(Co)homology of Crossed Products by Weak Hopf Algebras

TL;DR

This paper introduces a simplified mixed complex for computing cyclic homologies of crossed products by weak Hopf algebras, generalizing existing spectral sequences and providing new computational tools.

Contribution

It develops a new, simpler mixed complex with filtrations and spectral sequences for cyclic homology of crossed products involving weak Hopf algebras, extending prior spectral sequence results.

Findings

A new mixed complex simplifies cyclic homology computations.

The spectral sequence generalizes previous results by CGG.

Another filtration leads to a spectral sequence extending Feigin-Tsygan.

Abstract

We obtain a mixed complex simpler than the canonical one the computes the type cyclic homologies of a crossed product with invertible cocycle $A\times_{\rho}^f H$, of a weak module algebra $A$ by a weak Hopf algebra $H$. This complex is provided with a filtration. The spectral sequence of this filtration generalizes the spectral sequence obtained in \cite{CGG}. When $f$ takes its values in a separable subalgebra of $A$ that satisfies suitable conditions, the above mentioned mixed complex is provided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectral sequence.

Equations

$$\lambda \cdot (v_1{\otimes }_K\cdots {\otimes }_K{v}_n)\cdot {\lambda }^{\prime }:=\lambda \cdot v_1{\otimes }_K\cdots {\otimes }_K{v}_n\cdot {\lambda }^{\prime }.$$

$${\Delta }^2(1)=1^{(1)}{\otimes }_k{1}^{(2)}{1}^{(1^{\prime })}{\otimes }_k{1}^{(2^{\prime })}=1^{(1)}{\otimes }_k{1}^{(1^{\prime })}{1}^{(2)}{\otimes }_k{1}^{(2^{\prime })}$$

$$ϵ(hlm)=ϵ(h{l}^{(1)})ϵ(l^{(2)}m)=ϵ(h{l}^{(2)})ϵ(l^{(1)}m)\text{for all h,l,m∈H,}$$

$${\Pi }^L(h):=ϵ(1^{(1)}h)1^{(2)},{\Pi }^R(h):=1^{(1)}ϵ(h{1}^{(2)}),{\overline{\Pi }}^L(h):=1^{(1)}ϵ(1^{(2)}h),{\overline{\Pi }}^R(h):=ϵ(h{1}^{(1)})1^{(2)},$$

$${\overline{\Pi }}^R(hl)={\overline{\Pi }}^R({\overline{\Pi }}^R(h)l),{\Pi }^R(hl)={\Pi }^R({\overline{\Pi }}^R(h)l)\text{and}{\overline{\Pi }}^R(hm)={\overline{\Pi }}^R(h)m.$$

$$h^{(1)}S(h^{(2)})={\Pi }^L(h),S(h^{(1)})h^{(2)}={\Pi }^R(h)\text{and}S(h^{(1)})h^{(2)}S(h^{(3)})=S(h),$$

$${\Pi }^L=S\circ {\overline{\Pi }}^L\text{and}{\Pi }^R=S\circ {\overline{\Pi }}^R.$$

$$N$$

$$\displaystyle d_{s}\bigl{(}\overline{\mathbf{h}}_{1s}\otimes_{H^{\!L}}n\bigr{)}$$

$$+\sum _{i=1}^{s-1}(-1{)}^i{\overline{\mathbf{h}}}_{1,i-1}{\otimes }_{H^L}\overline{h_i{h}_{i+1}}{\otimes }_{H^L}{\overline{\mathbf{h}}}_{i+2,s}{\otimes }_{H^L}n+(-1{)}^s{\overline{\mathbf{h}}}_{1,s-1}{\otimes }_{H^L}{h}_s\cdot n.$$

$$N$$

$$\qquad\quad d_{s}(\beta)(\overline{\mathbf{h}}_{1s})\coloneqq\beta\bigl{(}\overline{\Pi}^{R}(h_{1})\cdot\overline{\mathbf{h}}_{2s}\bigr{)}+\sum_{i=1}^{s-1}(-1)^{i}\beta\bigl{(}\overline{\mathbf{h}}_{1,i-1}\otimes_{H^{\!L}}\overline{h_{i}h_{i+1}}\otimes_{H^{\!L}}\overline{\mathbf{h}}_{i+2,s}\bigr{)}+(-1)^{s}\beta\bigl{(}\overline{\mathbf{h}}_{1,s-1}\bigr{)}\cdot h_{s}.$$

