FINE HOCHSCHILD INVARIANTS OF DERIVED CATEGORIES FOR SYMMETRIC ALGEBRAS

ALEXANDER ZIMMERMANN

Abstract. Let A be a symmetric k-algebra over a perfect field k. Külshammer defined for any integer n a mapping ζ_{n} on the degree 0 Hochschild cohomology and a mapping κ_{n} on the degree 0 Hochschild homology of A as adjoint mappings of the respective p-power mappings with respect to the symmetrizing bilinear form. In an earlier paper it is shown that ζ_{n} is invariant under derived equivalences. In the present paper we generalize the definition of κ_{n} to higher Hochschild homology and show the invariance of κ and its generalization under derived equivalences. This provides fine invariants of derived categories.

Introduction

Let k be a commutative ring and let A be a k-algebra which is projective as a k-module. If B is a second k-algebra and if the derived categories D^{b}(A) of bounded complexes of A-modules and D^{b}(B) of B-modules are equivalent as triangulated categories, then the Hochschild cohomology of A is isomorphic to the Hochschild cohomology of B (Rickard [15]). Analogous statements hold for the cyclic homology (Keller [6]), the K-theory (Thomason-Trobaugh [19]), the fact of being symmetric algebra (cf Rickard [15] for fields and [20] in a more general situation) and others. This is one of the reasons why the derived category D^{b}(A) is now one of the main tools in representation theory. Nevertheless, most of the invariants are quite difficult to compute, except some small cases like the centre, or the Grothendieck group. Therefore, it is usually quite hard to distinguish two derived categories.

A symmetric k-algebra A is equipped with a symmetric non degenerate bilinear form ( , ) : A × A ⟶ k. Denote by KA the commutator subspace, that is the k-linear space generated by the set of ab − ba for a, b ∈ A. If k is a perfect field of characteristic p > 0, then Külshammer defined in [10] T_{n}(A)^{⊥} to be the orthogonal space to the set of x ∈ A so that x^{p^{n}} falls into KA. It turned out that T_{n}(A)^{⊥} is a decreasing sequence of ideals in the centre of A. If A and B are symmetric k-algebras with equivalent derived categories, then the centres of A and B are isomorphic, and in [21] we showed that this isomorphism maps T_{n}(A)^{⊥} to T_{n}(B)^{⊥}. This fine ideal structure of the centre of the algebra A gives valuable and computable derived invariants of A. In joint work with Thorsten Holm [5] we are able to apply the invariance of the ideals T_{n}(A)^{⊥} to tame blocks of group rings solving delicate questions whether certain parameters in the defining relations of particular algebras lead to different derived categories. Thorsten Holm and Andrzej Skowroński use this new fine invariant to classify all tame domestic symmetric algebras up to derived equivalence [4].

Alejandro Adem asked during the 2005 Oberwolfach conference ”Cohomology of finite groups: Interactions and applications” if it is possible to generalize Külshammer’s ideals of the centre of A to a derived invariant of higher degree Hochschild cohomology. The purpose of this paper is to answer to this question.

Külshammer shows in [10] that T_{n}(A)^{⊥} is the image of a certain mapping ζ_{n} : Z(A) ⟶ Z(A) which is defined by (z, a^{p^{n}}) = (ζ_{n}(z), a)^{p^{n}} for all z ∈ Z(A) and all a ∈ A. This is the way we view T_{n}(A)^{⊥} in [21]. Since KA = Z(A)^{⊥} the dual operation defines a mapping κ_{n} : A/KA ⟶ A/KA by the equation (z^{p^{n}}, a) = (z, κ_{n}(a))^{p^{n}} for all z ∈ Z(A) and all a ∈ A/KA. As a first result we show that κ_{n} as well is a derived invariant, observing that HH_{0}(A) = A/KA which is known to be a derived invariant.

Date: December 16, 2005; revised August 24, 2006.

2000 Mathematics Subject Classification. Primary 16E40, 18E30; Secondary 17B50 .

Key words and phrases. Gerstenhaber structure, derived categories of symmetric algebras, Hochschild homology operations, restricted Lie algebra.

As a first step we are going to show that for any symmetric algebra A one gets a non degenerate pairing

( , )_{m} : HH^{m}(A, A) × HH_{m}(A, A) ⟶ k .

Since HH_{∗}(A, A) does not have a multiplicative structure, it seems to be impossible to write down the defining relation for an analogue for ζ_{n}^{A} on higher Hochschild cohomology. Nevertheless, Hochschild cohomology is a graded commutative ring. Suppose either m is even and p odd or p even and m arbitrary. We are able to show for these parameters that the p^{n}- power mapping HH^{m}(A, A) ⟶ HH^{p^{n}m}(A, A) has 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} and moreover κ_{n}^{(0)} = κ_{n}. As a main result we show that any derived equivalence F of standard type between A and B induces an isomorphism HH_{m}(F) on the Hochschild homology and this in turn conjugates κ_{n}^{(m),A} to κ_{n}^{(m),B}.

Using a suggestion of Bernhard Keller we also study the p-power map by the Gerstenhaber Lie structure of the Hochschild cohomology. We show that this again is invariant under derived equivalences. Nevertheless, there is no obvious reason why this p-power map should be semilinear. Moreover, it is only defined from degree 1 Hochschild cohomology onwards. Furthermore, we expect that this will be somewhat harder to compute in examples. On the other hand this gives for p = 2 a richer structure to the set of derivations on A, and this structure is then a computable derived invariant.

Our paper is organised as follows. In Section 1 we recall the basic constructions concerning Hochschild (co-)homology and their invariance under derived equivalences. In Section 2 we show the derived invariance of Külshammer’s mapping κ_{n} as well as some consequences for derived equivalences between blocks of group rings. Section 3 is devoted to the definition of the generalization of κ_{n} to higher Hochschild homology, and to show its invariance under derived equivalence. In Section 4 we first recall the Gerstenhaber construction of a Lie algebra structure on the Hochschild cohomology, define for all primes p a structure of a restricted Lie algebra on odd degree Hochschild homology and on the entire Hochschild cohomology for p = 2, and show its invariance under derived equivalence.

