HOCHSCHILD HOMOLOGY INVARIANTS OF KÜLSHAMMER TYPE OF DERIVED CATEGORIES

ALEXANDER ZIMMERMANN

Herrn Klaus W. Roggenkamp zum 70. Geburtstag

Abstract. For a perfect field k of characteristic p > 0 and for a finite dimensional symmetric k-algebra A Külshammer studied a sequence of ideals of the centre of A using the p-power map on degree 0 Hochschild homology. In joint work with Bessenrodt and Holm we removed the condition to be symmetric by passing through the trivial extension algebra. If A is symmetric then the dual to the Külshammer ideal structure was generalised to higher Hochschild homology in earlier work [23]. In the present paper we follow this program and propose an analogue of the dual to the Külshammer ideal structure on the degree m Hochschild homology theory also to not necessarily symmetric algebras.

Introduction

Let k be a perfect field of characteristic p > 0 and let A be a finite dimensional kalgebra. For a symmetric algebra A Külshammer introduced in [14] a descending sequence of ideals T_{n}(A)^{⊥} of the centre of A satisfying various interesting properties. In [6] Héthelyi, Horváth, Külshammer and Murray continued to study this sequence of ideals, proved their invariance under Morita equivalence, and showed further properties linked amongst others to the Higman ideal of the algebra. In order to show these properties Külshammer introduced a sequence of mappings ζ_{n} : Z(A) ⟶ Z(A) with image being T_{n}(A)^{⊥}. In a completely dual procedure he introduced mappings κ_{n} : A/KA ⟶ A/KA again in case A is symmetric. These mappings encode many representation theoretic informations, in particular in case A is a group algebra. The reader may refer to Külshammer’s original articles [14] and [15] for more details. Moreover, in a recent development it could be proved in [22] that these mappings actually are an invariant of the derived category of A. This fact could be used to distinguish derived categories in very subtle situations, such as some parameter questions for blocks of dihedral or semidihedral type in joint work with Holm [10], such as very delicate questions to distinguish two families of symmetric algebras of tame domestic representation type by Holm and Skowroński [9], or in joint work with Holm [11] to distinguish derived equivalence classes of deformed preprojective algebras of generalised Dynkin type, as defined by Białkowski, Erdmann and Skowroński [3] (modified slightly for type E; cf [5, page 238]) with respect to different parameters. Most recently in joint work [17] with Yuming Liu and Guodong Zhou we gave a version which is invariant under stable equivalences of Morita type. This approach gives a link to Auslander’s conjecture on the invariance of the number of simple modules under stable equivalences of Morita type. König, Liu and Zhou [13] continued the work with focus being higher Hochschild and cyclic homology invariants.

In joint work with Bessenrodt and Holm [2] we could get rid of the assumption that A needs to be symmetric for the definition of the images of ζ_{n}. We used the trivial extension algebra 𝕋A of A, computed the image of ζ_{n} for 𝕋A and interpreted the result purely in terms of A. The derived invariant Külshammer’s ideal structure of the centre of A becomes available for any finite dimensional algebra over perfect fields of positive characteristic.

Date: January 6, 2010; revised May 10, 2010.

2000 Mathematics Subject Classification. Primary: 16E40, 18E30; Secondary: 16U70.

Key words and phrases. Hochschild homology operations, derived equivalences, Külshammer ideals.

I gratefully acknowledge financial support from a grant “partenariat Hubert Curien PROCOPE”.

Since the centre of the algebra is the degree 0 Hochschild cohomology and A/KA is the 0degree Hochschild homology, it is natural to try to understand the Külshammer construction for higher degree Hochschild (co-)homology. Denote by HH^{m}(A) the m-degree Hochschild cohomology of A and by HH_{m}(A) the m-degree Hochschild homology of A. We defined in [23] for a symmetric algebra A certain mappings κ_{n}^{(m);A} : HH_{p^{n}m}(A) ⟶ HH_{m}(A) in completely analogous manner as Külshammer’s κ_{n}. Moreover we showed that κ_{n}^{(0);A} = κ_{n}, and proved its derived invariance as well (cf Theorem 1 below).

In Definition 1 of Section 2 we propose a mapping κ̂_{n}^{(m);A}, denoted also κ̂_{n}^{(m)} if no confusion may occur, in analogy of κ_{n}^{(m);A} for algebras which are not necessarily symmetric. Further, we prove that κ̂_{n}^{(m)} is invariant under a derived equivalence: If F : D^{b}(A) ≃ D^{b}(B) is a standard equivalence then it induces coherent isomorphisms of Hochschild homology groups HH_{m}(F) : HH_{m}(A) ⟶ HH_{m}(B) and we get HH_{m}(F) ∘ κ̂_{n}^{(m);A} = κ̂_{n}^{(m);B} ∘ HH_{p^{n}m}(F) for all positive integers m and n.

Finally we give an elementary example in order to illustrate that these invariants are computable and to show that they are not trivial.

The paper is organised as follows. In Section 1.1 we recall some known facts about Hochschild (co-)homology. In Section 1.2 the known constructions concerning Külshammer ideals in degree 0 for symmetric and non symmetric algebras, using trivial extension algebras are recalled, as well as the m-degree Hochschild cohomology generalisation for symmetric algebras. Section 1.3 recalls that Hochschild homology is functorial, a statement which is used at a prominent point in our construction. In Section 2 finally we provide a Külshammer mapping for not necessarily symmetric algebras and higher Hochschild homology. We show in Theorem 2 that the new mapping κ̂_{n}^{(m)} is an invariant of the derived category. Section 3 is devoted to a detailed computation of an example.

Acknowledgement: Part of this work was done in October 2007 during a visit at Beijing Normal University and Nanjing Normal University. I wish to express my gratitude to Changchang Xi and to Jiaqun Wei for their invitation and their kind hospitality.

