & K & K \\ 0 & 0 & K \end{matrix} ) /( \begin{matrix}0 & 0 & K \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} )

the quotient of the upper triangular matrix ring by the ideal generated by the upper right component matrices have equivalent stable categories. Indeed, A has six indecomposable modules: four simple modules, two of which are projective, and the projective cover of the non projective simple. Hence the stable module category is equivalent to two copies of K − mod. The algebra B has five indecomposable modules: three simple modules, one of which is projective and the projective covers of the other two simple modules. Hence the stable module category is equivalent to two copies of K − mod as well.

The algebras A and B are hence stably equivalent and B is indecomposable whereas A is not. Neither A nor B has any simple direct factor.

Given two self-injective algebras A and B and suppose X is a complex of A−B-bimodules inducing a standard equivalence D^{b}(B) ⟶ D^{b}(A). The quasi-inverse is again a standard equivalence, given by a complex Y of B − A-bimodules. Then a somewhat technical construction on X and on Y produces an A − B-bimodule M, projective on either side, and a B − A-bimodule N, projective on either side, so that the A − A-bimodule M ⊗_{B} N is isomorphic to A ⊕ P for some projective A − A-bimodule P, and so that one has an isomorphism of B − B-bimodules N ⊗_{A} M ≃ B ⊕ Q for some projective B − B-bimodule Q. This motivated Broué to define a class of stable equivalences with nicer properties.

Definition 4.5. (Broué [14, Section 5]) Let K be a commutative ring, let A and B be two K-algebras and let M ∈ A ⊗_{K} B^{op} − mod and let N ∈ B ⊗_{K} A^{op} − mod. Then (M, N) is said to induce a stable equivalence of Morita type if

• M as well as N are projective as A-modules and as B-modules.

• M ⊗_{B} N ≃ A ⊕ P as A − A-bimodules for a projective A − A-bimodule P

• N ⊗_{A} M ≃ B ⊕ Q as B − B-bimodules for a projective B − B-bimodule Q.

Remark 4.6. Liu shows in [32, Theorem 2.2] that a stable equivalence of Morita type between two finite dimensional algebras with no separable summands restricts to a stable equivalence between their summands. Therefore the algebras A and B in Example 4.4 are not stably equivalent of Morita type.

In the meantime many properties have been shown to be invariant under stable equivalence of Morita type, whereas the general stable equivalences are still rather poorly understood.

In particular, the following definition will be of importance in our discussion. Let A be a K-algebra. Then

Z(A) = Hom_{A⊗_{K}A^{op}}(A, A)

and define the stable centre

Z^{st}(A) = \underline{Hom}_{A⊗_{K}A^{op}}(A, A)

The natural homomorphism

Hom_{A⊗_{K}A^{op}}(A, A) ⟶ \underline{Hom}_{A⊗_{K}A^{op}}(A, A)

has a kernel denoted Z^{pr}(A), the projective centre.

Proposition 4.7. (Broué [14, Proposition 5.4]) Let A and B be finite dimensional Kalgebras and let (M, N) be bimodules inducing a stable equivalence of Morita type. Then Z^{st}(A) ≃ Z^{st}(B) as algebras.

How can we determine Z^{pr}(A)? This is a result of Liu, Zhou and the author. The Cartan matrix of the algebra A is denoted by C_{A}. Recall that the Cartan matrix is square of size n, where n is the number of simple A-modules up to isomorphism. If one labels the rows and the columns by the isomorphism classes [S] of simple modules, then we have that the coefficient in position ([S], [T]) is Hom_{A}(P_{S}, P_{T}), where P_{S} denotes the projective cover of S. Hence C_{A} has integer coefficients and can therefore be interpreted as a linear endomorphism of the K-vector space K^{n}.

Proposition 4.8. (Liu, Zhou, Zimmermann [33, Proposition 2.4, Corollary 2.9, Lemma 7.8, Proposition 7.10]) Let K be an algebraically closed field and let A be a finite dimensional symmetric K-algebra. Then Z^{pr}(A) ⊆ Z(A) ∩ soc(A) = R(A) and dim_{K}(Z^{pr}(A)) = rank_{K}C_{A}.

In order to adapt the Külshammer ideal theory for stable equivalences of Morita type we need to replace the centre by the stable centre since we know that a stable equivalence gives an isomorphism of the stable centres and we do not have enough information about the centre.

Further we need to find a replacement of A/[A, A]. For this purpose we recall the Hattori-Stallings trace which was generalised by Bouc to a trace function on the whole Hochschild homology.

Definition 4.9. (Bouc [11] for higher dimensional Hochschild homology, Hattori-Stallings in degree 0) Let K be a field and let A and B be two finite dimensional K-algebras. Given an A − B-bimodule M which is projective as B-module, then there are elements m_{i} ∈ M, φ_{i} ∈ Hom_{B}(M, B), for i = 1, …, n so that the identity on M in End_{B}(M) ≃ M ⊗_{B}Hom_{B}(M, B) is mapped to ∑_{i=1}^{n} m_{i} ⊗ φ_{i}. The fact that M is an A − B-bimodule gives a mapping

A ⟶^{α_{M}} End_{B}(M) ≃ M ⊗_{B} Hom_{B}(M, B)
a ↦ ∑_{i=1}^{n}(am_{i}) ⊗ φ_{i}

We produce

eval : M ⊗_{B} Hom_{B}(M, B) ⟶ B/[B, B]

by eval(m ⊗ ψ) := ψ(m) + [B, B] for ψ ∈ Hom_{B}(M, B) and m ∈ M. The composition eval ∘ α factorises through A/[A, A] and the resulting mapping

A/[A, A] ⟶ B/[B, B]

is called the trace of M, denoted by tr_{M}. Similar statements hold if M is projective on the left.

Using the Hattori-Stallings trace we define

Definition 4.10. [33, Defintion 4.1] Let A be a finite dimensional K-algebra

HH_{0}^{st}(A) := ⋂_{P projective indecomposable A−mod} ker(tr_{P})

observing that any projective A-module P is a K −A-bimodule as required by Definition 4.9.

With this preparation we obtain that the dimension of HH_{0}^{st}(A) is an invariant under stable equivalence of Morita type. Denote by ℓ(A) the number of simple A-modules up to isomorphism.

Theorem 4. Let K be an algebraically closed field and let A and B be finite dimensional Kalgebras without any semisimple direct factor and suppose that A and B are stably equivalent of Morita type.

• (Liu, Zhou, Zimmermann [33, Theorem 6.1]) Then

dim_{K}(A/[A, A]) = dim_{K}(B/[B, B]) ⇔ ℓ(A) = ℓ(B).

Moreover dim(T_{n}(A)/K(A)) = dim(T_{n}(B)/K(B)).

• (Liu, Zhou, Zimmermann [33, Corollary 6.2]) If in addition A is symmetric then

dim_{K}(Z(A)) = dim_{K}(Z(B)) ⇔ ℓ(A) = ℓ(B).

and (König, Liu, Zhou [29, Proposition 5.8])

Z(A)/T_{n}(A)^{⊥} ≃ Z(B)/T_{n}(B)^{⊥}

for all n ≥ 1.

The proof of the first part uses first that HH_{0}^{st}(A) is an invariant under stable equivalences of Morita type. Then one shows [33, Theorem 4.4] that

dim(HH_{0}^{st}(A)) + rank_{K}(C_{A}) = dim(A/[A, A])

Further it is shown in [33, Section 5] that rank_{K}(C_{A}) − dim_{K}(K ⊗_{ℤ} K_{0}(A)) equals the dimension of the so-called stable Grothendieck group, which is known to be an invariant under stable equivalences of Morita type by work of Xi [46, Section 5].

We should mention the long standing Auslander-Reiten conjecture.

Conjecture. (Auslander-Reiten [3, page 409 Conjecture 5]) Let A and B be finite dimensional K-algebras. If A and B are stably equivalent, then the number of simple non projective A-module up to isomorphism equals the number of simple non projective B-modules up to isomorphism.

A priori I feel that there is no obvious reason why the invariance of the number of simple A-modules has anything to do with the commutator quotient. This fact appears somewhat surprisingly in this connection.

The conjecture has been verified for quite a few classes of algebras. Few general positive results are known so far.