1. Derived equivalences and Hochschild constructions revisited

1.1. Basic constructions and definitions for Hochschild homology and cohomology. In order to fix notation and also for convenience of the reader we recall in this section the basic notions for Hochschild cohomology and homology. Most of the material in this section can be found in Loday [12], Keller [7] and Stasheff [17].

Let k be a field and let A be a k-algebra. Consider a complex BA whose degree n homogeneous component is (BA)_{n} := A^{⊗(n+2)} and whose differential d_{n} : (BA)_{n} ⟶ (BA)_{n−1} is

d_{n}(x_{0} ⊗ ⋯ ⊗ x_{n+1}) := ∑_{j=0}^{n}(−1)^{j}(x_{0} ⊗ ⋯ ⊗ x_{j−1} ⊗ x_{j}x_{j+1} ⊗ x_{j+2} ⊗ ⋯ ⊗ x_{n+1}).

Then the complex (BA, d) is a projective resolution of A as A ⊗_{k} A^{op}-module. For a later application in Section 4 we need to extend this construction to

B(A) := k ⊕ A[1] ⊕ (A ⊗ A)[2] ⊕ (A ⊗ A ⊗ A)[3] ⊕ …

where the brackets indicate the degrees of the components. The mappings

Δ_{n} : B(A)_{n} ⟶ ⊕_{1≤j≤n}B(A)_{j} ⊗ B(A)_{n−j}
x_{1} ⊗ ⋯ ⊗ x_{n} ↦ ∑_{j=0}^{n}(x_{1} ⊗ ⋯ ⊗ x_{j}) ⊗ (x_{j+1} ⊗ ⋯ ⊗ x_{n})

(where it is understood that the boundary cases j = 0 and j = n correspond to the element 1_{k} in the first or the last bracket) compose to a graded co-algebra map Δ : B(A) ⟶ B(A) ⊗ B(A) and B(A) becomes a differential graded co-associative co-algebra. Observe the shift by degree 2 between BA and B(A).

We abbreviate in the sequel A^{e} := A ⊗ A^{op} and A^{∗} = Hom_{k}(A, k).

By definition, for all A^{e}-modules M one puts

HH_{n}(A, M) = H_{n}(BA ⊗_{A⊗A^{op}} M)

and

HH^{n}(A, M) = H^{n}(Hom_{A⊗A^{op}}(BA, M)).

1.2. Some facts on derived equivalences. Suppose two finite dimensional k-algebras A and B have equivalent derived categories D^{b}(A) ≃ D^{b}(B) as triangulated categories. Then, there is a complex Y in D^{b}(B ⊗ A^{op}) and a complex X in D^{b}(A ⊗ B^{op}) so that

F_{Y} := Y ⊗_{A}^{𝕃} − : D^{b}(A) ⟶ D^{b}(B)

is an equivalence and so that

X ⊗_{B}^{𝕃} − : D^{b}(B) ⟶ D^{b}(A)

is an equivalence quasi-inverse to F_{Y}. It is known that one may choose X and Y so that both complexes are formed by projective modules if restricted to either side, and that then the left derived tensor product can be replaced by the ordinary tensor product.

Moreover, doing so,

F_{Y}^{e} := Y ⊗_{A}^{𝕃} − ⊗_{A}^{𝕃} X : D^{b}(A ⊗ A^{op}) ⟶ D^{b}(B ⊗ B^{op})

is an equivalence of triangulated categories satisfying

F_{Y}^{e} (A) ≃ B .

In [20] we have shown that for quite general algebras, in particular for finite dimensional algebras over fields k, we have

F_{Y}^{e} (Hom_{k}(A, k)) ≃ Hom_{k}(B, k) .

Moreover, in [21] we have shown that for k a field of characteristic p > 0 denoting by k^{(n)} the k-vector space k twisted by the n-th power of the Frobenius automorphism F r,

F_{Y}^{e} (Hom_{k}(A, k^{(n)})) ≃ Hom_{k}(B, k^{(n)}).

In particular, if A ≃ Hom_{k}(A, k) in A^{e} − mod, then B ≃ Hom_{k}(B, k) in B^{e} − mod. Such algebras are called symmetric.

Another consequence (cf Rickard[15] or [9]) is that F_{Y}^{e} (and therefore F_{Y}) induces an isomorphism

HH^{m}(A, A) = Ext_{A^{e}}^{m}(A, A) = Hom_{D^{b}(A^{e})}(A, A[m])
⟶^{F^{e}} Hom_{D^{b}(B^{e})}(B, B[m]) = HH^{m}(B, B) .

Hochschild homology as well is an invariant of the derived category. This seems to be well known, but I could not find a reference in the literature, and therefore I include a proof. In particular, it is important to have an explicit isomorphism which we will need for our proof. In fact, BA is a free resolution of A in A^{e} − mod. Since F_{Y}^{e} is an equivalence,

F_{Y}^{e} (BA) = Y ⊗_{A} BA ⊗_{A} X

is a projective resolution of B in B^{e} − mod. The mapping

BA ⊗_{A^{e}} (X ⊗_{B} Y) ⟶ (Y ⊗_{A} BA ⊗_{A} Y) ⊗_{B^{e}} B
u ⊗ (x ⊗ y) ↦ (y ⊗ u ⊗ x) ⊗ 1

is well defined as is easily seen. Moreover, an inverse is

(Y ⊗_{A} BA ⊗_{A} Y) ⊗_{B^{e}} B ⟶ BA ⊗_{A^{e}} (X ⊗_{B} Y)
(y ⊗ u ⊗ x) ⊗ b ↦ u ⊗ (xb ⊗ y) = u ⊗ (x ⊗ by)

We take the degree m homology of the various complexes and observe since X ⊗_{B} Y ≃ A,

H_{m}(BA ⊗_{A^{e}} (X ⊗_{B} Y)) = T or_{m}^{A^{e}}(A, A) = HH_{m}(A, A) .