$$d_{si}^{\prime \prime }(\mathbf{x}):=\{\begin{array}{ll}{\overline{\Pi }}^R(h_1)h_2{\otimes }_k{h}_3{\otimes }_k\otimes \cdots {\otimes }_k{h}_{s+1}& \text{if i=0,}\\ h_1{\otimes }_k\cdots {\otimes }_k{h}_{i-1}{\otimes }_k{h}_i{h}_{i+1}{\otimes }_k{h}_{i+2}{\otimes }_k\cdots {\otimes }_k{h}_{s+1}& \text{if 1≤i≤s,}\end{array}$$

$$d_{si}^{\prime }(h_1{\otimes }_k\cdots {\otimes }_k{h}_{j}l{\otimes }_k{h}_{j+1}{\otimes }_k\cdots {\otimes }_k{h}_{s+1})=d_{si}^{\prime }(h_1{\otimes }_k\cdots {\otimes }_k{h}_j{\otimes }_{k}l{h}_{j+1}{\otimes }_k\cdots {\otimes }_k{h}_{s+1}).$$

$$H^R$$

$$d_{s-1,i}^{\prime \prime }\circ d_{sj}^{\prime \prime }=d_{s-1,j-1}^{\prime \prime }\circ d_{si}^{\prime \prime }$$

$$d_{s-1,0}^{\prime \prime }\circ d_{sj}^{\prime \prime }=d_{s-1,j-1}^{\prime \prime }\circ d_{s0}^{\prime \prime }$$

$$d^{\prime}_{s-1}\circ d^{\prime}_{s}\bigl{(}\overline{\mathbf{h}}_{1s}\otimes_{H^{\!L}}h_{s+1}\bigr{)}=\overline{\overline{\Pi}^{R}\bigl{(}\overline{\Pi}^{R}(h_{1})h_{2}\bigr{)}h_{3}}\otimes_{H^{\!L}}\overline{\mathbf{h}}_{4s}\otimes_{H^{\!L}}h_{s+1}-\overline{\overline{\Pi}^{R}(h_{1}h_{2})h_{3}}\otimes_{H^{\!L}}\overline{\mathbf{h}}_{4s}\otimes_{H^{\!L}}h_{s+1}=0,$$

$${\hslash }_0:H^R⟶H,{\hslash }_{s+1}:{\overline{H}}^{{\otimes }_{H^L}^s}{\otimes }_{H^L}H⟶{\overline{H}}^{{\otimes }_{H^L}^{s+1}}{\otimes }_{H^L}H\text{(s≥0),}$$

$${\Pi }^R\circ {\hslash }_0(h)=h,{\hslash }_0\circ {\Pi }^R(h)={\Pi }^R(h),d_1^{\prime }\circ {\hslash }_1(h)=-{\overline{\Pi }}^R(h)+h,$$

$${\hslash }_s\circ d_s^{\prime }({\overline{\mathbf{h}}}_{1s}{\otimes }_{H^L}{h}_{s+1})=(-1{)}^s\overline{{\overline{\Pi }}^R(h_1)h_2}{\otimes }_{H^L}{\overline{\mathbf{h}}}_{3,s+1}{\otimes }_{H^L}1$$

$$+\sum _{i=1}^s(-1{)}^{s+i}{\overline{\mathbf{h}}}_{1,i-1}{\otimes }_{H^L}\overline{h_i{h}_{i+1}}{\otimes }_{H^L}{\overline{\mathbf{h}}}_{i+2,s+1}{\otimes }_{H^L}1$$

$$d_{s+1}^{\prime }\circ {\hslash }_{s+1}({\overline{\mathbf{h}}}_{1s}{\otimes }_{H^L}{h}_{s+1})=(-1{)}^{s+1}\overline{{\overline{\Pi }}^R(h_1)h_2}{\otimes }_{H^L}{\overline{\mathbf{h}}}_{3,s+1}{\otimes }_{H^L}1$$

$$+\sum _{i=1}^s(-1{)}^{s+i+1}{\overline{\mathbf{h}}}_{1,i-1}{\otimes }_{H^L}\overline{h_i{h}_{i+1}}{\otimes }_{H^L}{\overline{\mathbf{h}}}_{i+2,s+1}{\otimes }_{H^L}1$$

$$+{\overline{\mathbf{h}}}_{1s}{\otimes }_{H^L}{h}_{s+1}.$$

$$h\cdot (a{a}^{\prime })=(h^{(1)}\cdot a)(h^{(2)}\cdot a^{\prime })\text{for all h∈H and a,a′∈A.}$$

$$a^{\prime }\cdot (a{\otimes }_{k}h):=a^{\prime }a{\otimes }_{k}h\text{and}(a{\otimes }_{k}h)\cdot a^{\prime }:=a(h^{(1)}\cdot a^{\prime }){\otimes }_k{h}^{(2)},$$