1. Setup of the basic tools

1.1. Hochschild homology and cohomology. Recall some well-known facts from Hochschild (co-)homology. We refer to Loday [18] for a complete presentation of this theory. Let A be a finitely generated k-algebra for a commutative ring k, and suppose A is projective as k-module. Then for every A ⊗_{k} A^{op}-module M we define the Hochschild homology of A with values in M as

HH_{n}(A, M) := T or_{n}^{A⊗_{k}A^{op}}(A, M)

and Hochschild cohomology of A with values in M as

HH^{n}(A, M) := Ext_{A⊗_{k}A^{op}}^{n}(A, M).

Often and frequently throughout the paper, for a k-algebra R we abbreviate R^{e} := R⊗_{k} R^{op}. There is a standard way to compute Hochschild (co-)homology by the bar resolution. We recall its definition in order to set up the notations.

The bar complex 𝔹A is given by

(𝔹A)_{n} := A^{(⊗_{k})^{n+1}} := \underbrace{A ⊗_{k} A ⊗_{k} ⋯ ⊗_{k} A}_{n+1 factors}

and differential

d(a_{0} ⊗ … a_{n}) = ∑_{j=0}^{n−1}(−1)^{j}a_{0} ⊗ ⋯ ⊗ a_{j−1} ⊗ a_{j}a_{j+1} ⊗ a_{j+2} ⊗ ⋯ ⊗ a_{n}.

One observes that d^{2} = 0, that (𝔹A)_{n} is a free A ⊗_{k} A^{op}-module for every n > 0 and that the complex (𝔹A, d) is a free resolution of A as a A⊗_{k} A^{op}-module, called the bar resolution. Hence,

HH_{n}(A, M) ≃ H_{n}(𝔹A ⊗_{A⊗A^{op}} M) and HH^{n}(A, M) ≃ H^{n}(Hom_{A⊗A^{op}}(𝔹A, M)).

Moreover, the bar complex is functorial in the sense that whenever f : A ⟶ B is a homomorphism of k-algebras, then

(𝔹A)_{n} ∋ a_{0} ⊗ ⋯ ⊗ a_{n} ↦ f(a_{0}) ⊗ ⋯ ⊗ f(a_{n}) ∈ 𝔹B

induces a morphism of complexes 𝔹A ⟶ 𝔹B.

1.2. The Külshammer ideals; constructions and their Hochschild generalisation. We recall the constructions introduced by Külshammer in [14] and [15] as well as their generalisation introduced in [23] to higher degree Hochschild cohomology and their generalisations to 0-degree Hochschild cohomology for not necessarily symmetric algebras.

1.2.1. The Külshammer ideal theory for algebras which are symmetric. Let k be a perfect field of characteristic p > 0 and let A be a k-algebra. Then

KA :=< ab − ba | a, b ∈ A >_{k−space}

is a Z(A)-module and for all n ∈ ℕ

T_{n}(A) := {x ∈ A | x^{p^{n}} ∈ KA}

is a Z(A) submodule of A. If A is symmetric with symmetrising form

< , >: A ⊗_{k} A ⟶ k

taking orthogonal spaces, since KA^{⊥} = Z(A), T_{n}(A)^{⊥} is an ideal of Z(A).

Given an equivalence of triangulated categories of standard type (cf Section 1.2.2 below) D^{b}(A) ⟶ D^{b}(B) between two symmetric k-algebras A and B, Rickard showed in [20] that this equivalence induces an isomorphism Z(A) ≃ Z(B) and if k is a perfect field, it is shown in [22] that T_{n}(A)^{⊥} is mapped by this isomorphism to T_{n}(B)^{⊥}.

Moreover A is symmetric if and only if A ≃ A^{∗} := Hom_{k}(A, k) as A ⊗_{k} A^{op}-modules. Since we supposed k to be perfect the Frobenius map is invertible and let k ∋ λ ↦ λ^{p^{−1}} ∈ k be its inverse. Denote the n-th iterate by p^{−n} := (p^{−1})^{n}. Since A/KA ∋ b ↦ b^{p} ∈ A/KA is a well-defined additive mapping, semilinear with respect to the Frobenius mapping, one defines an element in Hom_{k}(A, k) by

a ↦ (b ↦< a, b^{p^{n}} >^{p^{−n}})

and so for every a ∈ Z(A) there is a unique element ζ_{n}(a) ∈ A so that

< a, b^{p^{n}} >=< ζ_{n}(a), b >^{p^{n}} ∀ b ∈ A.

Similarly, using that Z(A)^{⊥} = KA, one gets a mapping κ_{n} : A/KA ⟶ A/KA satisfying

< a^{p^{n}}, b >=< a, κ_{n}(b) >^{p^{n}} ∀ a ∈ Z(A), b ∈ A/KA.

In [23] this construction was generalised to a structure on Hochschild homology which is invariant under equivalences of derived categories. More precisely, let A be a symmetric finite dimensional k-algebra for k a perfect field of characteristic p > 0. Then, denoting by 𝔹A the bar resolution of A and A^{e} = A ⊗_{k} A^{op},

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

and so

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

which induces a non degenerate pairing

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

By a construction analogous to the one for the pairing

< , >: Z(A) × A/KA ⟶ k

one gets a mapping

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

denoted simply by κ_{n}^{(m)} if no confusion may occur, satisfying

< z^{p^{n}}, x >_{p^{n}m}= (< z, κ_{n}^{(m)}(x) >_{m})^{p^{n}}

for all x ∈ HH_{p^{n}m} and z ∈ HH^{m}(A, A).

1.2.2. On derived equivalences and Hochschild (co-)homology. In [23] we studied invariance properties of these mappings with respect to equivalences between derived categories. Rickard showed in [20] that whenever A and B are k-algebras over a field k, and whenever D^{b}(A) and D^{b}(B) are equivalent as triangulated categories, then there is a complex X ∈ D^{b}(B ⊗_{k} A^{op}) so that X ⊗_{A}^{𝕃} − : D^{b}(A) ⟶ D^{b}(B) is an equivalence. Such an equivalence is called of standard type and X is called a two-sided tilting complex. It is shown in [20] that every equivalence of triangulated categories D^{b}(A) ⟶ D^{b}(B) coincides with a suitable standard one on objects.