Moreover,

H_{m}((Y ⊗_{A} BA ⊗_{A} Y) ⊗_{B^{e}} B) = T or_{m}^{B^{e}}(B, B) = HH_{m}(B, B) .

Therefore, also for two k-algebras A and B the Hochschild homology is an invariant of the derived category.

2. Symmetric algebras and the Külshammer-κ

Given a perfect field k of characteristic p and a symmetric k-algebra A with symmetrizing bilinear form ( , )_{A} = ( , ), Külshammer defined in [10] a mapping ζ_{n} = ζ_{n}^{A} : Z(A) ⟶ Z(A) by (z, a^{p^{n}})_{A} = (ζ_{n}(z), a)_{A}^{p^{n}}, for all z ∈ HH^{0}(A) and a ∈ HH_{0}(A).

The bilinear form ( , )_{A} induces an identification A ≃ Hom_{k}(A, k) as A ⊗_{k} A^{op}-bimodules. In [20] we showed that F^{e}(Hom_{k}(A, k)) ≃ Hom_{k}(B, k) and therefore we get an induced

non degenerate symmetric bilinear form ( , )_{B} on B, making B a symmetric algebra as well. In [21] we showed that ζ_{n} on A induces the mapping ζ_{n} on B by the derived equivalence F and F^{e}, if one chooses the symmetrizing form ( , )_{B} on B.

We would like to mention a consequence which is implied by [21].

Corollary 2.1. Suppose k is a perfect field of characteristic p, suppose G and H are finite groups, B_{G} is a block of kG with defect group D_{G} and suppose B_{H} is a block of kH of defect group D_{H}. If D^{b}(kG) ≃ D^{b}(kH), then the exponent of D_{G} and the exponent of D_{H} coincide.

Proof. This follows from the fact that the ideals im(ζ_{n}^{B_{G}}) and im(ζ_{n}^{B_{H}}) are mapped to each other by the isomorphism Z(B_{G}) ≃ Z(B_{H}), and that by a result due to Külshammer [11, formulae (17), (47) and (78)], the exponent of D_{G} is the smallest integer n so that im(ζ_{n}^{B_{G}}) = im(ζ_{n+1}^{B_{G}}), and likewise for H.

Remark 2.2. Suppose k is an algebraically closed field of characteristic p, suppose G and H are finite groups, suppose R is a complete discrete valuation domain of characteristic 0 with R/rad(R) = k and field of fractions K. Suppose B_{G} is a block of RG with defect group D_{G} and suppose B_{H} is a block of RH of defect group D_{H}. If D^{b}(B_{G}) ≃ D^{b}(B_{H}), then the orders of the defect groups coincide: |D_{G}| = |D_{H}|. The argument is the following construction^{1} of Cliff, Plesken and Weiss [2]. Let B be a block of RG with defect group D and let Λ be the centre of B. Define inductively Λ_{0} := Λ and

Λ_{i+1} := {x ∈ KΛ | x · rad(Λ_{i}) ⊆ rad(Λ_{i})} .

Cliff, Plesken and Weiss show for algebraically closed fields k [2, Theorem 3.4] that

min{s | Λ_{s} = Λ_{s+1}} = |D|.

Külshammer gives a second, in some sense dual mapping κ_{n} defined by the equation (z^{p^{n}}, a) = (z, κ_{n}(a))^{p^{n}}, for z ∈ HH^{0}(A), and for a ∈ HH_{0}(A).

Proposition 2.3. The mapping κ_{n}^{A} : HH_{0}(A) ⟶ HH_{0}(A) is invariant under a derived equivalence F : D^{b}(A) ⟶ D^{b}(B) in the sense that under the induced isomorphism HH_{0}(F) : HH_{0}(A) ⟶ HH_{0}(B) one has

HH_{0}(F) ∘ κ_{n}^{A} ∘ HH_{0}(F)^{−1} = κ_{n}^{B}

if one chooses the induced bilinear form ( , )_{B} on B.

1This argument arose during a discussion with Gabriele Nebe in Aachen in October 2005. I am very grateful for her kind hospitality and for giving me this reference

Proof. Examining this relation, κ_{n} fits in the commutative diagram

A ⊗_{A⊗_{k}A^{op}} A ≃ ((A ⊗_{A⊗_{k}A^{op}} A)^{∗})^{∗} ≃ Hom_{k}(Hom_{D^{b}(A⊗_{k}A^{op})}(A, A), k)
↓ ((F r^{k})_{∗})^{n}
↑ κ_{n}  Hom_{k}(Hom_{D^{b}(A⊗_{k}A^{op})}(A, A), k^{(n)})
↑ ((μ_{p})^{∗})^{n}
A ⊗_{A⊗_{k}A^{op}} A ≃ ((A ⊗_{A⊗_{k}A^{op}} A)^{∗})^{∗} ≃ Hom_{k}(Hom_{D^{b}(A⊗_{k}A^{op})}(A, A), k)

where F r is the Frobenius automorphism on k, and where

μ_{p} : Z(A) ∋ z ↦ z^{p} ∈ Z(A).

Using Z(A) = Hom_{A^{e}}(A, A), the mapping μ_{p} corresponds to

Hom_{A^{e}}(A, A) ∋ f ↦ \underbrace{f ∘ f ∘ ⋯ ∘ f ∘ f}_{p factors} ∈ Hom_{A^{e}}(A, A) .

Since F^{e} is a functor it is clear that F^{e}(μ_{p}^{A}) = μ_{p}^{B}. The rest of the proof is exactly analogous to the one in [21].

Hence F^{e} maps κ_{n}^{A} to κ_{n}^{B} with respect to the induced identification B ≃ Hom_{k}(B, k) and the proof is finished.

This result will be generalised to higher Hochschild homology in Theorem 1. Though the proof there covers the present proposition it seems to be useful to have a short independent proof here.