$$f(h{\otimes }_{k}l)=f(h^{(1)}{\otimes }_k{l}^{(1)})(h^{(2)}{l}^{(2)}\cdot 1_A)\text{for all h,l∈H.}$$

$$f(h^{(1)}{\otimes }_k{l}^{(1)})(h^{(2)}{l}^{(2)}\cdot a)=(h^{(1)}\cdot (l^{(1)}\cdot a))f(h^{(2)}{\otimes }_k{l}^{(2)})\text{for all h,l∈H and a∈A,}$$

$$f(h^{(1)}{\otimes }_k{l}^{(1)})f(h^{(2)}{l}^{(2)}{\otimes }_{k}m)=(h^{(1)}\cdot f(l^{(1)}{\otimes }_k{m}^{(1)}))f(h^{(2)}{\otimes }_k{l}^{(2)}{m}^{(2)})\text{for all h,l,m∈H.}$$

$${ȷ}_{\nu }(a)\gamma (h)=a\cdot {\nabla }_{\rho }(1_A{\otimes }_{k}h)={\nabla }_{\rho }(a{\otimes }_{k}h)=a\times h.$$

$$\gamma(h)\jmath_{\nu}(a)=\jmath_{\nu}(h^{(1)}\cdot a)\gamma(h^{(2)})\quad\text{and}\quad\gamma(h)\gamma(l)=\jmath_{\nu}\bigl{(}f(h^{(1)}\otimes_{k}l^{(1)})\bigr{)}\gamma(h^{(2)}l^{(2)}).$$

$$\delta_{E}\Bigl{(}\sum a_{i}\otimes_{k}h_{i}\Bigr{)}\coloneqq\sum\nabla_{\!\rho}\bigl{(}a_{i}\otimes_{k}h_{i}^{(1)}\bigr{)}\otimes_{k}h_{i}^{(2)}=\sum(a_{i}\times h_{i}^{(1)})\otimes_{k}h_{i}^{(2)}.$$

$$h\cdot (l\cdot a)=(hl)\cdot a\text{for all h,l∈H and all a∈A},$$

Full text

Jorge A. Guccione and Juan J. Guccione were supported by CONICET PIP 2021-2023 GI,11220200100423CO and CONCYTEC-FONDECYT within the framework of the contest “Proyectos de Investigación Básica 2020-01” [contract number 120-2020-FONDECYT]

Christian Valqui was supported by CONCYTEC-FONDECYT within the framework of the contest “Proyectos de Investigación Básica 2020-01” [contract number 120-2020-FONDECYT]