A tilting complex T is a complex in K^{b}(A), the homotopy category of bounded complexes of projective A-modules, satisfying Hom_{D^{b}(A)}(T, T [i]) = 0 for all i ≠ 0, and add(T) generates K^{b}(A) as triangulated category. Two k-algebras A and B have equivalent derived categories D^{b}(A) and D^{b}(B) if and only if there is a tilting complex T in D^{b}(A) with endomorphism ring B. Given an equivalence D^{b}(B) ⟶ D^{b}(A), the image of the regular B-module is a tilting complex. Hence, by the above, for every tilting complex T in D^{b}(A) there is a two-sided tilting complex X in D^{b}(A ⊗_{k} B^{op}) so that X ≃ T in D^{b}(A).

Let F_{X} = X ⊗_{A}^{𝕃} − : D^{b}(A) ⟶ D^{b}(B) be an equivalence of derived categories with quasiinverse Y ⊗^{𝕃} − (cf Rickard [20]). If A is symmetric, then B is symmetric as well (cf [21] for general base rings, and [20] for fields) and

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

is an equivalence again. This equivalence induces an isomorphism

HH_{∗}(F_{X}) : HH_{∗}(A) ⟶ HH_{∗}(B)

made explicit in [23]. Recall the precise mapping, defined on the level of complexes, which will be needed later.

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

Then HH_{∗}(F_{X}) is the mapping induced on the homology.

We recall one of the main results of [23]. This result will be one of our main ingredients in the proof of Theorem 2.

Theorem 1. [23] Let A be a finite dimensional symmetric k-algebra over the perfect field k of characteristic p > 0. Let B be a second algebra such that D^{b}(A) ≃ D^{b}(B) as triangulated categories, 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} = κ_{n}^{(m),B} ∘ HH_{p^{n}m}(F) .

We take the opportunity to mention that in [23] in the statement of the theorem the condition on the field k to be perfect is unfortunately missing. However, this was a general assumption throughout [23].

1.2.3. Trivial extension of an algebra; the Külshammer ideal theory in the general case. Let k be a commutative ring and let A be a k-algebra. Then Hom_{k}(A, k) is an A⊗_{k} A^{op}-module by the action

((a, b) · f) (x) := (a · f · b)(x) := f(bxa) ∀a, x ∈ A; b ∈ A^{op}; f ∈ Hom_{k}(A, k).

Now, the k-vector space

𝕋A := A ⊕ Hom_{k}(A, k)

becomes a k-algebra by putting

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

Then

A ⟶^{ι_{A}} 𝕋A
a ↦ (a, 0)

is a ring homomorphism. Moreover {0} ⊕ Hom_{k}(A, k) is a two-sided ideal of 𝕋A and the canonical projection

𝕋A ⟶^{π_{A}} A
(a, f) ↦ a

is a splitting for ι_{A}; i.e. π_{A} ∘ ι_{A} = id_{A}. The algebra 𝕋A is always symmetric via the bilinear form

𝕋A ⊗_{k} 𝕋A ∋< (a, f); (b, g) >↦ f(b) + g(a) ∈ k

which induces, for k a field and A finite dimensional over k, an isomorphism

𝕋A ⟶ Hom_{k}(𝕋A, k)
(a, f) ↦ ((b, g) ↦ f(b) + g(a))

of 𝕋A ⊗_{k} 𝕋A^{op}-modules.

In the joint paper [2] with Bessenrodt and Holm it is shown that the Külshammer ideals satisfy

T_{n}(𝕋A)^{⊥} = {0} ⊕ Ann_{Hom_{k}(A,k)}(T_{n}(A))

for all n ≥ 1, and therefore for all n ≥ 1 the Z(A)-modules Ann_{Hom_{k}(A,k)}(T_{n}(A)) are invariants under derived equivalences.

Remark 1.1. It should be noted that for algebras A which are already symmetric, via the symmetrising form < , > we get an isomorphism of vector spaces λ : Z(A) ⟶ Hom_{k}(A/KA, k). Now, every linear form on A equals a form of the shape < z, − > for some z ∈ Z(A) and so

λ^{−1}(Ann_{Hom_{k}(A,k)}(T_{n}(A))) = {z ∈ Z(A) | < z, T_{n}(A) >= 0} = T_{n}(A)^{⊥}.

We find back our original result.

1.3. Relating Hochschild homology of an algebra and of its trivial extension. Let R and S be k-algebras and let α : R ⟶ S and β : S ⟶ R be algebra homomorphisms. Then it is well-known (cf e.g. Loday [18, Chapter 1, Section 1.1.4]) that Hochschild homology is functorial, i.e. α and β induce mappings

HH_{∗}(α) : HH_{∗}(R) ⟶ HH_{∗}(S) and HH_{∗}(β) : HH_{∗}(S) ⟶ HH_{∗}(R)

so that

HH_{∗}(id_{R}) = id_{HH_{∗}(R)} as well as HH_{∗}(β) ∘ HH_{∗}(α) = HH_{∗}(β ∘ α).

In particular in case of the mappings ι_{A} : A ⟶ 𝕋A and π_{A} : 𝕋A ⟶ A from an algebra A to its trivial extension 𝕋A we get induced mappings giving a split projection

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

and a split injection

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

This can be defined on the level of complexes. As seen in Section 1.1 the algebra homomorphism α : R ⟶ S induces a morphism of complexes

𝔹α : 𝔹R ⟶ 𝔹S
(x_{0} ⊗ ⋯ ⊗ x_{n}) ↦ (α(x_{0}) ⊗ ⋯ ⊗ α(x_{n}))

and likewise for β : S ⟶ R. Also on the complex computing Hochschild homology an analogous morphism of complexes is defined by

𝔹R ⊗_{R^{e}} R ⟶ 𝔹S ⊗_{S^{e}} S
(x_{0} ⊗ ⋯ ⊗ x_{n}) ⊗ x_{n+1} ↦ (α(x_{0}) ⊗ ⋯ ⊗ α(x_{n})) ⊗ α(x_{n+1})

and likewise for β. Its homology induces a mapping H_{∗}𝔹α : HH_{∗}(R) ⟶ HH_{∗}(S) which is easily seen to be HH_{∗}(α) obtained above.