We use Külshammer’s description of the image and the kernel of κ_{n} to get a nice invariant of derived categories. Let P_{n}(ZA) :=< z^{p^{n}}| z ∈ Z(A) >_{k−space}. Then

P_{n}(ZA)^{⊥}/KA = {x ∈ A/KA| (z^{p^{n}}, x) = 0 ∀z ∈ Z(A)}

and

T_{n}(ZA)^{⊥}/KA = {x ∈ A/KA| ∀ z ∈ Z(A) : z^{p^{n}} = 0 ⇒ (z, x) = 0}.

Now, T_{n}(ZA)^{⊥}/KA is a Z(A)-submodule of the Z(A)-module A/KA. Indeed, for x ∈ T_{n}(ZA)^{⊥}/KA and y ∈ Z(A), one gets for any z ∈ Z(A) with z^{p^{n}} = 0 also (yz)^{p^{n}} = 0 and so, (z, yx) = (yz, x) = 0 as well.

Corollary 2.4. Let F : D^{b}(A) ⟶ D^{b}(B) be an equivalence of standard type between the derived categories of the symmetric k-algebras A and B over a perfect field k. Then

• the isomorphism HH_{0}(F) : A/KA ⟶ B/KB maps P_{n}(ZA)^{⊥}/KA to P_{n}(ZB)^{⊥}/KB.

• the isomorphism HH_{0}(F) : A/KA ⟶ B/KB maps T_{n}(ZA)^{⊥}/KA to T_{n}(ZB)^{⊥}/KB as submodules over the centres of the algebras.

Proof. This is a consequence of Proposition 2.3, the fact that our isomorphisms are functorial, and hence preserve the natural structure of A/KA = HH_{0}(A, A) as HH^{0}(A, A) = Z(A)- module, and the fact that the first module P_{n}(ZA)^{⊥}/KA is the kernel of κ_{n}^{A}, whereas T_{n}(ZA)^{⊥}/KA is the image of κ_{n}^{A}, as was shown by Külshammer [11, (52),(53)].

Remark 2.5. Since the analogous statement for the, in some sense dual, mapping ζ proved to be extremely useful for distinguishing derived categories, such as the classification of tame domestic symmetric algebras by Holm-Skowroński [4] or to fix some of the open parameters in the derived equivalence classification of tame blocks of group rings [5], there is quite some hope that this corollary is as useful as was the method in [21]. Indeed, this new quite sophisticated invariant is explicitly computable in case the Hochschild homology is known as vector space, but additional structure is lacking.

A first step for the generalisation is the following lemma. Instead of taking degree 0 homology, one takes higher degree homology, and finds again a bilinear form.

Lemma 2.6. Let A be a symmetric k-algebra with symmetrizing form ( , ). Then, there is a non degenerate bilinear form

( , )_{m} : HH^{m}(A, A) ⊗ HH_{m}(A, A) ⟶ k

so that ( , )_{0} = ( , )|_{Z(A)⊗A/KA} . This form gives an isomorphism

Hom_{k}(HH^{m}(A, A), k) ≃ HH_{m}(A, A)

and any such k-linear isomorphism induces a non degenerate bilinear form.

Proof. Since (− ⊗_{A⊗A^{op}} A, Hom_{k}(A, −)) is an adjoint pair of functors between k-vector spaces and A ⊗ A^{op}-modules,

Hom_{k}(BA ⊗_{A⊗A^{op}} A, k) ≃ Hom_{A⊗A^{op}}(BA, Hom_{k}(A, k))

Taking homology of these complexes, and using that Hom_{k}(−, k) is exact and contravariant, gives

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

Since A is symmetric, A ≃ Hom_{k}(A, k) as A ⊗ A^{op}-modules and we get an isomorphism

HH^{m}(A, A) ⟶^{φ_{n}} Hom_{k}(HH_{m}(A, A), k)

as k-vector spaces. Now, put for any f ∈ HH^{m}(A, A) and x ∈ HH_{m}(A, A)

(f, x)_{m} := (φ_{m}(f))(x) .

It is clear that ( , )_{m} is bilinear since φ_{m} is k-linear. The form ( , )_{m} is non degenerate since φ is an isomorphism. In case m = 0 we find back the form ( , ) which was used to identify A with Hom_{k}(A, k).

The last part of the statement is well known.

This finally proves the lemma.

3. The cup product κ

Now, HH^{∗}(A, A) carries a natural graded commutative ring structure given by cup product. We shall describe how this allows to define, just as in the construction of κ_{n}, a semilinear p-power map μ_{p}^{(m)} and by this means a higher degree κ_{n}^{(m)} : HH_{p^{n}m}(A, A) ⟶ HH_{m}(A, A) with κ_{n} = κ_{n}^{(0)}.

Let m ∈ 2ℕ. Then, for any n ∈ ℕ and any x ∈ HH_{p^{n}m}(A, A) one gets a k-linear map

HH^{m}(A, A) ⟶ k
f ↦ F r_{p}^{−n} ((f^{p^{n}}, x)_{p^{n}m}) .

This is hence an element in Hom_{k}(HH^{m}(A, A), k). Now, by Lemma 2.6

HH_{m}(A, A) ≃ Hom_{k}(HH^{m}(A, A), k)
x ↦ (f ↦ (f, x)_{m})

Hence, 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}} .

We defined for all n ∈ ℕ and all m ∈ 2ℕ a mapping

κ_{n}^{(m),A} = κ_{n}^{(m)} : HH_{p^{n}m}(A, A) ⟶ HH_{m}(A, A)

so that κ_{n}^{(0),A} = κ_{n}. In case A is clear from the context, we denote κ_{n}^{(m),A} = κ_{n}^{(m)}.