5. Applications

In the last five years Külshammer ideals were successfully employed to distinguish algebras up to derived and up to stable equivalences of Morita type for various classes of algebras which were extremely difficult to deal with previously. In particular if two algebras are defined by the same quiver Q and a set of relations I(c) subject to some parameter c in the base field, then the technique of computing Külshammer ideals and the quotient of the centre by the ideal proves to be fruitful in various cases.

5.1. Algebras of dihedral, semidihedral and quaternion type. Many authors during the last decades proved ring theoretic properties for group algebras, and still the question is an active field of research. In particular many properties are shown to hold for the Cartan matrices and the occurrence of certain components in the stable Auslander-Reiten quiver for blocks of group algebras with dihedral, semidihedral or quaternion defect groups. Moreover it was shown at that time that a block of a group algebra is of tame representation type (cf Section 5.3 below for the precise definition) if and only if the defect group is a dihedral, a semidihedral or a quaternion group.

Erdmann showed in [16] that these properties determine the Morita equivalence classes of these algebras as belonging to a finite number of families, given by quivers with relations, subject to certain parameters in the relations. Up to these parameters in the relations the algebras are classified in a finite number of classes up to Morita equivalences.

Holm [20] classified further these Morita equivalence classes up to derived equivalences. Many classes merge to a common derived equivalence class. However, Holm could not determine for a certain number of parameters if two algebras within one class but with different parameters are derived equivalent or not.

Külshammer ideals manage to distinguish derived equivalence classes in some of these cases.

We display Thorsten Holm’s list [20] of algebras of dihedral, semidihedral and quaternion type up to derived equivalences. Each of these types form a family. Each family is subdivided into three subclasses: algebras with one simple module, algebras with two simple modules and algebras with three simple modules. Each subfamily contains algebras defined by quivers and relations, depending on parameters.

18 ALEXANDER ZIMMERMANN

dihedral semidihedral

quaternion

1 simple K[X, Y]/(XY, X^{m} − Y^{n}), SD(1A)_{1}^{k}, k ≥ 2; Q(1A)_{1}^{k}, k ≥ 2;

m ≥ n ≥ 2, m + n > 4; D(1A)_{1}^{1} = K[X, Y]/(X^{2}, Y^{2});

(charK = 2)  (charK = 2) SD(1A)_{2}^{k}(c, d) (charK = 2) Q(1A)_{2}^{k}(c, d),
K[X, Y]/(X^{2}, Y X − Y^{2}); k ≥ 2, (c, d) ≠ (0, 0); k ≥ 2, (c, d) ≠ (0, 0);
D(1A)_{1}^{k}, k ≥ 2;

(charK = 2) D(1A)_{2}^{k}(d),

k ≥ 2, d = 0 or 1;

2 simples  D(2B)^{k,s}(c),  SD(2B)_{1}^{k,t}(c)  Q(2B)_{1}^{k,s}(a, c)
k ≥ s ≥ 1, c ∈ {0, 1}  k ≥ 1, t ≥ 2, c ∈ {0, 1};  k ≥ 1, s ≥ 3, a ≠ 0;
SD(2B)_{2}^{k,t}(c)
k ≥ 1, t ≥ 2,
k + t ≥ 4, c ∈ {0, 1};
3 simples  D(3K)^{a,b,c},  SD(3K)^{a,b,c}  Q(3K)^{a,b,c}
a ≥ b ≥ c ≥ 1;  a ≥ b ≥ c ≥ 1, a ≥ 2;  a ≥ b ≥ c ≥ 1, b ≥ 2,
(a, b, c) ≠ (2, 2, 1);
D(3R)^{k,s,t,u},  Q(3A)_{1}^{2,2}(d)
s ≥ t ≥ u ≥ k ≥ 1, t ≥ 2  d ∉ {0, 1}

All algebras with one simple module in the above list have the quiver of type 1A

[figure: •; X; Y]

The quivers of the algebras of type 2B, 3K, 3A and 3R are respectively:

[figure: type 3K; •; •; •; β; γ; δ; η; λ; κ; type 2B; •; •; α; η; β; γ]

[figure: type 3A; •; •; •; η; β; γ; δ; type 3R; •; •; •; β; δ; λ; α; ξ; ρ]

The relations are respectively

D(1A)_{1}^{k} : X^{2}, Y^{2}, (XY)^{k} − (Y X)^{k};
D(1A)_{2}^{k}(d) : X^{2} − (XY)^{k}, Y^{2} − d · (XY)^{k}, (XY)^{k} − (Y X)^{k}, (XY)^{k}X, (Y X)^{k}Y;
SD(1A)_{1}^{k} : (XY)^{k} − (Y X)^{k}, (XY)^{k}X, Y^{2}, X^{2} − (Y X)^{k−1}Y;
SD(1A)_{2}^{k}(c, d) : (XY)^{k} − (Y X)^{k}, (XY)^{k}X, Y^{2} − d(XY)^{k},
X^{2} − (Y X)^{k−1}Y + c(XY)^{k};
Q(1A)_{1}^{k} : (XY)^{k} − (Y X)^{k}, (XY)^{k}X, Y^{2} − (XY)^{k−1}X, X^{2} − (Y X)^{k−1}Y;
Q(1A)_{2}^{k}(c, d) : X^{2} − (Y X)^{k−1}Y − c(XY)^{k}, Y^{2} − (XY)^{k−1}X − d(XY)^{k},
(XY)^{k} − (Y X)^{k}, (XY)^{k}X, (Y X)^{k}Y.

as well as

D(2B)^{k,s}(c) : βη, ηγ, γβ, α^{2} − c(αβγ)^{k}, (αβγ)^{k} − (βγα)^{k}, η^{s} − (γαβ)^{k};
SD(2B)_{1}^{k,t}(c) : γβ, ηγ, βη, α^{2} − (βγα)^{k−1}βγ − c(αβγ)^{k}, η^{t} − (γαβ)^{k}, (αβγ)^{k} − (βγα)^{k};
SD(2B)_{2}^{k,t}(c) : βη − (αβγ)^{k−1}αβ, ηγ − (γαβ)^{k−1}γα, γβ − η^{t−1}, α^{2} − c(αβγ)^{k}, βη^{2}, η^{2}γ;
Q(2B)_{1}^{k,s}(a, c) : γβ − η^{s−1}, βη − (αβγ)^{k−1}αβ, ηγ − (γαβ)^{k−1}γα,
α^{2} − a(βγα)^{k−1}βγ − c(βγα)^{k}, α^{2}β, γα^{2};
D(3K)^{a,b,c} : βδ, δλ, λβ, γκ, κη, ηγ, (βγ)^{a} − (κλ)^{b}, (λκ)^{b} − (ηδ)^{c}, (δη)^{c} − (γβ)^{a};
D(3R)^{k,s,t,u} : αβ, βρ, ρδ, δξ, ξλ, λα, α^{s} − (βδλ)^{k}, ρ^{t} − (δγβ)^{k}, ξ^{u} − (λβδ)^{k};
SD(3K)^{a,b,c} : κη, ηγ, γκ, δγ − (γα)^{a−1}γ, βδ − (κλ)^{b−1}κ, λβ − (ηδ)^{c−1}η;
Q(3K)^{a,b,c} : βδ − (κλ)^{a−1}κ, ηγ − (λκ)^{a−1}λ, δλ − (γβ)^{b−1}γ, κη − (βγ)^{b−1}β,
λβ − (ηδ)^{c−1}η, γκ − (δη)^{c−1}δ, γβδ, δηγ, λκη;
Q(3A)_{1}^{2,2}(d) : βδη − βγβ, δηγ − γβγ, ηγβ − dηδη, γβδ − dδηδ, βδηδ, ηγβγ.

For the dihedral type algebras with two simple modules a result of Kauer and Roggenkamp [25, Corollary 5.3] show that the parameters c = 0 and c = 1 yield different derived equivalence classes of algebras. The method employed there is rather involved. The authors define graph algebras and show that being a graph algebra is invariant under derived equivalences. Further for one of the scalars the algebra is a graph algebra, for the other it is not. Holm and the author gave a much simpler proof in [23] avoiding graph algebras.

The semidihedral type case can be dealt with at least partially. Again we consider the derived equivalence classes of semidihedral type algebras with two simple modules, and again the question if the parameters c = 0 and c = 1 yield different derived equivalence classes was open.

Theorem 5. (Holm and Zimmermann [23, Theorem 1.1, Theorem 1.2 and Theorem 1.3]) Let K be an algebraically closed field of characteristic 2.