In particular in our situation

H_{∗}𝔹π_{A} ∘ H_{∗}𝔹ι_{A} = id_{𝔹A⊗_{A^{e}}A}

and it is clear that

H_{∗}𝔹π_{A} = HH_{∗}(π_{A}) as well as H_{∗}𝔹ι_{A} = HH_{∗}(ι_{A}).

We reproved the following proposition which was obtained in a much larger context and generality, and much more sophisticated methods, by Cibils, Marcos, Redondo and Solotar in [4, Theorem 5.8].

Proposition 1. [4, Theorem 5.8] Let k be a field and let A be a k-algebra. Then the canonical embedding A ⟶ 𝕋A induces a canonical embedding of HH_{∗}(A) as a direct factor of HH_{∗}(𝕋A).

2. Külshammer-like Hochschild invariants for non symmetric algebras

Let k be a perfect field of characteristic p > 0 and let A be a k-algebra. Recall from Section 1.2.1 that in case A is symmetric, we defined in [23] mappings

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

satisfying

(< x, κ_{n}^{(m)}(y) >_{m})^{p^{n}} =< x^{p^{m}}, y >_{p^{n}m} ∀ x ∈ HH^{m}(A); y ∈ HH_{p^{n}m}(A).

We should mention that the pairing < , >_{m} is defined only on the (co-)homology of finite dimensional symmetric algebras by the isomorphism

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

whose homology gives an isomorphism

HH_{∗}(A) ≃ Hom_{k}(HH^{∗}(A), k) .

Observe that the double dual is the identity for finite dimensional vector spaces only, and hence here we use the fact that finite dimensional algebras have projective bimodule resolutions with finite dimensional homogeneous components.

Definition 1. Let A be a finite dimensional (not necessarily symmetric) k-algebra. Then put

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

We shall prove now that the invariant κ̂_{n}^{(m)} is an invariant of the derived category of A in the same sense as it was proved for symmetric algebras (cf Theorem 1).

Rickard showed in [19] that an equivalence between the derived categories of two kalgebras induces an equivalence between the derived categories of their trivial extension algebras. We need the following improvement of his result, which seems to be of interest in its own right.

Proposition 2. Let A and B be finite dimensional k algebras and let T be a tilting complex in D^{b}(A) with endomorphism ring B.

(1) (Rickard [19]) Then 𝕋A ⊗_{A} T is a tilting complex in D^{b}(𝕋A) with endomorphism ring 𝕋B.

(2) Let X be a two-sided tilting complex in D^{b}(𝕋A ⊗_{k} 𝕋B) so that 𝕋A ⊗_{A} T ≃ X in D^{b}(𝕋A). Then A⊗_{𝕋A}X and X ⊗_{𝕋B}B are two-sided tilting complexes in D^{b}(A⊗_{k} B). Moreover, T ≃ A ⊗_{𝕋A} X.

Proof. Let T be a tilting complex in D^{b}(A) with endomorphism ring B. Then by Rickard’s theorem [19, Corollary 5.4] the complex 𝕋A ⊗_{A} T is a tilting complex in D^{b}(𝕋A) with endomorphism ring 𝕋B. By Keller’s theorem [12] there is a two-sided tilting complex X in D^{b}(𝕋A ⊗_{k} 𝕋B^{op}) so that _{𝕋A}|X ≃ 𝕋A ⊗_{A} T.

Now,

Hom_{D^{b}(A)}(A ⊗_{𝕋A} X, A ⊗_{𝕋A} X) = Hom_{D^{b}(A)}(A ⊗_{𝕋A} _{𝕋A}|X, A ⊗_{𝕋A} _{𝕋A}|X)
≃ Hom_{D^{b}(A)}(A ⊗_{𝕋A} 𝕋A ⊗_{A} T, A ⊗_{𝕋A} 𝕋A ⊗_{A} T)
≃ Hom_{D^{b}(A)}(T, T)
≃ B

as rings. But, we know that 𝕋B acts on A ⊗_{𝕋A} X by multiplication on the right. Hence, we get a ring homomorphism

𝕋B ⟶ Hom_{D^{b}(A)}(A ⊗_{𝕋A} X, A ⊗_{𝕋A} X).

Since we have seen that the endomorphism ring of A⊗_{𝕋A}X is isomorphic to B, the mapping

𝕋B ⟶ Hom_{D^{b}(A)}(A ⊗_{𝕋A} X, A ⊗_{𝕋A} X)

factorises through the canonical projection 𝕋B ⟶ B. Hence, the action of 𝕋B on A⊗_{𝕋A}X has B^{∗} in the kernel.

Now, the 𝕋B ⊗_{k} 𝕋B^{op}-module structure of Hom_{D^{b}(A)}(A⊗_{𝕋A}X, A⊗_{𝕋A}X) is the following. The action of 𝕋B on the first argument gives the 𝕋B-action from the left on Hom_{D^{b}(A)}(A⊗_{𝕋A} X, A ⊗_{𝕋A} X) and the action of 𝕋B on the right comes from the action of 𝕋B on the second argument. Both have the degree 2 nilpotent ideal B^{∗} in the kernel and so, the natural action of 𝕋B ⊗_{k} 𝕋B^{op} is actually an action of B ⊗_{k} B^{op}.

Therefore,

Hom_{D^{b}(A)}(A ⊗_{𝕋A} X, A ⊗_{𝕋A} X) ≃ B

as B ⊗_{k} B^{op}-modules.

Hence A ⊗_{𝕋A} X is invertible from the left in the sense that

Hom_{D^{b}(A)}(A ⊗_{𝕋A} X, A ⊗_{𝕋A} X) ≃ Hom_{A}(A ⊗_{𝕋A} X, A) ⊗_{A} (A ⊗_{𝕋A} X) ≃ B

as B ⊗_{k} B^{op}-modules.