1. 1 Preliminaries

2. 1.1 Weak Hopf algebras

3. 1.2 (Co)homology of weak Hopf algebras

4. 1.3 Crossed products by weak Hopf algebras

5. 1.4 Weak crossed products of weak module algebras

6. 1.5 Mixed complexes

7. 2 Hochschild homology of cleft extensions

8. 2.1 Technical results

9. 2.2 Main results

10. 3 Hochschild cohomology of cleft extensions

11. 4 The cup and cap products for cleft extensions

12. 5 Cyclic homology of cleft extensions

Introduction

Given a differential or algebraic manifold $M$, each group $G$ acting on $M$ acts in a natural way on the ring $A$ of regular functions of $M$, and the algebra ${}^{G\!}A$ of invariants of this action consists of the functions that are constants on each of the orbits of $M$. This suggest to consider ${}^{G\!}A$ as a replacement for $M/G$ in noncommutative geometry. Under suitable conditions the invariant algebra ${}^{G\!}A$ and the smash product $A\#k[G]$, associated with the action of $G$ on $A$, are Morita equivalent. Since $K$-theory, Hochschild homology and cyclic homology are Morita invariant, there is no loss of information if ${}^{G\!}A$ is replaced by $A\#k[G]$. In the general case the experience has shown that smash products are better choices than invariant rings for algebras playing the role of noncommutative quotients. In fact, except when the invariant algebra and the smash product are Morita equivalent, the first one never is considered in noncommutative geometry. The problem of developing tools to compute the cyclic homology of smash products algebras $A\#k[G]$, where $A$ is an algebra and $G$ is a group, was considered in [FT, GJ, N]. For instance, in the first paper the authors obtained a spectral sequence converging to the cyclic homology of $A\#k[G]$, and in [GJ] this result was derived from the theory of paracyclic modules and cylindrical modules developed by the authors. The main tool for this computation is a version for cylindrical modules of the Eilenberg-Zilber theorem. More recently, and also due to its connections with noncommutative geometry, the cyclic homology of algebras obtained through more general constructions involving Ore extensions, twisted tensor products or Hopf algebras (Hopf crossed products, Hopf Galois extensions, Braided Hopf crossed products, etcetera) has been extensively studied. See for instance [AK, CGG, CGG1, CGG2, CGGV, FGG, GG2, GG3, JS, KR, KS, R, V, W, ZH]. Weak Hopf algebras (also called quantum groupoids) are an important generalization of Hopf algebras in which the counit is not required to be an algebra homomorphism and the unit is not required to be a coalgebra homomorphism, these properties being replaced by weaker axioms. Examples of weak Hopf algebras are groupoid algebras and their duals, face algebras [Ha], quantum groupoids constructed from subfactors [NV] and generalized Kac algebras of Yamanouchi [Y]. It is natural to try to extend the results of [FT] to noncommutative quotients $A\#H$ of algebras $A$ by actions of weak Hopf algebras $H$. More generally, in this paper we use the results obtained in [GGV2] to study the Hochschild (co)homology and the cyclic type homologies of weak crossed products with invertible cocycle. Specifically, for a unitary crossed product $E\coloneqq A\times_{\rho}^{f}H$, of an algebra $A$ by a weak Hopf algebra $H$ and a subalgebra $K$ of $A$ satisfying suitable conditions, we construct a chain complex, a cochain complex and a mixed complex, simpler than the canonical ones (and also simpler that the ones constructed in [GGV2]), that compute the Hochschild (co)homology of $E$ with coefficients in an $E$-bimodule $M$, relative to $K$, and the cyclic, negative and periodic homologies of $E$, relative to $K$ (see [GS]). It is well known that when $K$ is separable, then these relative groups coincide with the absolute groups. These complexes are endowed with canonical filtrations whose spectral sequences generalize the Hochschild-Serre spectral sequence and the Feigin and Tsygan spectral sequence. By example, applying these results, we obtain that when $A=K$, the Hochschild homology of $E$ with coefficients in $M$, relative to $K$, coincide with the homology of $H$ with coefficients in the group $M/[M,K]$, considered as a left $H$-module via conjugation. A similar result is obtained for the Hochschild cohomology (see Subsection 1.2 and Examples 2.15 and 3.9).

The paper is organized as follows:

In Section 1 we review the notions of weak Hopf algebra and of crossed products of algebras by weak Hopf algebras, and we recall the concept of mixed complex and the perturbation lemma. In sections 2 and 3 we obtain complexes that compute the Hochschild homology and the Hochschild cohomology of a weak crossed product with invertible cocycle $E\coloneqq A\times_{\rho}^{f}H$ with coefficients in an $E$-bimodule $M$. Then, in Section 4 we study the cup and the cap products of $E$, and in Section 5 we obtain a mixed complex that computes the cyclic type homologies of $E$.

Remark. In this paper we consider the notion of weak crossed products introduced in [AG], but this is not the unique concept of weak crossed product of algebras by weak Hopf algebras in the literature. There is a notion of crossed product of an algebra $A$ with a Hopf algebroid introduced by Böhm and Brzeziński in [BB]. It is well known that weak Hopf algebras $H$ provide examples of Hopf algebroids. The crossed products considered by us in this paper are canonically isomorphic to the Böhm-Brzeziński crossed products $A\#_{f}H$ with invertible cocycle whose action satisfies $h\cdot(l\cdot 1_{A})=hl\cdot 1_{A}$ for all $h\in H$ and $l\in H^{L}$ (see [GG1]). So, our results also apply to these algebras.

We thank the referee for a careful reading of the paper and for his indications that helped improve the writing. We also thank the referee for pointing out the paper [KS], in which the Hochschild homology of the twisted tensor product of algebras is studied, obtaining complete calculations for several examples. There is some intersection between the results of [KS] and some results obtained in [GG2] and [GG3]. More precisely, [KS]*Proposition 2.1 is [GG2]*Theorem 1.7 and the results in [KS]*Subsections 3.3.1 to 3.3.3 are the cases $r=n=2$, $r=1,n=2$ and $r=0,n=2$ of [GG3]*Proposition 1.9.

1   Preliminaries

In this article we work in the category of vector spaces over a field $k$. Hence we assume implicitly that all the maps are $k$-linear maps. The tensor product over $k$ is denoted by $\otimes_{k}$. Given an arbitrary algebra $K$, a $K$-bimodule $V$ and an $n\geq 0$, we let $V^{\otimes_{K}^{n}}$ denote the $n$-fold tensor product $V\otimes_{K}\cdots\otimes_{K}V$, which is considered as a $K$-bimodule via

[TABLE]

Given $k$-vector spaces $U$, $V$, $W$ and a map $g\colon V\to W$ we write $U\otimes_{k}g$ for $\operatorname{id}_{U}\otimes_{k}g$ and $g\otimes_{k}U$ for $g\otimes_{k}\operatorname{id}_{U}$. We assume that the reader is familiar with the notions of weak Hopf algebra introduced in [BNS1, BNS2] and of weak crossed products introduced in [AG] and studied in a series of papers (see for instance [AFGR1, AFGR2, AFGR3, FGR, GGV1, Ra]). We are specifically interested in the case in which $A$ is a weak module algebra and the cocycle of the crossed product is convolution invertible (see [GGV1]*Sections 4–6).