• For any given integers k, s ≥ 1 consider the algebras of dihedral type D(2A)^{k,s}(c) for the scalars c = 0 and c = 1. Suppose that k ≥ 2. Suppose if k = 2 then s ≥ 3 is odd, and if s = 2 then k ≥ 3 is odd.

Put A_{0}^{k,s} := D(2A)^{k,s}(0) and A_{1}^{k,s} := D(2A)^{k,s}(1).

Then the factor rings Z(A_{0}^{k,s})/T_{1}(A_{0}^{k,s})^{⊥} and Z(A_{1}^{k,s})/T_{1}(A_{1}^{k,s})^{⊥} are not isomorphic as rings.

In particular, the algebras D(2A)^{k,s}(0) and D(2A)^{k,s}(1) are not derived equivalent and are not stably equivalent of Morita type.

• For any given integers k ≥ 1 and s ≥ 1, consider the algebras of semidihedral type SD(2B)_{1}^{k,s}(c) for the scalars c = 0 and c = 1. Suppose that k ≥ 2. Suppose that if k = 2 then s ≥ 3 is odd, and if s = 2 then k ≥ 3 is odd.

Put B_{0}^{k,s} := SD(2B)_{1}^{k,s}(0) and B_{1}^{k,s} := SD(2B)_{1}^{k,s}(1).

Then the factor rings Z(B_{0}^{k,s})/T_{1}(B_{0}^{k,s})^{⊥} and Z(B_{1}^{k,s})/T_{1}(B_{1}^{k,s})^{⊥} are not isomorphic as rings.

In particular, the algebras SD(2B)_{1}^{k,s}(0) and SD(2B)_{1}^{k,s}(1) are not derived equivalent and are not stably equivalent of Morita type in these cases.

• For any given integers k ≥ 1 and s ≥ 1, consider the algebras of semidihedral type SD(2B)_{2}^{k,s}(c) for the scalars c = 0 and c = 1. Suppose k ≥ 2. Put C_{0}^{k,s} := SD(2B)_{2}^{k,s}(0) and C_{1}^{k,s} := SD(2B)_{2}^{k,s}(1).

If the parameters k and s are both odd, then the factor rings Z(C_{0}^{k,s})/T_{1}(C_{0}^{k,s})^{⊥} and Z(C_{1}^{k,s})/T_{1}(C_{1}^{k,s})^{⊥} are not isomorphic as rings.

In particular, the algebras SD(2B)_{2}^{k,s}(0) and SD(2B)_{2}^{k,s}(1) are not derived equivalent and are not stably equivalent of Morita type in these cases.

Remark 5.1. (1) We should mention that actually the dimension of the quotients of the centres modulo the Külshammer ideals do not depend on the scalar c. The algebra structure of the quotient is needed.

(2) It is worth noticing that these algebras are all symmetric and so [52] applies directly. Moreover, the dimension of the centre of the algebra equals the dimension of the quotient of the algebra by the commutator subspace. This immediate consequence of the fact that the algebras are symmetric is not clear in case the algebra is selfinjective only. Example 2.18 gives an easy example for how complicated the situation might become already for very small selfinjective algebras. For general selfinjective algebras a rather sophisticated theory needs to be developed in order to compute the commutator subspace.

Remark 5.2. I would like to mention that in [23, Theorem 1.1] the condition that k ≥ 2 in case of algebras of dihedral type in Theorem 5 is unfortunately missing. The condition is necessary. A recent result of Frauke Bleher [8, Theorem 2] determines the parameter c in the relations for a specific group by completely different methods. If k = 1 would be allowed, then the parameter would be different than determined by Bleher. This observation is due to Zhou.

In recent work Zhou and the author studied if the derived equivalence classification of Holm of dihedral, semidihedral and quaternion type algebras also gives a classification up to stable equivalence. Some partial statements are already given in Theorem 5.

One has to deal with several additional problems for stable equivalences of Morita type. The first problem is that the Auslander-Reiten conjecture is open, that is we might a priori

have a stable equivalence of Morita type between two algebras of dihedral, semidihedral or quaternion type with different number of simple modules. This does not happen for derived equivalences since there the rank of the Grothendieck group is an invariant.

The second problem is that a derived equivalence between an algebra and a local algebra is in fact a Morita equivalence. This was shown by Roggenkamp and the author [50, Section 5]. The statement is false for stable equivalences of Morita type. Given a finite group G and a field K of characteristic p dividing the order of G a KG-module M is endotrivial if the KG-module End_{K}(M) has the property End_{K}(M) ≃ K ⊕ P for K being the trivial K-module and P a projective KG-module. Every endotrivial module over a p-group gives a stable self-equivalence of Morita type for the group ring over this p-group. The set of endotrivial modules over a fixed p-group up to some equivalence relation form a group, whose structure was completely determined by Carlson and Thévenaz, and which is non trivial free abelian in most cases [15].

Since the statement of the result might be technical for the non specialist reader we illustrate the result in a coarser form.

Recall that we have the following rough classification of algebras up to derived equivalences.

dihedral semidihedral

quaternion

1 simple five types of algebras two types of algebras

two types of algebras

depending on parameters depending on parameters

depending on parameters

2 simples one type of algebras two types of algebras

one type of algebras

depending on parameters depending on parameters

depending on parameters

3 simples two types of algebras one type of algebras

two types of algebras

depending on parameters depending on parameters

depending on parameters

Theorem 6 below states mainly that the columns are preserved under stable equivalences of Morita type and that the rows are preserved under stable equivalences of Morita type.

The actual statement is finer than this, but this scheme gives a relatively good approximation of what is proved in Theorem 6.

The details we obtain are given in the following result.

Theorem 6. (Zhou and Zimmermann [48, Theorem 7.1]) Let K be an algebraically closed field.

Suppose A and B are indecomposable algebras which are stably equivalent of Morita type.

• If A is an algebra of dihedral type, then B is of dihedral type. If A is of semidihedral type, then B is of semidihedral type. If A is of quaternion type then B is of quaternion type.

• If A and B are of dihedral, semidihedral or quaternion type, then A and B have the same number of simple modules.

• Let A be an algebra of dihedral type.

(1) If A is local, then A is stably equivalent of Morita type to one and exactly one

algebra in the following list:

– A_{1}(n, m) with m ≥ n ≥ 2 and m + n > 4;

– C_{1};

– D(1A)_{1}^{k} with k ≥ 2;

– if p = 2, B_{1} and D(1A)_{2}^{k}(d) with k ≥ 2 and d ∈ {0, 1}, except that we don’t know whether D(1A)^{k}(0) and D(1A)^{k}(1) are stably equivalent of Morita type or not.

(2) If A has two simple modules, then A is stably equivalent of Morita type to one and exactly one of the following algebras: D(2B)^{k,s}(0) with k ≥ s ≥ 1 or if p = 2, D(2B)^{k,s}(1) with k ≥ s ≥ 1.

(3) If A has three simple modules then A is stably equivalent of Morita type to one and exactly one of the following algebras: D(3K)^{a,b,c} with a ≥ b ≥ c ≥ 1 or D(3R)^{k,s,t,u} with s ≥ t ≥ u ≥ k ≥ 1 and t ≥ 2.

• Let A be an algebra of semidihedral type.

(1) If A has one simple module then A is stably equivalent of Morita type to one of the following algebras: SD(1A)_{1}^{k} for k ≥ 2 or SD(1A)_{2}^{k}(c, d) for k ≥ 2 and (c, d) ≠ (0, 0) if the characteristic of K is 2. Different parameters k yield algebras in different stable equivalence classes of Morita type.

(2) If A has two simple modules then A is stably equivalent of Morita type to SD(2B)_{1}^{k,s}(c) for k ≥ 1, s ≥ 2, c ∈ {0, 1} or to SD(2B)_{2}^{k,s}(c) for k ≥ 1, s ≥ 2, c ∈ {0, 1}, k + s ≥ 4.

(3) If A has three simple modules, then A is stably equivalent of Morita type to one

and only one algebra of the type SD(3K)^{a,b,c} for a ≥ b ≥ c ≥ 1.

• Let A be an algebra of quaternion type.