We still need to show that A ⊗_{𝕋A} X is invertible from the right as well. Since 𝕋A and 𝕋B are both symmetric, and since X is a two-sided tilting complex in D^{b}(𝕋A ⊗_{k} 𝕋B^{op}), its inverse complex is given by its k-linear dual Hom_{k}(X, k) =: X̌. We claim that X̌ ⊗_{𝕋A} A is a right inverse of A ⊗_{𝕋A} X. Indeed, by the previous paragraph, we see that 𝕋B acts on the right of A ⊗_{𝕋A} X via the projection 𝕋B ⟶ B, i.e. the nilpotent ideal B^{∗} is in the kernel of this action. Analogously, the same holds for X̌ ⊗_{𝕋A} A. Hence, the tensor product over 𝕋B equals the tensor product over B only.

(A ⊗_{𝕋A} X) ⊗_{B} (X̌ ⊗_{𝕋A} A) ≃ (A ⊗_{𝕋A} X) ⊗_{𝕋B} (X̌ ⊗_{𝕋A} A)
≃ A ⊗_{𝕋A} (X ⊗_{𝕋B} X̌) ⊗_{𝕋A} A
≃ A ⊗_{𝕋A} 𝕋A ⊗_{𝕋A} A
≃ A

as A ⊗_{k} A^{op}-modules.

As a consequence A⊗_{𝕋A}X is a two-sided tilting complex with restriction to the left being T. Moreover, it is clear that π_{A} will map X to A ⊗_{𝕋A} X.

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).

Proof. Let T be a tilting complex in D^{b}(A) with endomorphism ring B and let X be a twosided tilting complex in D^{b}(𝕋A ⊗_{k} 𝕋B^{op}) with 𝕋A ⊗_{A} T ≃ X in D^{b}(𝕋A). By Proposition 2 we get A ⊗_{𝕋A} X is a two-sided tilting complex in D^{b}(A ⊗_{k} B).

We now use Theorem 1:

HH_{m}(F_{X}) ∘ κ_{n}^{(m);𝕋A} = κ_{n}^{(m);𝕋B} ∘ HH_{p^{n}m}(F_{X}).

Multiplying with HH_{p^{n}m}(ι_{A}) from the right and with HH_{m}(π_{B}) from the left gives then

HH_{m}(π_{B})∘HH_{m}(F_{X})∘κ_{n}^{(m);𝕋A}∘HH_{p^{n}m}(ι_{A}) = HH_{m}(π_{B})∘κ_{n}^{(m);𝕋B}∘HH_{p^{n}m}(F_{X})∘HH_{p^{n}m}(ι_{A}).

We now claim that

HH_{p^{n}m}(F_{X}) ∘ HH_{p^{n}m}(ι_{A}) = HH_{p^{n}m}(ι_{B}) ∘ HH_{p^{n}m}(F_{A⊗_{𝕋A}X})

and

HH_{m}(π_{B}) ∘ HH_{m}(F_{X}) = HH_{m}(F_{A⊗_{𝕋A}X}) ∘ HH_{m}(π_{A}).

Recall from Section 1.2.2 how HH_{m}(F_{X}) is defined. Then, the commutativity relation will be proven as soon as we have that the diagrams

(X̌ ⊗_{𝕋A} 𝔹(𝕋A) ⊗_{𝕋A} X) ⊗_{(𝕋B)^{e}} 𝕋B  ⟵  𝔹𝕋A ⊗_{(𝕋A)^{e}} (X ⊗_{𝕋B} X̌)
↑ HH(ι_{B})  HH(ι_{A}) ↑
((X̌ ⊗_{𝕋A} A) ⊗_{A} 𝔹A ⊗_{A} (A ⊗_{𝕋A} X)) ⊗_{B^{e}} B ⟵ 𝔹A ⊗_{A^{e}} ((A ⊗_{𝕋A} X) ⊗_{B} (X̌ ⊗_{𝕋A} A))

as well as

(X̌ ⊗_{𝕋A} 𝔹(𝕋A) ⊗_{𝕋A} X) ⊗_{(𝕋B)^{e}} 𝕋B ⟵ 𝔹𝕋A ⊗_{(𝕋A)^{e}} (X ⊗_{𝕋B} X̌)
↓ HH(π_{B})  HH(π_{A}) ↓
((X̌ ⊗_{𝕋A} A) ⊗_{A} 𝔹A ⊗_{A} (A ⊗_{𝕋A} X)) ⊗_{B^{e}} B ⟵ 𝔹A ⊗_{A^{e}} ((A ⊗_{𝕋A} X) ⊗_{B} (X̌ ⊗_{𝕋A} A))

are commutative.

We use the canonical isomorphisms

(X̌ ⊗_{𝕋A} A) ⊗_{A} 𝔹A ⊗_{A} (A ⊗_{𝕋A} X) ≃ X̌ ⊗_{𝕋A} 𝔹A ⊗_{𝕋A} X

and

𝔹A ⊗_{A^{e}} (A ⊗_{𝕋A} X ⊗_{B} X̌ ⊗_{𝕋A} A) ≃ 𝔹A ⊗_{(𝕋A)^{e}} (X ⊗_{𝕋B} X̌),

which proves that we only need to show that the diagrams

(X̌ ⊗_{𝕋A} 𝔹(𝕋A) ⊗_{𝕋A} X) ⊗_{(𝕋B)^{e}} 𝕋B ⟵ 𝔹𝕋A ⊗_{(𝕋A)^{e}} (X ⊗_{𝕋B} X̌)
↑ HH(ι_{B})  HH(ι_{A}) ↑
(X̌ ⊗_{𝕋A} 𝔹A ⊗_{𝕋A} X) ⊗_{B^{e}} B  ⟵ 𝔹A ⊗_{(𝕋A)^{e}} (X ⊗_{𝕋B} X̌)

and

(X̌ ⊗_{𝕋A} 𝔹(𝕋A) ⊗_{𝕋A} X) ⊗_{(𝕋B)^{e}} 𝕋B ⟵ 𝔹𝕋A ⊗_{(𝕋A)^{e}} (X ⊗_{𝕋B} X̌)
↓ HH(π_{B})  HH(π_{A}) ↓
(X̌ ⊗_{𝕋A} 𝔹A ⊗_{𝕋A} X) ⊗_{B^{e}} B  ⟵ 𝔹A ⊗_{(𝕋A)^{e}} (X ⊗_{𝕋B} X̌)

are commutative. But this is clear since the unit element of 𝕋B (and of 𝕋A resp.) is the image of the unit element of B (and of A resp.) under ι.