1.1   Weak Hopf algebras

Weak bialgebras and weak Hopf algebras are generalizations of bialgebras and Hopf algebras, introduced in [BNS1, BNS2], in which the axioms about the unit, the counit and the antipode are replaced by weaker properties. Next we give a brief review of the basic properties of these structures.

Definition 1.1**.**

Let $k$ be a field. A weak bialgebra is a $k$-vector space $H$, endowed with an algebra structure and a coalgebra structure, such that $\Delta(hl)=\Delta(h)\Delta(l)$ for all $h,l\in H$, and the equalities

[TABLE]

are fulfilled, where we are using the Sweedler notation for the coproduct, with the summation symbol omitted. A weak bialgebra morphism is a function $g\colon H\to L$ that is an algebra and a coalgebra map.

In the rest of this subsection we assume that $H$ is a weak bialgebra.

It is well known that the maps $\Pi^{L},\Pi^{R},\overline{\Pi}^{L},\overline{\Pi}^{R}\in\operatorname{End}_{k}(H)$,

defined by

[TABLE]

respectively, are idempotent (for a proof see [BNS1, CDG]). We set $H^{\!L}\coloneqq\operatorname{Im}(\Pi^{L})$ and $H^{\!R}\coloneqq\operatorname{Im}(\Pi^{R})$. In [CDG] it was also proven that $\operatorname{Im}(\overline{\Pi}^{R})=H^{\!L}$ and $\operatorname{Im}(\overline{\Pi}^{L})=H^{\!R}$.

Proposition 1.2**.**

For all $h,l\in H$ and $m\in H^{\!L}$, we have

[TABLE]

Proof.

Left to the reader. ∎

An antipode of $H$ is a map $S\colon H\to H$ (or $S_{H}$ if necessary), such that

[TABLE]

for all $h\in H$. As it was shown in [BNS1], if an antipode $S$ exists, then it is unique. It was also shown in [BNS1] that $S$ is antimultiplicative, anticomultiplicative and leaves the unit and counit invariant. A weak Hopf algebra is a weak bialgebra that has an antipode. By [BNS1]*equalities (2.24a) and (2.24b),

[TABLE]

A morphism of weak Hopf algebras $g\colon H\to L$ is simply a bialgebra morphism from $H$ to $L$. In [AFGLV]*Proposition 1.4 it was proven that if $g\colon H\to L$ is a weak Hopf algebra morphism, then $g\circ S_{H}=S_{L}\circ g$.

1.2   (Co)homology of weak Hopf algebras

Consider $H^{\!R}$ as a right $H$-module via $l\cdot h\coloneqq\Pi^{R}(lh)$ (by [BNS1]*equality (2.5b) this is an action and the map $\Pi^{R}\colon H\to H^{R}$ is a morphism of right $H$-modules). By definition, the homology of $H$ with coefficients in a left $H$-module $N$ is $\operatorname{H}_{*}(H,N)\coloneqq\operatorname{Tor}^{H}_{*}(H^{\!R},N)$, while the cohomology of $H$ with coefficients in a right $H$-module $N$ is $\operatorname{H}^{*}(H,N)\coloneqq\operatorname{Ext}_{H}^{*}(H^{\!R},N)$.

Notations 1.3**.**

We will use the following notations:

1. (1)

We set $\overline{H}\coloneqq H/H^{\!L}$. Moreover, given $h\in H$ we let $\overline{h}$ denote its class in $\overline{H}$. 2. (2)

Given $h_{1},\dots,h_{s}\in H$ we set $\overline{\mathbf{h}}_{1s}\coloneqq\overline{h_{1}}\otimes_{H^{\!L}}\cdots\otimes_{H^{\!L}}\overline{h_{s}}$.

Clearly $\overline{H}^{\otimes_{\!H^{\!L}}^{s}}$ is an $H^{\!L}$-bimodule via $l\cdot\overline{\mathbf{h}}_{1s}\cdot l^{\prime}\coloneqq\overline{lh_{1}}\otimes_{\!H^{\!L}}\overline{\mathbf{h}}_{2,s-1}\otimes_{\!H^{\!L}}\overline{h_{s}l}$.