(1) If A has one simple modules, then A is stably equivalent of Morita type to one of the algebras Q(1A)_{1}^{k} for k ≥ 2 or Q(1A)_{2}^{k}(c, d) for k ≥ 2, (c, d) ≠ (0, 0) if characteristic if the K is 2. Different parameters k yield algebras in different stable equivalence classes of Morita type.

(2) If A has two simple modules then A is stably equivalent of Morita type to one of the algebras Q(2B)_{1}^{k,s}(a, c) for k ≥ 1, s ≥ 3, a ≠ 0.

(3) If A has three simple modules, then A is stably equivalent of Morita type to one of the algebras Q(3K)^{a,b,c} for a ≥ b ≥ c ≥ 1, b ≥ 2, (a, b, c) ≠ (2, 2, 1) or Q(3A)_{1}^{2,2}(d) for d ∈ K \ {0, 1}. Different parameters a, b, c yield algebras in different stable equivalence classes of Morita type.

One particularly nice consequence should be mentioned though.

Corollary 5.3. (Zhou and Zimmermann [48, Corollary 7.3]) The Auslander-Reiten conjecture 4.3 is true for algebras of dihedral, semidihedral or quaternion type.

5.2. Białkowski-Erdmann-Skowroński deformation of preprojective algebras. Recently Białkowski, Erdmann and Skowroński classified in [5] all selfinjective algebras with the property that for all simple modules S the third syzygy of S is again isomorphic to S. A recent survey on the circle around these questions was given by Erdmann and Skowroński in [17].

The problem of classifying algebras so that Ω^{2}(S) ≃ S for all simple modules was completely solved before and the next most interesting case is Ω^{3}(S) ≃ S for all simple modules. In order to formulate the result [5] of Białkowski, Erdmann and Skowroński we need to introduce deformed preprojective algebras as defined in [5].

The preprojective algebra of type A_{n} is given by the quiver

[figure: •; a_{1}; ^{\overline{a}^{1}}; •; a_{2}; ^{\overline{a}^{2}}; •; a_{3}; ^{\overline{a}^{3}}; •; a_{4}; ^{\overline{a}^{4}}; ^{. . . . . .}; a_{n−1}; ^{\overline{a}^{n−1}}; •]

subject to the relations

a_{1}\overline{a}_{1} = \overline{a}_{n−1}a_{n−1} = 0 and \overline{a}_{i}a_{i} = a_{i+1}\overline{a}_{i+1}∀ i ∈ \{1, 2, …, n − 2\}.

The deformed preprojective algebra of type D_{n+1} is given by the quiver

[figure: •; a_{1}; ^{\overline{a}^{1}}; •; ^{\overline{a}^{0}}; a_{0}; •; a_{2}; ^{\overline{a}^{2}}; •; a_{3}; ^{\overline{a}^{3}}; •; a_{4}; ^{\overline{a}^{4}}; ^{. . . . . .}; a_{n−1}; ^{\overline{a}^{n−1}}; •]

subject to the relations

a_{0}\overline{a}_{0} = a_{1}\overline{a}_{1} = \overline{a}_{n−1}a_{n−1} = \overline{a}_{1}a_{1} + \overline{a}_{0}a_{0} + a_{2}\overline{a}_{2} + f(\overline{a}_{0}a_{0}, \overline{a}_{1}a_{1}) = (\overline{a}_{1}a_{1} + \overline{a}_{0}a_{0})^{n−2} = 0

and \overline{a}_{i}a_{i} = a_{i+1}\overline{a}_{i+1}∀ i ∈ \{2, …, n − 2\}.

for some element

f(X, Y) ∈ rad^{2}(K < X, Y > /(X^{2}, Y^{2}, (X + Y)^{n−1}).

The algebra of type L_{n} for n ≥ 2 was already displayed in Example 2.19 and is given by the quiver

[figure: •; 0; •; 1; •; 2; •; 3; •; n − 2; •; n − 1; ⋯; ⋯; ⋯; ϵ; a_{0}; a_{1}; a_{2}; a_{n−2}; ^{\overline{a}^{0}}; ^{\overline{a}^{1}}; ^{\overline{a}^{2}}; ^{\overline{a}^{n−2}}]

subject to the relations

a_{i}\overline{a}_{i} + \overline{a}_{i−1}a_{i−1} = 0 for all i ∈ \{1, …, n − 2\} ,
\overline{a}_{n−2}a_{n−2} = 0 , ϵ^{2n} = 0 , ϵ^{2} + a_{0}\overline{a}_{0} + ϵ^{3}p(ϵ) = 0

for a polynomial p(X) ∈ K[X]. Denote by L_{n}^{p} the deformed preprojective algebra of type L with deformation polynomial p(X) and abbreviate L_{n}^{j} := L_{n}^{X^{2j}} for simplicity when no confusion may occur.

The deformed preprojective algebra of type E_{n} for n ∈ {6, 7, 8} is given by the quiver

[figure: •; a_{0}; ^{\overline{a}^{0}}; •; a_{1}; ^{\overline{a}^{1}}; •; a_{2}; ^{\overline{a}^{2}}; •; a_{3}; ^{\overline{a}^{3}}; •; a_{4}; ^{\overline{a}^{4}}; ^{. . . . . .}; a_{n−2}; ^{\overline{a}^{n−2}}; •]

subject to the relations

a_{0}\overline{a}_{0} = \overline{a}_{n−2}a_{n−2} = \overline{a}_{2}a_{2} = 0, \overline{a}_{i}a_{i} = a_{i+1}\overline{a}_{i+1}∀ i ∈ \{5, …, n − 2\}
\overline{a}_{1}a_{1} + a_{2}\overline{a}_{2} + a_{3}\overline{a}_{3} + f(\overline{a}_{1}a_{1}, a_{2}\overline{a}_{2}) = (\overline{a}_{1}a_{1} + a_{2}\overline{a}_{2})^{n−3} = 0

for some

f ∈ rad^{2}(K < X, Y > /(X^{3}, Y^{2}, (X + Y)^{n−3})

so that

(X + Y + f(X, Y))^{n−3} = 0.

For all deformed preprojective algebras we number the vertices by the condition that the vertex a_{i} starts at v_{i} and ends at a vertex of higher label. This convention numbers the vertices in a unique way.

Remark 5.4. Observe that in [17, page 238] for type E only ”admissible deformations” may be applied, which is the condition that (X +Y +f(X, Y))^{n−3} = 0. However one relation is missing in [17] for type D and type E whereas the relation is correctly displayed in [5]. I am grateful to Karin Erdmann for a clarifying email on this subject.

The result is the following.

Theorem 7. (Białkowski, Erdmann and Skowroński [5, Theorem 1.2]) Let A be a finite dimensional selfinjective K-algebra. Then Ω^{3}(S) ≃ S for every simple A-module if and only if A is preprojective of type A_{n} for (n ≥ 1) or deformed preprojective of type D_{n}^{f} for (n ≥ 4), E_{6}^{f}, E_{7}^{f}, E_{8}^{f} or L_{n}^{p} for (n ≥ 1).

It is a non trivial task to determine when deformations f actually lead to non isomorphic algebras. In a lecture at the ICRA XIV conference in Tokyo in August 2010 Białkowski annonce that in characteristic 2 the deformed preprojective algebras of type L_{n}^{X^{2j}} for j ∈ {0, 1, …, n − 1} form a complete set of Morita equivalence classes of these algebras, a fact that Skowroński pointed out in an email to the author from March 2007. Skowroński announces in another email to the author in October 2008 that all algebras L_{n}^{p} are symmetric and that moreover, in characteristic different from 2 the algebra L_{n}^{p} is Morita equivalent to L_{n}^{X^{n−1}}, the non deformed preprojective algebra of type L_{n}. The content of Białkowski’s ICRA lecture are available in the conference abstract volume.

For type D_{n}^{f} Białkowski, Erdmann and Skowroński [5, Proposition 6.2] show that the algebras D_{n}^{(XY)^{j}} are not Morita equivalent for different values of j.

No statement is known for type E preprojective algebras.

In joint work with Holm we computed the Külshammer ideals for the algebras L_{n}^{X^{2j}}. One main difficulty was to determine the commutator subspace. It is not very difficult to get a generating set for the quotient of the algebra modulo the commutator space, it is much more complicated to prove that the commutators one found really generate the commutator space. In order to do so we apply the method described in Section 2.3 and Section 2.4.