Remark 3.1. Observe that since the Hochschild cohomology ring is graded commutative, for p odd the mapping

HH^{2m+1}(A, A) ∋ f ↦ f^{p} ∈ HH^{p·(2m+1)}(A, A)

is 0 for m ∈ ℕ. Of course, the 0-mapping is semilinear as well. If p = 2, then being graded commutative is just the same as being commutative, and so the restriction on m is not necessary.

Theorem 1. Let A be a finite dimensional symmetric k-algebra over the 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} .

Proof. By Remark 3.1 in case m odd we may assume p = 2.

Again let BA be the bar resolution. Then the complex computing the Hochschild homology is BA ⊗_{A^{e}} A. Since A is finite dimensional, we compute for any integer ℓ

H_{ℓ}(BA ⊗_{A^{e}} A) ≃ Hom_{k}(H^{ℓ}(Hom_{k}(BA ⊗_{A^{e}} A, k)), k)
≃ Hom_{k}(H^{ℓ}(Hom_{A^{e}}(BA, Hom_{k}(A, k))), k)

just using the standard adjointness formulas between hom and tensor functors. Finally, using the bilinear form ( , ) on A we get

H_{ℓ}(BA ⊗_{A^{e}} A) ≃ Hom_{k}(H^{ℓ}(Hom_{A^{e}}(BA, Hom_{k}(A, k))), k)
≃ Hom_{k}(H^{ℓ}(Hom_{A^{e}}(BA, A)), k) .

We discover

HH_{ℓ}(A, A) ≃ Hom_{k}(Ext_{A^{e}}^{ℓ}(A, Hom_{k}(A, k)), k) ≃ Hom_{k}(HH^{ℓ}(A, A), k)

and get diagram

HH_{p^{n}m}(A, A) ⟶^{≃} Hom_{k}(Ext_{A^{e}}^{p^{n}m}(A, Hom_{k}(A, k)), k) ⟶^{≃} Hom_{k}(HH^{p^{n}m}(A, A), k)
((μ_{p}^{(m)})^{n})^{∗} ↓
↓ κ_{n}^{(m)}  Hom_{k}(HH^{m}(A, A), k^{(n)})
(F r_{p}^{n})_{∗} ↑
HH_{m}(A, A) ⟶^{≃} Hom_{k}(Ext_{A^{e}}^{m}(A, Hom_{k}(A, k)), k) ⟶^{≃} Hom_{k}(HH^{m}(A, A), k)

which we easily see to be commutative by what we observed previously.

We need to show that applying HH_{∗}(F) (resp. HH^{∗}(F)) to the various mapping spaces for the Hochschild (co-)homology of A gives the analogous mapping for B.

Let X ∈ D^{b}(A ⊗ B^{op}) be a two-sided tilting complex with inverse Y ∈ D^{b}(B ⊗ A^{op}). We may and will assume that X and Y are complexes being projective on the left and projective on the right. Then, we may replace the left derived tensor product by the ordinary tensor product. Let

F_{X} := Y ⊗_{A} − ⊗_{A} X : D^{b}(A ⊗ A^{op}) ⟶ D^{b}(B ⊗ B^{op})

Then, F_{X} is a triangle equivalence, and in particular, F_{X}(BA) is a resolution of B in D^{b}(B ⊗ B^{op}). So, F_{X} induces a commutative diagram

Hom_{k}(HH_{p^{n}m}(A, A), k) ⟵ Ext_{A^{e}}^{p^{n}m}(A, A)
↓  ↓
Hom_{k}(HH_{p^{n}m}(B, B), k) ⟵ Ext_{B^{e}}^{p^{n}m}(B, B)

where the bottom row is again given by the adjointness formula between Hom and tensor functors.

Moreover, F_{X}(Hom_{k}(A, k)) = Hom_{k}(B, k) as was shown in [20]. This implies that an isomorphism A ≃ Hom_{k}(A, k) induces an isomorphism B ≃ Hom_{k}(B, k) as bimodules. Therefore, the induced diagram

Hom_{k}(Ext_{A^{e}}^{p^{n}m}(A, Hom_{k}(A, k)), k) ⟶ Hom_{k}(Ext_{A^{e}}^{p^{n}m}(A, A), k)
↓ F_{X}  ↓ F_{X}
Hom_{k}(Ext_{B^{e}}^{p^{n}m}(B, Hom_{k}(B, k)), k) ⟶ Hom_{k}(Ext_{B^{e}}^{p^{n}m}(B, B), k)

is commutative.

Now, we know that the cup product on HH^{∗}(A, A) is the composition of mappings in Ext_{A^{e}}^{∗}(A, A) = Hom_{D^{b}(A⊗A^{op})}(A, A[∗]). Therefore, applying F_{X} to any of this composition of mappings is going to give again the composition of mappings on Ext_{B^{e}}^{∗}(B, B) = Hom_{D^{b}(B⊗B^{op})}(B, B[∗]). Since the mapping μ_{p}^{(m)} is the k-linear dual of this, again F_{X} induces a commutative diagram

Hom_{k}(HH^{p^{n}m}(A, A), k) ⟶^{μ_{p}^{(m),A}} Hom_{k}(HH^{m}(A, A), k^{(n)})
↓ F_{X}  ↓ F_{X}
Hom_{k}(HH^{p^{n}m}(B, B), k) ⟶^{μ_{p}^{(m),B}} Hom_{k}(HH^{m}(B, B), k^{(n)})

Since F_{X} acts on the contravariant variable of Hom_{k}(HH^{m}(A, A), k) and in the space of semilinear mappings Hom_{k}(HH^{m}(A, A), k^{(n)}), we get

Hom_{k}(F_{X}, k) ∘ Hom_{k}(HH^{m}(A, A), (Fr_{k})) =
= Hom_{k}(HH^{m}(A, A), (Fr_{k})) ∘ Hom_{k}(F_{X}, k) .

This shows the theorem.

We observe first properties analogous to those in Külshammer [11]. For this put

T_{n}^{(m)}(HH^{m}(A, A)) := {x ∈ HH^{m}(A, A) | x^{p^{n}} = 0}.