Hence the claim is proven.

This implies now

HH_{m}(F_{A⊗_{𝕋A}X}) ∘ κ̂_{n}^{(m);A} = HH_{m}(F_{A⊗_{𝕋A}X}) ∘ HH_{m}(π_{A}) ∘ κ_{n}^{(m);𝕋A} ∘ HH_{p^{n}m}(ι_{A})
= HH_{m}(π_{B}) ∘ HH_{m}(F_{X}) ∘ κ_{n}^{(m);𝕋A} ∘ HH_{p^{n}m}(ι_{A})
= HH_{m}(π_{B}) ∘ κ_{n}^{(m);𝕋B} ∘ HH_{p^{n}m}(F_{X}) ∘ HH_{p^{n}m}(ι_{A})
= HH_{m}(π_{B}) ∘ κ_{n}^{(m);𝕋B} ∘ HH_{p^{n}m}(ι_{B}) ∘ HH_{p^{n}m}(F_{A⊗_{𝕋A}X})
= κ̂_{n}^{(m);B} ∘ HH_{p^{n}m}(F_{A⊗_{𝕋A}X})

which proves Theorem 2.

3. Dual numbers as an example

It should be noted that the mappings κ_{n}^{(m)} are not zero in general. The dual numbers provide an example. In this section we shall show this fact and prove moreover that κ_{m}^{(n)} = κ̂_{m}^{(n)} in this case.

Holm computed the cohomology ring of rings k[X]/(f(X)) for all polynomials f(X) and all fields k. In particular for f(X) = X^{2} one obtains a symmetric algebra, the algebra of dual numbers, whose Hochschild homology and cohomology are isomorphic. For fields of characteristic zero Lindenstrauss [16] computed in general the Hochschild homology of algebras k[X_{1}, …, X_{n}]/𝔪^{m} for 𝔪 being the maximal ideal corresponding to the point (0, 0, …, 0) by exhibiting an explicit projective resolution.

We shall reprove parts of these statements since we will need quite detailed information about the (co-)cycles that represent each element in the Hochschild structure.

Let A = k[ϵ]/(ϵ^{2}) throughout this section and let k be a field of characteristic p > 0.

Lemma 3. 𝕋A ≃ A ⊗_{k} A.

Proof. Clearly, since A is commutative, also 𝕋A is commutative. Moreover, there is a kbasis A = k·1⊕k·ϵ and since A is symmetric, the symmetrising form being ⟨x+yϵ, x′+y′ϵ⟩ := xy′ + x′y, we get a K-basis of 𝕋A = A^{∗} ×A by

𝕋A = k · (⟨1, −⟩, 0) ⊕ k · (⟨ϵ, −⟩, 0) ⊕ k · (⟨0, −⟩, 1) ⊕ k · (⟨0, −⟩, ϵ).

Let δ := (⟨1, −⟩, 0), σ := (⟨ϵ, −⟩, 0), ε := (⟨0, −⟩, ϵ) and 1 := (⟨0, −⟩, 1) (remarking that this is still the unit element of the trivial extension algebra). We verify immediately that ε^{2} = δ^{2} = 0. Furthermore, ε · δ = δ · ε = σ. Hence, 𝕋A is the quotient of the quiver algebra with one vertex and two loops by the relations saying that the two loops are nilpotent of order 2 and that they commute. This describes exactly the algebra A ⊗_{k} A.

Remark 3.1. If p = 2, then A is isomorphic to the group algebra of the cyclic group of order 2. Then A ⊗_{k} A^{op} is isomorphic to the group algebra of the Klein four group and the Hochschild cohomology ring is known. Holm showed [8, Theorem 3.2.1] that in this case

HH^{∗}(A ⊗_{k} A^{op}) ≃ A[X, Y]

where X and Y are algebraically independent of degree 1.

For any p > 0 we know how to determine the Hochschild homology in this more general case as well by the Künneth formula. Indeed,

HH_{∗}(A ⊗_{k} A) ≃ HH_{∗}(A) ⊗_{k} HH_{∗}(A)

by the Künneth formula.

We have to study the injection

HH_{∗}(A) ⟶ HH_{∗}(𝕋A) ≃ HH_{∗}(A ⊗_{K} A) ≃ HH_{∗}(A) ⊗_{K} HH_{∗}(A)

given by the isomorphism 𝕋A ≃ A ⊗_{K} A, the injection A ⟶ 𝕋A and the Künneth formula, as well as the projection

HH_{∗}(A) ⊗_{k} HH_{∗}(A) ≃ HH_{∗}(A ⊗_{k} A) ≃ HH_{∗}(𝕋A) ⟶ HH_{∗}(A).

The algebra A is symmetric and hence

Hom_{k}(HH_{m}(A), k) ≃ HH^{m}(A)

by an isomorphism induced by the symmetrising bilinear form. Therefore there is a non degenerate pairing

HH^{m}(A) × HH_{m}(A) ⟶ K

as usual.

Remark 3.2. The algebra A ⊗ A is the quotient of the quiver algebra with one vertex and two loops by the relations saying that the two loops are nilpotent of order 2 and that they commute. Hence, we may replace any loop by a non trivial linear combination of these two loops, completing by another linear combination of the two loops so that the determinant of the coefficient matrix is non zero. Hence, we may suppose that the inclusion A ⟶ 𝕋A is given by the inclusion A ⟶ A ⊗ A defined by a ↦ 1 ⊗ a. Indeed, the Hochschild homology computation by means of the corresponding double complex does not depend on the choice of a basis.