A first step is

Lemma 5.5. [24, Lemma 3.9] Let K be any field. Then L_{n+1}^{p}/[L_{n+1}^{p}, L_{n+1}^{p}] has a K-linear generating set

{e_{0}, e_{1}, …, e_{n}, ϵ, ϵ^{3}, ϵ^{5}, ϵ^{7}, …, ϵ^{2n+1}}.

We first need to fix a basis.

Proposition 5.6. [24, Proposition 3.1] Let K be any field. A K-basis of L_{n}^{p} is given by the following paths between the vertices i and j, where i, j ∈ {0, 1, …, n − 1}.

(1) a_{i}a_{i+1} … a_{j−1}  for i < j
(2) a_{i}a_{i+1} … a_{j−1}a_{j} … a_{ℓ}\overline{a}_{ℓ}\overline{a}_{ℓ−1} … \overline{a}_{j}  for i < j and some j ≤ ℓ ≤ n − 2
(3) \overline{a}_{i−1}\overline{a}_{i−2} … \overline{a}_{j}  for i ≥ j
(4) a_{i}a_{i+1} … a_{ℓ}\overline{a}_{ℓ}\overline{a}_{ℓ−1} … \overline{a}_{i} … \overline{a}_{j}  for i ≥ j and some i ≤ ℓ ≤ n − 2
(5) \overline{a}_{i−1}\overline{a}_{i−2} … \overline{a}_{0}ϵa_{0}a_{1} … a_{j−1}  for any i, j
(6) \overline{a}_{i−1}\overline{a}_{i−2} … \overline{a}_{0}ϵa_{0}a_{1} … a_{ℓ−1}a_{ℓ}\overline{a}_{ℓ}\overline{a}_{ℓ−1} … \overline{a}_{j} for i < j and some j ≤ ℓ ≤ n − 2
(7) a_{i}a_{i+1} … a_{ℓ}\overline{a}_{ℓ}\overline{a}_{ℓ−1} … \overline{a}_{1}\overline{a}_{0}ϵa_{0}a_{1} … a_{j−1} for i ≥ j and some i ≤ ℓ ≤ n − 2

Now we may define a Frobenius form with respect to this basis using Proposition 2.15. It turns out that this form is in fact symmetric, non degenerate and associative.

Theorem 8. [24, Theorem 3.5] Let K be any field. Then the algebra L_{n}^{p} is symmetric.

We get many elements in the centre of L_{n}^{p} by

Lemma 5.7. [24, Lemma 3.13] Let K be any field. Then

Z(L_{n}^{p}) ∋ ϵ^{2} + ϵ^{3}p(ϵ) + ∑_{j=0}^{n−2}(−1)^{j+1}\overline{a}_{j}a_{j}.

Hence

\{(ϵ^{2} + ϵ^{3}p(ϵ) + ∑_{j=0}^{n−2}(−1)^{j+1}\overline{a}_{j}a_{j})^{ℓ} | ℓ ∈ \{0, 1, …, n − 1\}\} ⊆ Z(L_{n}^{p})

is a K-free subset. Moreover soc(Z(L_{n}^{p})) ⊆ Z(L_{n}^{p}).

Lemma 5.7 provides a large space in Z(L_{n}^{p}) of dimension 2n. Lemma 5.5 shows that L_{n}^{p}/[L_{n}^{p}, L_{n}^{p}] has dimension at most 2n, whereas Proposition 2.12 show that the two vector spaces are isomorphic. Hence the elements displayed in Lemma 5.5 form a basis of L_{n}^{p}/[L_{n}^{p}, L_{n}^{p}].

Using this result it is then possible to show

Theorem 9. (Holm and Zimmermann [24, Theorem 4.1]) Let K be a perfect field of characteristic 2. Then for 0 ≤ j < n we get

dim (T_{i}(L_{n}^{X^{2j}})) − dim ([L_{n}^{X^{2j}}, L_{n}^{X^{2j}}]) = n − max (⌈\frac{2n − (2^{i+1} − 2)j − (2^{i+1} − 1)}{2^{i+1}}⌉ , 0)

The attentive reader remarks that for the theorem one needs to suppose that the field K is perfect, whereas this was not supposed in the auxiliary steps. The reason for this assumption comes from the technicalities in the proof of Theorem 9. We need to find elements x so that x^{2^{n}} ∈ [L_{n}^{p}, L_{n}^{p}]. By what preceded and rather easy arguments it is necessary to consider this question only for x being a K-linear combination of odd powers of ϵ. Now, the 2^{n} powers of x will give an element which is expressed in 2^{n}-th powers of the original coefficients, and from there it is not hard to imagine that one needs to take 2^{n}-th roots of the solutions, in order to get the original coefficients from some expression one obtains from some solution one got by a linear algebra argument.

Moreover, we remark that the Morita equivalence classification of the algebras L_{n}^{p} is finer than what can be done by Külshammer ideals. However, Külshammer ideals distinguish algebras up to derived equivalences and even up to stable equivalences of Morita type (cf Theorem 4).

5.3. Algebras of polynomial growth and domestic weakly symmetric algebras. Let K be an algebraically closed field and let A be a finite dimensional K-algebra.

• The algebra A is called of finite representation type if A admits only a finite number of indecomposable A-modules up to isomorphism.

• The algebra A is called of tame representation type if A is not of finite representation type and if for every positive integer d there are a finite number of A ⊗_{K} K[X]- modules M_{1}(d), M_{2}(d), …, M_{n(d)}(d), which are free as K[X]-modules and so that for each d all but a finite number of d-dimensional indecomposable A-modules are isomorphic to a module of the form M_{i}(d) ⊗_{K} K[X]/(X − λ) for some λ ∈ K and some i ∈ {1, …, n(d)}.

– The tame algebra A is called of domestic representation type if, taking n(d) as small as possible, there is an integer m so that n(d) ≤ m for all d.

– The tame algebra A is called of polynomial growth if, taking n(d) as small as possible, there is an integer m so that

lim_{n→∞} \frac{n(d)}{d^{m}} = 0.

• The algebra A is called of wild representation type if for every algebra B there is a functor B − mod ⟶ A − mod which is exact, preserves isomorphism classes and carries indecomposable objects to indecomposable objects.

Of course, tame domestic algebras are polynomial growth tame algebras.

A fundamental result of Drozd says that A is either tame or wild or of finite representation type. There is intensive research aiming a possible classification of algebras of tame representation type, though the goal seems still to be very far. Nevertheless Bocian, Holm and Skowroński classified tame domestic weakly symmetric algebras [9, 10, 21] and tame weakly symmetric algebras of polynomial growth [6, 7, 22] up to derived equivalences in a series of papers.

We present some of the details.

Definition 5.8. A selfinjective algebra of tame representation type is called standard if its basic algebra admits simply connected Galois coverings. Else a selfinjective algebra is called non standard.

Theorem 10. [9, Theorem 1] For an algebra A the following statements are equivalent:

(1) A is a representation-infinite domestic selfinjective algebra having simply connected Galois coverings and the Cartan matrix C_{A} is singular.

(2) A is derived equivalent to the trivial extension T (C) of a canonical algebra C of Euclidean type.

(3) A is stably equivalent to the trivial extension T (C) of a canonical algebra C of

Euclidean type. Moreover, the trivial extensions T (C) and T (C′) of two canonical algebras C and C′ of Euclidean type are derived equivalent (respectively, stably equivalent) if and only if the algebras C and C′ are isomorphic.

In order to be able to formulate Bocian, Holm and Skowroński’s result for weakly symmetric algebras of domestic representation type with nonsingular Cartan matrices, we need to define the following algebras.