Proposition 3.2. Suppose k is a perfect field of characteristic p > 0 and that A is a symmetric finite dimensional k-algebra. Then, denoting by ⊥_{m} the orthogonality with respect to the pairing ( , )_{m},

(1) κ_{n}^{(m)} is k-semilinear,

(2) κ_{n+ℓ}^{(m)} = κ_{ℓ}^{(m)} ∘ κ_{n}^{(p^{ℓ}m)}

(3) im(κ_{n}^{(m)}) = (T_{n}^{(m)})^{⊥_{m}}.

(4) ker κ_{n}^{(m)} = {x^{p^{n}} | x ∈ HH^{m}(A, A)}^{⊥_{p^{n}m}}

Proof. The first statement comes from the construction in the proof of Theorem 1 of κ as composition of semilinear mappings.

The second statement again is implied by the following argument.

(f, κ_{n+ℓ}^{(m)}(x))_{m} = (f^{p^{n+ℓ}}, x)_{p^{n+ℓ}m}
= (f^{p^{ℓ}}, κ_{n}^{(p^{ℓ}m)}(x))_{p^{ℓ}m}
= (f, κ_{ℓ}^{(m)}(κ_{n}^{(p^{ℓ}m)}(x)))_{m}

The third statement is shown as follows: The defining equation

(x^{p^{n}}, y)_{p^{n}m} = ((x, κ_{n}^{(m)}(y))_{m})^{p^{n}}

and the fact that ( , )_{p^{n}m} is non degenerate show that im(κ_{n}^{(m)})^{⊥_{m}} = T_{n}^{(m)}. Since ( , )_{m} is non degenerate, we may take the orthogonal spaces of these and get the result.

The fourth statement comes directly from the defining equation

(x^{p^{n}}, y)_{p^{n}m} = ((x, κ_{n}^{(m)}(y))_{m})^{p^{n}}

as well.

Remark 3.3. We see that the kernel and the image of κ_{n}^{(m)} are very much linked to the set of nilpotent elements of the Hochschild cohomology. Snashall and Solberg conjectured [16] that the Hochschild cohomology ring of any finite dimensional algebra is finitely generated modulo the ideal generated by nilpotent elements.

Corollary 3.4. We get dim(im(κ_{n}^{(m)})) = dim(HH^{m}(A, A)) − dim(T_{n}^{(m)}).

Proof. This is an immediate consequence of the third statement of Proposition 3.2.

In general even degree Hochschild cohomology rings of symmetric algebras contain nilpotent elements, but are not necessarily entirely nilpotent. As an example I refer to the article Erdmann and Holm [3, Section 4] where Hochschild cohomology rings of self-injective Nakayama algebras, which includes the Hochschild cohomology of Brauer tree algebras, are computed. There nilpotent elements arise in even Hochschild degrees, though the even degree Hochschild cohomology modulo the nilpotent radical is not zero in general. In particular, κ_{n}^{(m)} is neither zero nor surjective in general.

Remark 3.5. In the joint paper [1] with Bessenrodt and Holm we showed that for the degree zero Hochschild homology one may pass from a possibly non-symmetric algebra A to its trivial extension 𝕋A. Rickard showed in [14] that whenever the algebras A and B are derived equivalent then also the trivial extension algebras 𝕋A and 𝕋B are derived equivalent. In degree 0 it is then possible to interpret the mappings κ and ζ on the degree 0 (co-)homology of 𝕋A in terms of A only. One might ask if an analogous construction is possible for κ_{n}^{(m)} as well. The obvious fact that the Hochschild homology of A is a direct factor of the Hochschild homology of 𝕋A might give a natural definition. Nevertheless, there are quite a number of technical problems, such as the fact that Rickard gives a one-sided tilting complex only whereas a two-sided complex is needed for our method. Moreover, on a more practical level, in order to be able to compute κ_{n}^{(m)} via the trivial extension method, one needs at the present stage at least parts of the multiplicative structure of the Hochschild cohomology of 𝕋A. Even for rather small algebras A its trivial extension 𝕋A usually will have quite complicate cohomology. A significant simplification is needed and at the moment I do not see clearly how one can cope with these difficulties.

4. Stasheff-Quillen’s construction of the Gerstenhaber structure and the Gerstenhaber κ

We recall first a most helpful construction of the Gerstenhaber bracket appearing in a slightly implicit fashion in Quillen [13] and very explicitly in Stasheff [17]. I learned the construction in discussions from Bernhard Keller [7, Section 4.7]. This construction shows that the Gerstenhaber bracket can be defined using a homological construction on the bar complex. For the reader’s convenience we give the the construction in some detail.

4.1. Stasheff-Quillen’s construction. 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. Now (cf e.g. Stasheff [17, Proposition]),

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

The isomorphism is induced by composing an f ∈ Coder(B(A), B(A)) with the projection τ on the degree 1 component A of B(A) so that γ_{A}(f) := τ∘f ∈ Hom_{A⊗A^{op}}(BA, A)[1]. So, there is a unique d_{A} ∈ Coder^{1}(B(A), B(A)) with τ ∘d_{A} = m_{A} for m_{A} being the multiplication map m_{A} : A⊗_{k}A ⟶ A. m_{A} being associative is equivalent to d_{A}^{2} = 0. Hence, Coder(B(A), B(A)) is a differential graded Lie algebra.

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)).