BACH [1] and Holm [7] computed an explicit resolution of bimodules for a monogenic algebra, such as the dual numbers. Abbreviate for simplicity A = K[ϵ]/ϵ^{2} (as usual) and A^{2} := A ⊗_{k} A. Then a free resolution C of A is periodic of period 2 and is given by

C : (A ⟵)A^{2} ⟵^{d_{1}} A^{2} ⟵^{d_{2}} A^{2} ⟵^{d_{1}} A^{2} ⟵^{d_{2}} A^{2} ⟵ ⋯

where d_{1} is multiplication by 1 ⊗ ϵ − ϵ ⊗ 1 and d_{2} is multiplication by 1 ⊗ ϵ + ϵ ⊗ 1. Applying the functor − ⊗_{A^{2}} A gives a complex

A ⟵^{(d_{1})_{⊗}} A ⟵^{(d_{2})_{⊗}} A ⟵^{(d_{1})_{⊗}} A ⟵^{(d_{2})_{⊗}} A ⟵ …

with (d_{1})_{⊗} = 0 and (d_{2})_{⊗} = 2ϵ.

Now, applying the functor Hom_{A^{2}}(−, A) gives a complex Hom_{A^{2}}(C, A)

… ⟶ A ⟶^{(d_{1})_{h}} A ⟶^{(d_{2})_{h}} A ⟶^{(d_{1})_{h}} A ⟶^{(d_{2})_{h}} A

where again (d_{1})_{h} = 0 and (d_{2})_{h} = 2ϵ.

The Künneth formula gives that the tensor product of the cohomology is the cohomology of the tensor product

H (Hom_{A^{2}}(C, A) ⊗ Hom_{A^{2}}(C, A)) = H (Hom_{A^{2}}(C, A)) ⊗ H (Hom_{A^{2}}(C, A)) .

We observe that

Hom_{A^{2}}(C, A) ⊗ Hom_{A^{2}}(C, A) ≃ Hom_{A^{2}⊗A^{2}}(C ⊗ C, A ⊗ A)

and this is the complex computing the Hochschild cohomology of A ⊗_{k} A, and whence for 𝕋A.

As usual the Hochschild (co-)homology depends on weather p = 2 or p > 2. Our arguments use the explicit structure of the Hochschild (co-)homology and therefore we shall need to treat these two cases separately.

3.1. The case p > 2. Holm [7] shows that

HH^{∗}(A) ≃ A[U, Z]/(Zϵ, Uϵ, U^{2})

for an element U in degree 1 and an element Z in degree 2. Hence HH^{m}(A) is onedimensional, generated by Z^{n} if m = 2n or UZ^{n} if m = 2n + 1, for m > 0 and isomorphic to A in degree 0.

We may therefore choose for each n ∈ ℕ \ {0} an element z_{n} in HH_{2n}(A) which corresponds to Z^{n} in HH^{∗}(A) under the above isomorphism and we get HH_{2n}(A) = kz_{n}. Moreover, choose an element uz_{n} ∈ HH_{2n+1}(A) which corresponds to UZ^{n} under the above isomorphism. Hence, HH_{2n+1}(A) = kuz_{n}.

Observe that

(HH_{∗}(A) ⊗ HH_{∗}(A))^{∗} ≃ HH_{∗}(A)^{∗} ⊗ HH_{∗}(A)^{∗} ≃ HH^{∗}(A) ⊗ HH^{∗}(A)

where the first isomorphism is canonical and the second one is induced by the isomorphism HH_{∗}(A)^{∗} ≃ HH^{∗}(A). Hence using this chain of isomorphisms we choose the k-basis

x_{i}y_{j} in bidegree (2i, 2j) of HH_{2i+2j}(A ⊗ A),
x_{i}vy_{j} in bidegree (2i + 1, 2j) of HH_{2i+1+2j}(A ⊗ A),
x_{i}y_{j}w in bidegree (2i, 2j + 1) of HH_{2i+2j+1}(A ⊗ A),
x_{i}vy_{j}w in bidegree (2i + 1, 2j + 1) of HH_{2i+1+2j+1}(A ⊗ A),

which is the dual basis element to the corresponding monomial in X^{i}Y^{j}, the monomial X^{i}V Y^{j}, the monomial X^{i}Y^{j}W or the monomial X^{i}V Y^{j}W of HH^{∗+∗}(A ⊗ A) under that isomorphism. The p-th power of all these basis elements is zero, except the element X^{i}Y^{j}, whose p-th power is X^{pi}Y^{pj}.

The (minimal) projective resolution of A^{2} used above can most easily be expressed as a double complex C ⊗ C, as was shown above.

A^{2}  ⟵  A^{2}  ⟵  A^{2}  ⟵  A^{2}  ⟵  A^{2}  ⟵
↑  ↑  ↑  ↑  ↑
A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵
↑ ↑ ↑ ↑ ↑ ↑
A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵
↑ ↑ ↑ ↑ ↑ ↑
A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵
↑ ↑ ↑ ↑ ↑ ↑
A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵ A^{2} ⊗ A^{2} ⟵
↑  ↑  ↑  ↑  ↑  ↑

In order to view the image of z_{n} ∈ HH_{2n}(A) in HH_{2n}(𝕋A) we need to first establish the multiplicative structure of HH^{∗}(A ⊗ A) in terms of maps in the double complex.

The element X is the degree (2, 0) mapping consisting of the identity on all homogeneous components except on the borders of this bi-complex, and the element Y is the degree (0, 2) mapping identity on all homogeneous components except on the borders of this bi-complex.

The element V is the degree (1, 0) mapping consisting of ϵ ⊗ 1 in all degrees, except on the borders of the bi-complex, where it is 0. Likewise the element W is the degree (0, 1) mapping consisting of ϵ ⊗ 1 in all degrees, except on the borders of the bi-complex, where it is 0.