[figure: •; β; α; A(λ); λ ∈ K\{0}; α^{2} = 0, β^{2} = 0, αβ = λβα]

[figure: β_{1}; β_{2}; β_{3}; β_{4}; β_{q−3}; β_{q−2}; β_{q−1}; β_{q}; α_{1}; α_{2}; α_{3}; α_{4}; α_{p}; α_{p−1}; α_{p−2}; α_{p−3}; A(p, q); 1 ≤ p ≤ q; p + q ≥ 3; α_{1}α_{2} ⋯ α_{p}β_{1}β_{2} ⋯ β_{q} = β_{1}β_{2} ⋯ β_{q}α_{1}α_{2} ⋯ α_{p}; α_{p}α_{1} = 0, β_{q}β_{1} = 0; α_{i}α_{i+1} ⋯ α_{p}β_{1} ⋯ β_{q}α_{1} ⋯ α_{i−1}α_{i} = 0, 2 ≤ i ≤ p; β_{j}β_{j+1} ⋯ β_{q}α_{1} ⋯ α_{p}α_{1} ⋯ β_{i−1}β_{i} = 0, 2 ≤ j ≤ q]

[figure: β_{1}; β_{2}; β_{3}; β_{4}; β_{n−3}; β_{n−2}; β_{n−1}; β_{n}; α; Λ(n); n ≥ 2; α^{2} = (β_{1}β_{2} ⋯ β_{n})^{2}, αβ_{1} = 0, β_{n}α = 0; β_{j}β_{j+1} ⋯ β_{n}β_{1} ⋯ β_{n}α_{1} ⋯ β_{i−1}β_{i} = 0, 2 ≤ j ≤ n]

[figure: β_{1}; β_{2}; β_{3}; β_{4}; β_{n−3}; β_{n−2}; β_{n−1}; β_{n}; γ_{1}; α_{2}; γ_{2}; α_{1}; Γ(n); n ≥ 1; α_{1}α_{2} = (β_{1}β_{2} ⋯ β_{n})^{2} = γ_{1}γ_{2},; α_{2}β_{1} = 0, γ_{2}β_{1} = 0, β_{n}α_{1} = 0; β_{n}γ_{1} = 0, α_{2}γ_{1} = 0, γ_{2}α_{1} = 0; β_{j}β_{j+1} ⋯ β_{n}β_{1} ⋯ β_{n}α_{1} ⋯ β_{i−1}β_{i} = 0, 2 ≤ j ≤ n]

Theorem 11. (Bocian, Holm and Skowroński [9, Theorem 2]) For a domestic standard selfinjective algebra A the following statements are equivalent:

(1) A is weakly symmetric and the Cartan matrix C_{A} is nonsingular.

(2) A is derived equivalent to an algebra of the form A(λ), A(p, q), Λ(n), Γ(n).

(3) A is stably equivalent to an algebra of the form A(λ), A(p, q), Λ(n), Γ(n).

Moreover, two algebras of the forms A(λ), A(p, q), Λ(n), Γ(n) are derived equivalent (respectively, stably equivalent) if and only if they are isomorphic.

For algebras of non standard type we need to introduce the following algebra Ω(n). The quiver with relations of Ω(n) is as follows.

[figure: β_{1}; β_{2}; β_{3}; β_{4}; β_{n−3}; β_{n−2}; β_{n−1}; β_{n}; α; Ω(n); n ≥ 1; α^{2} = αβ_{1}β_{2} ⋯ β_{n}, αβ_{1}β_{2} ⋯ β_{n} + β_{1}β_{2} ⋯ β_{n}α = 0,; β_{n}β_{1} = 0, β_{j}β_{j+1} ⋯ β_{n}β_{1} ⋯ β_{n}α_{1} ⋯ β_{i−1}β_{i} = 0, 2 ≤ j ≤ n]

Theorem 12. (Bocian, Holm and Skowroński [10, Theorem 1]) Any nonstandard representation infinite selfinjective algebra of domestic type is derived equivalent (resp. stably equivalent) to an algebra Ω(n) with n ≥ 1. Moreover, two algebras Ω(n) and Ω(m) are derived equivalent (respectively, stably equivalent) if and only if m = n.

Bocian, Holm and Skowroński show that standard and non standard domestic algebras cannot be derived equivalent. We are able to improve partially the result.

Lemma 5.9. (Zhou and Zimmermann [49, Lemma 2.3]) A standard weakly symmetric algebra of domestic type cannot be stably equivalent to a nonstandard one.

The method of proof is to compare stable Auslander Reiten quivers and to use the fact that the class of special biserial algebras is closed under stable equivalences, a result due to Pogorzały [38, Theorem 7.3].

Observe that a Morita equivalence classification of standard self-injective domestic algebras is not given in the results of Bocian, Holm and Skowroński. This is the reason why we suppose that the domestic algebras are weakly symmetric and not only selfinjective.

We show

Theorem 13. (Zhou and Zimmermann [49, Theorem 2.5])

(1) Two weakly symmetric algebras of domestic representation type are derived equivalent if and only if they are stably equivalent.

(2) The class of weakly symmetric algebras of domestic representation type is closed under stable equivalences.

Bocian, Holm and Skowroński give a classification of symmetric algebras of polynomial growth up to derived equivalences [6], [7], [22]. They get a finite list of algebras which are defined by quivers and relations, and where the relations involve some parameters. We are not completely able to distinguish the algebras with the same quiver and relations and different parameters. We call this problem the scalar problem.

As for symmetric algebras of polynomial growth we get the following result.

Theorem 14. (Zhou and Zimmermann [49, Theorem 3.5]) The classification of indecomposable non-domestic weakly symmetric algebras of polynomial growth up to stable equivalences of Morita type coincide with the derived equivalence classification, modulo the scalar problems.

The method of proof uses, among other general arguments, a computation of Külshammer ideals.

Concerning the Auslander Reiten conjecture we get

Corollary 5.10. (Zhou and Zimmermann [49, Corollary 2.7, Theorem 3.6, Theorem 3.7])

• Let A be an indecomposable algebra stably equivalent to an indecomposable symmetric algebra B of domestic type. Then A and B have the same number of isomorphism classes of simple modules.

• Let A be an indecomposable algebra stably equivalent to an indecomposable weakly symmetric standard algebra B of domestic type. Then A and B have the same number of isomorphism classes of simple modules.

• Let A and B be indecomposable algebras which are stably equivalent of Morita type. If A is tame symmetric with only Ω-periodic modules, then A and B have the same number of isomorphism classes of simple modules.

• Let A be an indecomposable algebra and suppose A is stably equivalent of Morita type to an indecomposable non-domestic symmetric algebra B of polynomial growth. Then A and B have the same number of isomorphism classes of simple modules.

• Let A and B be two indecomposable algebras which are standard non-domestic weakly symmetric algebra of polynomial growth or non-standard non-domestic selfinjective algebra of polynomial growth. If they are stably equivalent of Morita type, then A and B have the same number of isomorphism classes of simple modules.

6. Hochschild (co-)homology

It is well-known (cf e.g. Loday [34, Paragraphs 1.1.6 and 1.5.2]), and actually we already used this fact implicitly in e.g. in Section 2.3 and Section 4.3, that A/[A, A] = HH_{0}(A) is the degree 0 Hochschild homology and Z(A) = HH^{0}(A) is the degree 0 Hochschild cohomology. A natural question becomes now to generalise Külshammer’s ideals to higher Hochschild (co-)homology. This was done by the author in [53] and [54].

The symmetrising form on A induces a non degenerate pairing HH^{0}(A)×HH_{0}(A) ⟶ K. Hence, in order to generalise to higher Hochschild (co-)homology we first need to produce a non degenerate bilinear form

HH^{n}(A) × HH_{n}(A) ⟶ K

for symmetric algebras induced by the symmetrising form. Let 𝔹A be the bar resolution (i.e. a specific projective resolution; and actually projective resolution will do at this place) of A as A ⊗_{K} A^{op}-modules.

Hom_{K}(HH_{n}(A), K) = Hom_{K}(H_{n}(𝔹A ⊗_{A⊗A^{op}} A), K)
= H^{n}(Hom_{K}(𝔹A ⊗_{A⊗A^{op}} A, K))
= H^{n}(Hom_{A⊗_{K}A^{op}}(𝔹A, Hom_{K}(A, K))
≃ H^{n}(Hom_{A⊗_{K}A^{op}}(𝔹A, A))
= HH^{n}(A)

where the second last isomorphism is induced by the symmetrising form A ≃ Hom_{K}(A, K) as A ⊗_{K} A^{op} modules.

This isomorphism yields a non degenerate bilinear form

⟨ , ⟩_{n} : HH^{n}(A) × HH_{n}(A) ⟶ K

which extends the symmetrising form on A.

In order to define Külshammer ideals we used the p-power map on HH_{0}(A) = A/[A, A]. However, how to get a p-power map on HH_{∗}(A) is not completely clear. Nevertheless, the multiplicative structure on HH^{∗}(A) defined by the cup product can be used instead. By adjointness with respect to the bilinear form we get then an analogue of the Külshammer ideal structure on Hochschild homology instead of cohomology. This dual construction on A/[A, A] = HH_{0}(A) was studied by Külshammer as well in [30, Part IV].