Proposition 1.4**.**

The following facts hold:

1. (1)

The homology of $H$ with coefficients in a left $H$-module $N$ is the homology of the chain complex

[TABLE]

where d_{1}\bigl{(}h_{1}\otimes_{H^{\!L}}n\bigr{)}\coloneqq\overline{\Pi}^{R}(h_{1})\cdot n-h_{1}\cdot n and, for $s>1$,

[TABLE] 2. (2)

The cohomology of $H$ with coefficients in a right $H$-module $N$ is the cohomology of the cochain complex

[TABLE]

where $d^{1}(n)(h_{1})\coloneqq n\cdot\overline{\Pi}^{R}(h_{1})-n\cdot h_{1}$ and, for $s>1$,

[TABLE]

Proof.

For $s\in\mathds{N}$ and $0\leq i\leq s$, let $d^{\prime\prime}_{si}\colon H^{\otimes_{k}^{s+1}}\to H^{\otimes_{k}^{s+1}}$ be the map defined by

[TABLE]

where $\mathbf{x}\coloneqq h_{1}\otimes_{k}\cdots\otimes_{k}h_{s+1}$. For each $s>1$ and $0\leq i\leq s$, we let $d^{\prime}_{si}\colon H^{\otimes_{k}^{s+1}}\to\overline{H}^{\otimes_{\!H^{\!L}}^{s-1}}\otimes_{H^{\!L}}H$ denote the morphism induced by $d^{\prime\prime}_{si}$. Moreover we set $d^{\prime}_{10}=d^{\prime\prime}_{10}$ and $d^{\prime}_{11}=d^{\prime\prime}_{11}$. We claim that

1. (1)

If $s=1$ and $h_{1}\in H^{\!L}$, then $d^{\prime}_{10}(\mathbf{x})=d^{\prime}_{11}(\mathbf{x})$, 2. (2)

If $s>1$ and $h_{1}\in H^{\!L}$, then $d^{\prime}_{s0}(\mathbf{x})=d^{\prime}_{s1}(\mathbf{x})$ and $d^{\prime}_{si}(\mathbf{x})=0$ for $1<i$, 3. (3)

If $s>1$ and $h_{j}\in H^{\!L}$ for some $j>1$, then $d^{\prime}_{s,j-1}(\mathbf{x})=d^{\prime}_{sj}(\mathbf{x})$ and $d^{\prime}_{si}(\mathbf{x})=0$ for $i\notin\{j-1,j\}$, 4. (4)

$d^{\prime}_{si}(h_{1}l\otimes_{k}h_{2}\otimes_{k}\cdots\otimes_{k}h_{s+1})=d^{\prime}_{si}(h_{1}\otimes_{k}lh_{2}\otimes_{k}\cdots\otimes_{k}h_{s+1})$ for $0\leq i\leq s$ and $l\in H^{\!L}$, 5. (5)

For $0\leq i\leq s$, $1\leq j\leq s$ and $l\in H^{\!L}$, we have

[TABLE]

Items (1) and the first assertion in item (2) hold since $\overline{\Pi}^{R}(h)=h$, for all $h\in H^{\!L}$; while the second assertion in item (2) is clear. We next prove item (3). The fact that $d^{\prime}_{sj}(\mathbf{x})=0$, for $i>0$ and $i\notin\{j-1,j\}$, is trivial; while the equality $d^{\prime}_{s,j-1}(\mathbf{x})=d^{\prime}_{sj}(\mathbf{x})$ is straightforward. Finally if $j=2$, then $d^{\prime}_{s0}(\mathbf{x})=0$, since $\overline{\Pi}^{R}(h_{1})h_{2}\in H^{\!L}$. The case $i=0$ in item (4) holds, since $\overline{\Pi}^{R}(h_{1}l)h_{2}=\overline{\Pi}^{R}(h_{1})lh_{2}$, by the third identity in Proposition 1.2; while the cases $i>0$ are straightforward. It remains to check item (4). Equality (1.4) for $i=0$ and $j=1$ holds, since $\overline{\Pi}^{R}(h_{1}l)h_{2}=\overline{\Pi}^{R}(h_{1})lh_{2}$, by the third identity in Proposition 1.2; while the other cases are trivial. This ends the proof of the claim. Consequently, $\sum_{i=0}^{s}(-1)^{1}d^{\prime}_{si}$ induces a morphism $d^{\prime}_{s}\colon\overline{H}^{\otimes_{\!H^{\!L}}^{s}}\otimes_{H^{\!L}}H\to\overline{H}^{\otimes_{\!H^{\!L}}^{s-1}}\otimes_{H^{\!L}}H$, for each $s\geq 1$. Consider the diagram