We claim that the element X^{i}Y^{j} is represented by the degree (2i, 2j) mapping consisting of the identity on all homogeneous components except on the borders of this bi-complex. The proof of this statement is an easy induction on i and j. Actually, the composition of a morphism of bi-complexes of degree (2i, 2j) of the given shape by a morphism of complexes of degree (2, 0), or (0, 2) respectively, corresponds to a morphism of bi-complexes of degree (2(i + 1), 2j), or (2i, 2(j + 1)) respectively. This corresponds to the cup product of X^{i}Y^{j} with X, or Y respectively.

The element X^{i}V Y^{j} is represented by the degree (2i + 1, 2j) mapping consisting of the mapping (ϵ ⊗ id) ⊗ (id ⊗ id) on all homogeneous components except on the borders of this bi-complex. Likewise the element X^{i}Y^{j}W is represented by the degree (2i, 2j + 1) mapping consisting of the mapping (id ⊗ id) ⊗ (ϵ ⊗ id) and the element X^{i}V Y^{j}W is represented by the degree (2i + 1, 2j + 1) mapping consisting of the mapping (ϵ ⊗ id) ⊗ (ϵ ⊗ id).

The cup product on HH^{∗}(A) is similar: Z corresponds to the degree 2 mapping being the identity on all homogeneous of C, except the degree 0 and degree 1 component, where the mapping clearly is 0. Now, again by an analogous induction as in the bi-complex case, the element Z^{n} corresponds to the degree 2n mapping being the identity on all homogeneous of C, except the degrees up to 2n − 1 component, where the mapping clearly is 0. The element U corresponds to the degree 1 mapping ϵ ⊗ 1 in all degrees except the degree 0, where it is 0. The cup product with Z^{n} is just an additional shift in degree, the cup product of U with U is 0, since (ϵ ⊗ 1)^{2} = 0.

We now need to compare this mapping to the dual of the Hochschild homology side. Hence, we dualise the bi-complex and the cocycle representing X^{i}Y^{j}. This just reverses the arrows, identifying again A^{∗} with A. The dual of the cocycle X^{i}Y^{j} can also be obtained by first applying Hom_{A^{2}⊗A^{2}}(−, A^{2}) to the double complex and then K-dualising the result. We obtain the double complex D_{HH(A^{2})}.

.  .  .  .  .
↑  ↑  ↑  ↑  ↑
. ⟵ A^{2} ⟵^{0} A^{2} ⟵ A^{2} ⟵^{0} A^{2} ⟵ A^{2} ⟵
↑ 0 ↑ 0 ↑ 0 ↑ 0 ↑ 0
. ⟵ A^{2} ⟵^{0} A^{2} ⟵ A^{2} ⟵^{0} A^{2} ⟵ A^{2} ⟵
↑ ↑ ↑ ↑ ↑
. ⟵ A^{2} ⟵^{0} A^{2} ⟵ A^{2} ⟵^{0} A^{2} ⟵ A^{2} ⟵
↑ 0 ↑ 0 ↑ 0 ↑ 0 ↑ 0
. ⟵ A^{2} ⟵^{0} A^{2} ⟵ A^{2} ⟵^{0} A^{2} ⟵ A^{2} ⟵
↑ ↑ ↑ ↑ ↑

which now represents exactly the complex computing Hochschild homology of A⊗A. Half of the arrows represent the 0-morphism, the rest, without a side- or superscript in the diagram, represent the mapping 2ϵ. The cycle representing X^{n} is the 2n-th mapping identity in the upper line. The complex computing the Hochschild homology of A is simpler:

C_{HH(A)} : A ⟵^{0} A ⟵ A ⟵^{0} A ⟵ A ⟵^{0} ⋯

with the same convention on non superscribed arrows.

The injection A ⟶ A ⊗ A ≃ 𝕋A maps a ↦ 1 ⊗ a and the projection A ⊗ A ⟶ A maps a ⊗ b ↦ ab. Hence, the projective resolution P_{•} of A as A^{2}-modules maps to the projective resolution P_{•} ⊗ P_{•} of A^{2} as A^{2} ⊗ A^{2}-modules as x ↦ 1 ⊗ x. The second degree component is 0, except in degree 0, where it is constantly 1 ⊗ 1. Hence, the Hochschild homology complex of A injects by the identity into the first line of the Hochschild homology (double-) complex of A^{2}. In other words, the injection produces the mapping of complexes from C_{HH(A)} to the first line of the bi-complex D_{HH(A^{2})} by the mapping A ∋ a ↦ 1 ⊗ a ∈ A ⊗ A on the level of each homogeneous component.

Likewise, the projection produces a mapping of complexes in the inverse order, which compose to the identity on C_{HH(A)}. Hence z_{n} is mapped to x_{n} for all n > 0 and z_{n}u is mapped to x_{n}v for all n > 0.

Finally, since p is odd, p^{n}m is odd if and only if m is odd. We proved the following

Proposition 4. Let k be a perfect field of characteristic p > 2. Then κ̂_{n}^{(m),k[ϵ]/ϵ^{2}} = κ_{n}^{(m),k[ϵ]/ϵ^{2}} for all n and m. Moreover, κ_{n}^{(m)}(z_{p^{n}m}) = z_{m} and κ_{n}^{(m)}(z_{p^{n}m−1}u) = 0.

3.2. The case p = 2. The case of even characteristic is completely analogous, with a few exceptions. The generator Z of HH^{∗}(A) is in degree 1, Z is not annihilated by ϵ, and the differentials for the homology and the cohomology complex are all 0. The Hochschild cohomology ring does only contain nilpotency coming from the centre. The rest is immediate.

Proposition 5. Let k be a perfect field of characteristic 2. Then κ̂_{n}^{(m),k[ϵ]/ϵ^{2}} = κ_{n}^{(m),k[ϵ]/ϵ^{2}} for all n and m. Moreover, κ_{n}^{(m)}(z_{p^{n}m}) = z_{m} for all m and n.

Proof. The proof is a straightforward analogue of the proof in the p > 2-case.

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

   E-mail address: alexander.zimmermann@u-picardie.fr
   URL: http://www.lamfa.u-picardie.fr/alex/azim.html