For the p^{n}-power mapping by cup product HH^{m}(A, A) ⟶ HH^{p^{n}m}(A, A) one gets a right adjoint κ_{n}^{(m),A} : HH_{p^{n}m}(A, A) ⟶ HH_{m}(A, A) with respect to ⟨ , ⟩_{m} and ⟨ , ⟩_{p^{n}m} Observe that, if p is odd, the cup product square is 0 in odd degree cohomology. Hence for p odd the p-power map as well as the adjoint is 0 in odd degree cohomology.

Proposition 6.1. [53, Lemma 2.6, beginning remarks of Section 3] Let A be a finite dimensional symmetric K algebra over a perfect field K of characteristic p > 0. Then, for all n ∈ ℕ and for all x ∈ HH_{p^{n}m}(A, A) there is a unique κ_{n}^{(m)}(x) ∈ HH_{m}(A, A) so that for all f ∈ HH^{m}(A, A) one has

⟨f^{p^{n}}, x⟩_{p^{n}m} = (⟨f, κ_{n}^{(m)}(x)⟩_{m} )^{p^{n}} .

Using the mapping κ_{n}^{(m)} one gets the compatibility with derived equivalences.

Theorem 15. [53, Theorem 1] Let A be a finite dimensional symmetric K-algebra over a perfect field K of characteristic p > 0. Let B be a second algebra so that D^{b}(A) ≃ D^{b}(B) as triangulated categories. Let p be a prime and let m ∈ ℕ. Then, there is a standard equivalence F : D^{b}(A) ≃ D^{b}(B), and any such standard equivalence induces an isomorphism HH_{m}(F) : HH_{m}(A, A) ⟶ HH_{m}(B, B) of all Hochschild homology groups satisfying

HH_{m}(F) ∘ κ_{n}^{(m),A} ∘ HH_{p^{n}m}(F)^{−1} = κ_{n}^{(m),B} .

The proof is very much like the one in degree 0 for Hochschild cohomology. Nevertheless a clear definition of an isomorphism on Hochschild homology induced by a standard derived equivalence was not published explicitly before. The construction was somewhat implicit in Rickard’s work, but an explicit clarification seems to appear in [53, Section 1.2] for the first time.

One should notice however that a derived equivalence may be non standard, and then it is not clear how to define an induced mapping on the category of bimodules, and in the sequel on the Hochschild homology. One needs the standard equivalence in order to control the way it acts on Hochschild homology.

What happens if A is not symmetric? Already for the non degenerate pairing between Hochschild homology and cohomology it is not clear how to to define it properly. We may again use trivial extension algebras. Then there are ring homomorphisms A ⟶ 𝕋A and 𝕋A ⟶ A by projection, injection from and to the second component. Hochschild homology is functorial, contrary to Hochschild cohomology. Hence we get mappings

HH_{n}(ι_{A}) : HH_{n}(A) ⟶ HH_{n}(𝕋A)

and

HH_{n}(π_{A}) : HH_{n}(𝕋A) ⟶ HH_{n}(A).

Now, defining

κ̂_{n}^{(m)} := HH_{m}(π_{A}) ∘ κ_{n}^{(m)} ∘ HH_{p^{n}m}(ι_{A}) : HH_{p^{n}m}(A) ⟶ HH_{m}(A)

one obtains an invariant under derived equivalences.

Theorem 16. [54, Theorem 2] Let K be a perfect field of characteristic p > 0, let A and B be finite dimensional K-algebras and suppose that D^{b}(A) ≃ D^{b}(B) as triangulated categories. Let F be an explicit standard equivalence between D^{b}(A) and D^{b}(B). Then, F induces a sequence of isomorphisms HH_{m}(F) : HH_{m}(A) ⟶ HH_{m}(B) so that

HH_{m}(F) ∘ κ̂_{n}^{(m);A} = κ̂_{n}^{(m);B} ∘ HH_{p^{n}m}(F).

Obviously Theorem 16 generalises Theorem 15 to non symmetric algebras. Since Hochschild homology is often better understood than Hochschild cohomology we expect that this generalisation will bear use in the future.

Nevertheless there is a p-power map available in some cases coming from the Gerstenhaber structure on Hochschild homology. The construction is due to Stasheff and Quillen.

Let

Coder(𝔹(A), 𝔹(A)) := {D ∈ End_{A⊗A^{op}}(𝔹(A))|Δ ∘ D = (id_{𝔹(A)} ⊗ D + D ⊗ id_{𝔹(A)}) ∘ Δ} be the coderivations. Since 𝔹(A) is graded, Coder(𝔹(A), 𝔹(A)) is graded as well. Denote by Coder^{n}(𝔹(A), 𝔹(A)) the degree n coderivations. The vector space Coder(𝔹(A), 𝔹(A)) is a graded Lie algebra with Lie bracket being the commutator. Moreover (cf Stasheff [44, Proposition]),

Coder(𝔹(A), 𝔹(A)) ≃ Hom_{A⊗A^{op}}(𝔹(A), A)[1] .

The key observation is

Lemma 6.2. [53, Lemma 4.1] (Keller, personal communication)

• Suppose K is a field. Then,

D ∈ Coder^{2n+1}(𝔹(A), 𝔹(A)) ⇒ D^{2} ∈ Coder^{2·(2n+1)}(𝔹(A), 𝔹(A)).

• Suppose K is a field of characteristic p > 0. Then,

D ∈ Coder^{2n}(𝔹(A), 𝔹(A)) ⇒ D^{p} ∈ Coder^{2pn}(𝔹(A), 𝔹(A)).

This p-power structure carries over to Hochschild cohomology.

Lemma 6.3. [53, Lemma 4.2] Let K be a field of characteristic p > 0 and let D ∈ Coder^{n}(𝔹(A), 𝔹(A)).

(1) If p = 2 and n ∈ ℕ, then the mapping D ↦ D^{2} induces a mapping

HH^{n+1}(A, A) ⟶ HH^{2n+1}(A, A)

(2) If p > 2 and n = 2m ∈ 2ℕ, then the mapping D ↦ D^{p} induces a mapping

HH^{2m+1}(A, A) ⟶ HH^{2pm+1}(A, A)

Hence, for p = 2 the Hochschild cohomology becomes a 2-restricted Lie algebra with the Gerstenhaber bracket and the 2-power map. For p > 2 the odd degree Hochschild cohomology becomes a p-restricted Lie algebra with the Gerstenhaber bracket and the ppower mapping.

Using these constructions we get

Theorem 17. [53, Proposition 4.4] Let A and B be K-algebras over a field K. Suppose D^{b}(A) ≃ D^{b}(B) as triangulated categories.

• If the characteristic of K is 2 then HH^{∗}(A, A) and HH^{∗}(B, B) are isomorphic as restricted Lie algebras.

• If the characteristic of K is p > 2, then the Lie algebras consisting of odd degree Hochschild cohomologies ⨁_{n∈ℕ} HH^{2n+1}(A, A) and ⨁_{n∈ℕ} HH^{2n+1}(B, B) are isomorphic as restricted Lie algebras.

However we fail to prove that the so-defined Gerstenhaber p-power map is additive, neither semilinear. Hence it does not seem to be clear how a Külshammer structure could be built from there.

Acknowledgement. I wish to express my thanks to Javad Asadollahi and all mathematicians at the IPM in Teheran for their great hospitality during the conference ”Representation theory of Algebras” in June 2008.

I also wish to thank Karin Erdmann and Andrzej Skowroński for very useful and clarifying discussions, and the referee for a very detailed, careful and most helpful report.

References

[1] Estefanía Andreu Juan, The Hochschild cohomology ring of preprojective algebras of type L_{n}, preprint 2010; arXiv math.RT 1010.2790v1

[2] Maurice Auslander and Idun Reiten, Stable equivalence of Artin algebras. Proceedings of the Conference on Orders, Group Rings and Related Topics (Ohio State Univ., Columbus, Ohio, 1972), 8-71. Springer Lecture Notes in Mathematics 353 (1973).

[3] Maurice Auslander, Idun Reiten and Sverre Smålø, Representation Theory of Artin Algebras, Cambridge University Press 1995.