[TABLE]

where $\overline{H}^{\otimes_{\!H^{\!L}}^{s}}\otimes_{H^{\!L}}H$ is a right $H$-module via the canonical action. Clearly the $d^{\prime}_{s}$’s are right $H$-linear maps. Next we will prove that (1.5) is a chain complex. Assume that $s>2$. Since

[TABLE]

the composition $d^{\prime}_{s-1}\circ d^{\prime}_{s}$ is the map induced by $d^{\prime\prime}_{s-1,0}\circ d^{\prime\prime}_{s0}-d^{\prime\prime}_{s-1,0}\circ d^{\prime\prime}_{s1}$. Thus,

[TABLE]

where the last equality follows from the first identity in Proposition 1.2. In order to finish the proof that (1.5) is a chain complex, we must check that $\Pi^{R}\circ d^{\prime}_{1}=0$ and $d^{\prime}_{1}\circ d^{\prime}_{2}=0$. But, this follows from the first two identities in Proposition 1.2. We claim that the family of morphisms

[TABLE]

given by $\hbar_{0}(h)\coloneqq h$ and $\hbar_{s+1}(\overline{\mathbf{h}}_{1s}\otimes_{H^{\!L}}h_{s+1})\coloneqq(-1)^{s+1}\overline{\mathbf{h}}_{1,s+1}\otimes_{H^{\!L}}1$ for $s\geq 0$, is a contracting homotopy of (1.5) as a complex of right $H^{\!L}$-modules. In fact, this follows immediately from the equalities

[TABLE]

Note that, since $H^{\!L}$ is separable, the right $H^{\!L}$-modules $\overline{H}^{\otimes_{\!H^{\!L}}^{s}}$ are projective . Hence, $\overline{H}^{\otimes_{\!H^{\!L}}^{s}}\otimes_{H^{\!L}}H$ is a right projective $H$-module for each $s\geq 0$ and, consequently, (1.5) is a projective resolution of $H^{\!R}$ as a right $H$-module. Now items (1) and (2) follows immediately from this and the fact that \bigl{(}\overline{H}^{\otimes_{\!H^{\!L}}^{s}}\otimes_{H^{\!L}}H\bigr{)}\otimes_{H}N\simeq\overline{H}^{\otimes_{\!H^{\!L}}^{s}}\otimes_{H^{\!L}}N and \operatorname{Hom}_{H}\bigl{(}\overline{H}^{\otimes_{\!H^{\!L}}^{s}}\otimes_{H^{\!L}}H,N\bigr{)}\simeq\operatorname{Hom}_{\!H^{\!L}}\bigl{(}\overline{H}^{\otimes_{\!H^{\!L}}^{s}},N\bigr{)}. ∎

1.3   Crossed products by weak Hopf algebras

Let $H$ be a weak-Hopf algebra, $A$ an algebra and $\rho\colon H\otimes_{k}A\to A$ a linear map. For $h\in H$ and $a\in A$, we set $h\cdot a\coloneqq\rho(h\otimes_{k}a)$. We say that $\rho$ is a weak measure of $H$ on $A$ if

[TABLE]

From now on $\rho$ denotes a weak measure of $H$ on $A$. Let $\chi_{\rho}\colon H\otimes_{k}A\longrightarrow A\otimes_{k}H$ be the map defined by $\chi_{\rho}(h\otimes_{k}a)\coloneqq h^{(1)}\cdot a\otimes_{k}h^{(2)}$. By equality (1.6) the triple $(A,H,\chi_{\rho})$ is a twisted space (see [GGV1]*Definition 1.6). By [GGV1]*Subsection 1.2 we know that $A\otimes_{k}H$ is a non unitary $A$-bimodule via

[TABLE]

and that the map $\nabla_{\!\rho}\colon A\otimes_{k}H\longrightarrow A\otimes_{k}H$, defined by $\nabla_{\!\rho}(a\otimes_{k}h)\coloneqq a\cdot\chi_{\rho}(h\otimes_{k}1_{A})$, is a left and right $A$-linear idempotent. In the sequel we will write $a\times h\coloneqq\nabla_{\!\rho}(a\otimes_{k}h)$. It is easy to check that the $A$-subbimodule $A\times H\coloneqq\nabla_{\!\rho}(A\otimes_{k}H)$ of $A\otimes_{k}H$ is unitary. Let $\gamma\colon H\to A\times H$, $\nu\colon k\to A\otimes_{k}H$ and $\jmath_{\nu}\colon A\to A\times H$, be the maps defined by $\gamma(h)\coloneqq 1_{A}\times h$, $\nu(\lambda)\coloneqq\lambda 1_{A}\times 1$ and $\jmath_{\nu}(a)\coloneqq a\times 1$, respectively. Given a morphism $f\colon H\otimes_{k}H\to A$, we define $\mathcal{F}_{f}\colon H\otimes_{k}H\longrightarrow A\otimes_{k}H$ by $\mathcal{F}_{f}(h\otimes_{k}l)\coloneqq f(h^{(1)}\otimes_{k}l^{(1)})\otimes_{k}h^{(2)}l^{(2)}$. Assume that