Proof of the first statement:

Δ ∘ D^{2} = (id_{B(A)} ⊗ D + D ⊗ id_{B(A)})^{2} ∘ Δ
= id_{B(A)} ⊗ D^{2} + (D ⊗ id_{B(A)})(id_{B(A)} ⊗ D) + (id_{B(A)} ⊗ D)(D ⊗ id_{B(A)}) + D^{2} ⊗ id_{B(A)}
= id_{B(A)} ⊗ D^{2} − D ⊗ D + D ⊗ D + D^{2} ⊗ id_{B(A)}

Proof of the second statement:

Δ ∘ D^{p} = (id_{B(A)} ⊗ D + D ⊗ id_{B(A)})^{p} ∘ Δ
= (id_{B(A)} ⊗ D^{p} + (∑_{j=1}^{p−1} \binom{p}{j} · (D^{j} ⊗ D^{p−j})) + D^{p} ⊗ id_{B(A)}) ∘ Δ
= (id_{B(A)} ⊗ D^{p} + D^{p} ⊗ id_{B(A)}) ∘ Δ

Lemma 4.2. Let k be a field of characteristic p. 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)

Proof. Let D ∈ Coder^{n}(B(A), B(A)). Then, D^{p} ∈ Coder^{pn}(B(A), B(A)). The differential in the Hochschild cohomology complex Hom_{A^{e}}(BA, A) corresponds to the commutator [d_{A}, −] where Coder^{1}(B(A), B(A)) ∋ d_{A} comes from τ ∘ d_{A} = m_{A} : A ⊗ A ⟶ A being the multiplication in the algebra A. We would like to show that the p-power operation induces a genuine operation on Hochschild cohomology.

For this, it is immediate that

[d_{A}, D] = 0 ⇒ [d_{A}, D^{p}] = 0.

Hence, the p-power operation induces an operation on the cycles of the Hochschild cohomology complex.

We need to show moreover that for any E ∈ Coder^{n−1}(A, A) one has

(D + [d_{A}, E])^{p} ∈ D^{p} + im([d_{A}, −]).

(1) Let p = 2. Then, since d_{A}^{2} = 0, we get [d_{A}, E]^{2} = [d_{A}, E[d_{A}, E]] and therefore

(D + [d_{A}, E])^{2} = D^{2} + D[d_{A}, E] + [d_{A}, E]D + [d_{A}, E[d_{A}, E]].

We need to show that D[d_{A}, E] + [d_{A}, E]D ∈ im([d_{A}, −]). But,

D[d_{A}, E] + [d_{A}, E]D = [d_{A}, [D, E]] + [[d_{A}, D], E]

and whenever D is a Hochschild cocycle, then [d_{A}, D] = 0 and therefore

(D + [d_{A}, E])^{2} = D^{2} + [d_{A}, [D, E] + E[d_{A}, E]] .

This shows the statement for p = 2.

(2) Let p > 2. Again using that d_{A}^{2} = 0 one sees that for any positive integer n one has

[d_{A}, E]^{n} = (d_{A}E)^{n} − (∑_{j=1}^{n−1}(d_{A}E)^{j}(Ed_{A})^{n−j}) + (−1)^{n}(Ed_{A})^{n}

But, using as well that d_{A}^{2} = 0, for the n-fold Lie-bracket one gets the same result

[d_{A}, E[d_{A}, …, E[d_{A}, E] …]] = (d_{A}E)^{n} − (∑_{j=1}^{n−1}(d_{A}E)^{j}(Ed_{A})^{n−j}) + (−1)^{n}(Ed_{A})^{n}

So, there is an element X_{n}(E) with [d_{A}, E]^{n} = [d_{A}, X_{n}(E)] and hence [d_{A}, E]^{p} ∈ im([d_{A}, −]). Moreover, just as in the case p = 2, for any Hochschild cocycle D one has [d_{A}, D] = 0 and one gets

(D + [d_{A}, E])^{p} ∈ D^{p} + im([d_{A}, −]) .

This finishes the proof.

4.2. The p-restricted Lie structure and its derived invariance. We recall the definition of a restricted Lie algebra (cf e.g.[18, Chapter 2 Section 1]).

Definition 1. Let L be a Lie algebra over a field k of characteristic p > 0. Denote

(ad (a ⊗ X + b ⊗ 1))^{p−1} (a ⊗ 1) = ∑_{i=1}^{p−1} i · s_{i}(a, b) ⊗ X^{i} ∈ L ⊗ k[X].

A mapping [p] : L ⟶ L is called a p-mapping if (1) ad a^{[p]} = (ad a)^{p}

(2) (αa)^{[p]} = α^{p}a^{[p]}

(3) (a + b)^{[p]} = a^{[p]} + b^{[p]} + ∑_{i=1}^{p−1} s_{i}(a, b) A Lie algebra L together with a p-mapping [p] is then called a p-restricted Lie algebra.

The following proposition should be well known to the experts, but I could not find a reference. So, I include the short proof.

Proposition 4.3. • For any field k of characteristic 2 the differential graded Lie algebra Coder^{∗}(B(A), B(A)) and its homology HH^{∗+1}(A, A) are 2-restricted Lie algebras under the Gerstenhaber construction.

• For any field k of characteristic p > 2 the sum of the odd degree Hochschild cohomology groups ⨁_{n∈ℕ} HH^{2n+1}(A, A) is a p-restricted Lie algebra under the Gerstenhaber construction.

Proof. As we have seen in Lemma 4.1, for p = 2 the square of any coderivation is a coderivation. Moreover by Lemma 4.1, for any prime p the p-power of an even degree coderivation is a coderivation. So, the Gerstenhaber p-power induces a mapping

Hom_{A^{e}}((BA)^{2m+1}, A) ⟶ Hom_{A^{e}}((BA)^{2pm+1}, A).

Let us show the first property (ad a)^{p} = ad a^{[p]}. The Lie structure on the Hochschild cohomology is given by the commutator bracket on the coderivations Coder(B(A), B(A)).

The mapping a ↦ a^{[p]} is given by taking the ordinary composition of mappings. Hence,

(ad a)^{p}(y) = \underbrace{[a, [a, …, [a}_{p factors}, y \underbrace{] …]]}_{p}

= [a^{p}, y]
= (ad a^{[p]}) (y)

The second property is trivial, since (αa)^{p} = α^{p}a^{p} in Coder(B(A), B(A)).

The third property is done exactly analogously to the first example following Lemma 1.2 in [18, Chapter 2].

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.