[4] Christine Bessenrodt, Thorsten Holm and Alexander Zimmermann, Generalized Reynolds ideals for non-symmetric algebras, Journal of Algebra 312 (2007) 985-994.

[5] Jerzy Białkowski, Karin Erdmann and Andrzej Skowroński, Deformed preprojective algebras of generalized Dynkin type, Transactions of the American Mathematical Society, 359 (2007) 2625-2650.

[6] Jerzy Białkowski, Thorsten Holm and Andrzej Skowroński, Derived equivalences for tame weakly symmetric algebras having only periodic modules. Journal of Algebra 269 (2003), no. 2, 652-668.

[7] Jerzy Białkowski, Thorsten Holm and Andrzej Skowronski, On nonstandard tame selfinjective algebras having only periodic modules. Colloquium Mathematicum 97 (2003), no. 1, 33-47.

[8] Frauke Bleher, Dihedral blocks with two simple modules, Proceedings of the American Mathematical Society 138 (2010) 3467-3479.

[9] Rafał Bocian, Thorsten Holm and Andrzej Skowroński, Derived equivalence classification of weakly symmetric algebras of Euclidean type. Journal of Pure and Applied Algebra 191 (2004), no. 1-2, 43-74.

[10] Rafał Bocian, Thorsten Holm and Andrzej Skowroński, Derived equivalence classification of nonstandard selfinjective algebras of domestic type. Communications in Algebra 35 (2007), no. 2, 515–526.

[11] Serge Bouc, Bimodules, trace généralisée, et transferts en homologie de Hochschild, preprint available at http://www.lamfa.u-picardie.fr/bouc/transfer.pdf

[12] Richard Brauer, Zur Darstellungstheorie der Gruppen endlicher Ordnung, Mathematische Zeitschrift 63 (1956) 406-444.

[13] Thomas Breuer, László Héthelyi, Erzsébet Horváth, Burkhard Külshammer, John Murray, Cartan invariants and central ideals of group algebras, Journal of Algebra 296 (2006) 177-195.

[14] Michel Broué, Equivalences of blocks of group algebras. In: Finite dimensional algebras and related topics. V.Dlab and L.L.Scott (eds.), Kluwer, 1994, 1-26.

[15] Jon Carlson and Jacques Thévenaz, The cassification of torsion endo-trivial modules, Annals of Mathematics 162 (2005) 823-883

[16] Karin Erdmann, Blocks of tame representation type and related algebras, Springer Lecture Notes in Mathematics 1428 (1990).

[17] Karin Erdmann and Andrzej Skowroński, Periodic algebras, in ”Trends in representation theory of algebras and related topics”, European Mathematical Society, Series of Congress Reports, (2008) pp 201-251

[18] Walter Feit, The representation theory of finite groups, North-Holland Amsterdam, 1982.

[19] László Héthelyi, Erszébet Horváth, Burkhard Külshammer and John Murray, Central ideals and Cartan invariants of symmetric algebras, Journal of Algebra 293 (2005) 243-260.

[20] Thorsten Holm, Blocks of tame representation type and related algebras: derived equivalences and Hochschild cohomology, Habilitationsschrift (2001) Universität Magdeburg.

[21] Thorsten Holm and Andrzej Skowroński, Derived equivalence classification of symmetric algebras of domestic type. Journal of the Mathematical Society of Japan 58 (2006), no. 4, 1133–1149

[22] Thorsten Holm and Andrzej Skowroński, Derived equivalence classification of symmetric algebras of polynomial growth. to appear in Glasgow Journal of Mathematics.

[23] Thorsten Holm and Alexander Zimmermann, Generalized Reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type, Journal of Algebra 320 (2008) 3425-3437.

[24] Thorsten Holm and Alexander Zimmermann, Deformed preprojective algebras of type L: Külshammer spaces and derived equivalences, preprint 2010.

[25] Michael Kauer, Klaus W. Roggenkamp, Higher-dimensional orders, graph-orders, and derived equivalences. Journal of Pure and Applied Algebra 155 (2001), 181-202.

[26] Bernhard Keller, Deriving DG categories, Annales Scientifiques Ecole Normale Supérieure 27 (1994), 63 - 102.

[27] Bernhard Keller and Dieter Vossieck, Sous les catégories dérivées. Comptes Rendus de l’Académie des Sciences Paris, Série I Mathématique 305 (1987), no. 6, 225–228.

[28] Steffen König and Alexander Zimmermann, Derived Equivalences for Group Rings, with contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphaël Rouquier; Springer Lecture Notes in Mathematics 1685 (1998).

[29] Steffen König, Yuming Liu and Guodong Zhou, Transfer maps in Hochschild (co-)homology and applications to stable and derived invariants and to the Auslander-Reiten conjecture, to appear in Transactions of the American Mathematical Society.

[30] Burkhard Külshammer, Bemerkungen über die Gruppenalgebra als symmetrische Algebra I, II, III, IV, Journal of Algebra 72 (1981), 1–7; Journal of Algebra 75 (1982), 59–69; Journal of Algebra 88 (1984), 279–291; Journal of Algebra 93 (1985), 310–323.

[31] Burkhard Külshammer, Group-theoretical descriptions of ring theoretical invariants of group algebras, Progress in Mathematics, 95 (1991), 425–441.

[32] Yuming Liu, Summands of stable equivalences of Morita type. Communications in Algebra 36(10) (2008), 3778-3782.

[33] Yuming Liu, Guodong Zhou and Alexander Zimmermann, Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture, preprint (2008).

[34] Jean-Louis Loday, Cyclic Homology, Springer Verlag, Berlin 1992.

[35] John Murray, On a certain ideal of Külshammer in the centre of a group algebra, Archiv der Mathematik 77 (2001) 373-377.

[36] John Murray, Squares in the centre of the group algebra of a symmetric group, Bulletin of the London Mathematical Society 34 (2002) 155-164.

[37] Tadasi Nakayama, On Frobeniusean algebras I, II, Annals of Mathematics 40 (1939) 611-633; Annals of Mathematics 42 (1941) 1-21.

[38] Zygmunt Pogorzały, Algebras stably equivalent to self-injective special biserial algebras, Communications in Algebra 22 (1994), no. 4, 1127-1160

[39] Daniel Quillen, Cyclic cohomology and algebra extensions, K-Theory 3 (1989) 205-246.

[40] Irving Reiner, Maximal Orders, Academic Press London, New York, San Francisco 1975

[41] William F. Reynolds, Sections and ideals of centres of group algebras, Journal of Algebra 20 (1972) 176-181.

[42] Jeremy Rickard, Derived categories and stable equivalence. Journal of Pure and Applied Algebra 61 (1989), no. 3, 303–317.

[43] Jeremy Rickard, Derived equivalences as derived functors, Journal of the London Mathematical Society 43 (1991) 37-48.

[44] Jim Stasheff, The intrinsic bracket on the deformation complex of an associative algebra, Journal of Pure and Applied Algebra 89 (1993) 231-235.

[45] Jean-Louis Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 Société mathématique de France 1996.

[46] Chang-Chang Xi, Stable equivalences of adjoint type. Forum Mathematicum 20 (2008), 81-97.

[47] Kunio Yamagata, Frobenius algebras, in Handbook of algebra Vol.1, 1996, pages 841-887, North-Holland Amsterdam.

[48] Guodong Zhou and Alexander Zimmermann, Classifying tame blocks and related algebras up to stable equivalences of Morita type, preprint 2010

[49] Guodong Zhou and Alexander Zimmermann, Auslander-Reiten conjecture for symmetric algebras of polynomial growth, preprint 2010

[50] Alexander Zimmermann, Derived Equivalences of Orders, in Proceedings of the ICRA VII, Mexico, eds: Bautista, Martinez, de la Peña. Canadian Mathematical Society Conference Proceedings 18, (1996) 721-749.

[51] Alexander Zimmermann, Tilted symmetric orders are symmetric orders, Archiv der Mathematik 73 (1999) 15-17.

[52] Alexander Zimmermann, Invariance of generalized Reynolds ideals under derived equivalences. Mathematical Proceedings of the Royal Irish Academy 107A (1) (2007) 1-9.

[53] Alexander Zimmermann, Fine Hochschild invariants of derived categories for symmetric algebras, Journal of Algebra 308 (2007) 350-367

[54] Alexander Zimmermann, Hochschild homology invariants of Külshammer type of derived categories, to appear in Communications in Algebra.

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

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