Proof. The fact that the Gerstenhaber structure is preserved is shown by Keller in [7]. We need to show that the isomorphism maps the p-power maps to each other. Let X ∈

D^{b}(A ⊗_{k} B^{op}) be a twosided tilting complex which we will assume to be formed by modules projective on either side. Let Y = Hom_{k}(X, k) ∈ D^{b}(B ⊗_{k} A^{op}) be the inverse complex.

We first suppose p = 2. Then, as above, the functor

Y ⊗_{A} − ⊗_{A} X : D^{b}(A ⊗_{k} A^{op}) ⟶ D^{b}(B ⊗_{k} B^{op})

is an equivalence. So, this functor induces an isomorphism

φ_{X} : Hom_{A^{e}}(BA, A) ⟶ Hom_{B^{e}}(Y ⊗_{A} BA ⊗_{A} X, Y ⊗_{A} A ⊗_{A} X)
∥
Hom_{B^{e}}(BB, B)

since Y ⊗_{A} BA ⊗_{A} X is a projective resolution of B in the category of B ⊗ B^{op}-modules, and therefore homotopy equivalent to BB, and since Y ⊗_{A} A ⊗_{A} X ≃ B in D^{b}(B ⊗_{k} B^{op}). We know by the discussion in Section 4.1 that there is an isomorphism γ_{A}

Coder(B(A), B(A))[−1] ⟶^{γ_{A}} Hom_{A⊗_{k}A^{op}}(BA, A)
D ↦ τ ∘ D

where τ is the projection mapping B(A) ⟶ A on the degree 1 component A of B(A). Denote by σ_{A} the square map on Coder(B(A), B(A)) and likewise by σ_{B} the squaring on Coder(B(B), B(B)).

We need to show that

γ_{B}^{−1} ∘ σ_{B} ∘ γ_{B} = φ_{X} ∘ γ_{A}^{−1} ∘ σ_{A} ∘ γ_{A} ∘ φ_{X}^{−1}

or in other words that the diagram below is commutative.

[figure: Coder(B(A), B(A))[−1]; Coder(B(A), B(A))[−1]; Coder(B(B), B(B))[−1]; Coder(B(B), B(B))[−1]; Hom_{A^{e}}(BA, A); Hom_{A^{e}}(BA, A); Hom_{B^{e}}(BB, B); Hom_{B^{e}}(BB, B); φ_{X}; φ_{X}; γ_{A}; γ_{A}; γ_{B}; γ_{B}; σ_{A}; σ_{B}]

But this is obvious. Suppose now p > 2. Then, the p-power operation σ_{p} is only defined on the space

Coder^{2ℕ}(B(A), B(A)). Restricting therefore to only odd degree Hochschild cocycles, the same proof then holds.

This finishes the proof of the Proposition.

Remark 4.5. (1) In [8] Bernhard Keller shows that a derived equivalence of standard

type even preserves the structure of Hom_{A^{e}}(BA, A) as B_{∞}-algebra.

(2) Since there is no obvious reason why the p-power mapping should be additive in general, neither semilinear, it seems to be difficult to get an analogue for Gerstenhaber structures to Külshammer’s mappings κ_{n}.

Acknowledgement: I want to thank Bernhard Keller for his explanations concerning the Gerstenhaber bracket and I want to thank Gabriele Nebe for her kind hospitality at the RWTH Aachen as well as for bringing my attention to [2].

Furthermore I want to thank the referee for pointing out the references [13], [17] and very helpful remarks.

References

[1] Christine Bessenrodt, Thorsten Holm and Alexander Zimmermann, Generalized Reynolds ideals for nonsymmetric algebras, preprint (2006) 9 pages; arXiv: math.RA/0603189

[2] Gerald H. Cliff, Wilhelm Plesken, Alfred Weiss, Order theoretic properties of the center of a block, Proceedings of Symposia in Pure Mathematics, 47 (1987) 413-420.

[3] Karin Erdmann and Thorsten Holm, Twisted bimodules and Hochschild cohomology for self-injective algebras algebras of class A_{n}, Forum Math. 11 (1999) 177-201.

[4] Thorsten Holm and Andrzej Skowroński, Derived equivalence classification of symmetric algebras of domestic type, to appear in Journal of the Mathematical Society of Japan, arXiv: math.RA/0511227.

[5] Thorsten Holm and Alexander Zimmermann, Generalized Reynolds ideals and the scalar problem for derived equivalences of tame blocks of finite groups, preprint (2005)

[6] Bernhard Keller, Invariance and localization for cyclic homology of DG-algebras, Journal of pure and applied Algebra 123 (1998) 223-273.

[7] Bernhard Keller, Hochschild cohomology and derived Picard groups, Journal of pure and applied algebra 190 (2004) 177-196

[8] Bernhard Keller, Derived invariance of higher structures on the Hochschild complex, preprint (2003)

[9] Steffen König and Alexander Zimmermann, Derived Equivalences for Group Rings; Lecture Notes in Mathematics 1685, Springer Verlag Berlin-Heidelberg 1998.

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

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

[12] Jean-Louis Loday, Cyclic homology, third edition, Springer Verlag, Berlin-Heidelberg 1998

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

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

[16] Nicole Snashall and Oyvind Solberg, Support varieties and Hochschild cohomology rings, Proceedings of the London Mathematical Society 88 (2004) 705-732.

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

[18] Helmut Strade and Rolf Farnsteiner, Modular Lie algebras and their representations, Marcel Decker, New-York and Basel 1988

[19] Robert W. Thomason and Thomas F. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift (a collection of papers to honor Grothendieck’s 60’th birthday) vol. 3, Birkhäuser, 1990, pp. 247-435.

[20] Alexander Zimmermann, Tilted symmetric orders are symmetric orders, Arch. Math. 73 (1999) 15-17 [21] Alexander Zimmermann, Invariance of generalized Reynolds ideals under derived equivalences., to appear

in Mathematical Proceedings of the Royal Irish Academy; arXiv: math.RA/0509580

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

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