[TABLE]

By [GGV1]*Proposition 2.4 we know that $(A,H,\chi_{\rho},\mathcal{F}_{f})$ is a crossed product system (see [GGV1]*Definition 1.7). We say that $f$ satisfis the twisted module condition if

[TABLE]

and that $f$ is a cocycle if

[TABLE]

Let $E$ be $A\times H$ endowed with the multiplication map $\mu_{E}$ introduced in [GGV1]*Notation 1.9. A direct computation using the twisted module condition shows that $(a\times h)(b\times l)=a(h^{(1)}\cdot b)f(h^{(2)}\otimes_{k}l^{(1)})\times h^{(3)}l^{(2)}$.

Theorem 1.5**.**

Assume that

1. (1)

$f(h\otimes_{k}l)=f(h^{(1)}\otimes_{k}l^{(1)})(h^{(2)}l^{(2)}\cdot 1_{A})$* for all $h,l\in H$,* 2. (2)

$h\cdot 1_{A}=(h^{(1)}\cdot(1^{(1)}\cdot 1_{A}))f(h^{(2)}\otimes_{k}1^{(2)})$* for all $h\in H$,* 3. (3)

$h\cdot 1_{A}=(1^{(1)}\cdot 1_{A})f(1^{(2)}\otimes_{k}h)$* for all $h\in H$,* 4. (4)

$a\times 1=1^{(1)}\cdot a\otimes_{k}1^{(2)}$* for all $a\in A$,* 5. (5)

$f$* is a cocycle that satisfies the twisted module condition.*

Then,

1. (a)

$(A,H,\chi_{\rho},\mathcal{F}_{f},\nu)$* is a crossed product system with preunit (see **[GGV1]**Definition 1.11), 2. (b)

$\mathcal{F}_{f}$* is a cocycle that satisfies the twisted module condition (see **[GGV1]**Definitions 1.8), 3. (c)

$\mu_{E}$* is left and right $A$-linear, associative and has unit $1_{A}\times 1$,* 4. (d)

The morphism $\jmath_{\nu}\colon A\to E$ is left and right $A$-linear, multiplicative and unitary, 5. (e)

$\jmath_{\nu}(a)\mathbf{x}=a\cdot\mathbf{x}$* and $\mathbf{x}\jmath_{\nu}(a)=\mathbf{x}\cdot a$, for all $a\in A$ and $\mathbf{x}\in E$,* 6. (f)

$\chi_{\rho}(h\otimes_{k}a)=\gamma(h)\jmath_{\nu}(a)$* and $\mathcal{F}_{f}(h\otimes_{k}l)=\gamma(h)\gamma(l)$, for all $h,l\in H$ and $a\in A$.*

Proof.

By [GGV1]*Propositions 2.4, 2.6 and 2.10, and Theorems 1.12(7) and 2.11. ∎

In the rest of this subsection we assume that the hypotheses of Theorem 1.5 are fulfilled, and we say that $E$ is the unitary crossed product of $A$ with $H$ associated with $\rho$ and $f$. Note that by item (10) of that theorem we have

[TABLE]

Moreover, by [GGV2]*Equalities (1.6) and (1.7) and the definitions of $\chi_{\rho}$ and $\mathcal{F}_{f}$, we have

[TABLE]

1.3.1   The weak comodule structure of $\bm{E}$

Let $B$ be a right $H$-comodule. The tensor product $B\otimes_{k}B$ is a (not necessarily counitary) right $H$-comodule, via $\delta_{B\otimes_{k}B}(b\otimes_{k}c)\coloneqq b^{(0)}\otimes_{k}c^{(0)}\otimes_{k}b^{(1)}c^{(1)}$.

Definition 1.6**.**

A unitary associative algebra $B$, which is also a counitary right $H$-comodule, is a right $H$-comodule algebra if $\mu_{B}$ is right $H$-colinear and one of the following equivalent conditions is satisfied:

1. (1)

$1_{B}^{(0)}\otimes_{k}1_{B}^{(1)}\otimes_{k}1_{B}^{(2)}=1_{B}^{(0)}\otimes_{k}1_{B}^{(1)}1^{(1)}\otimes_{k}1^{(2)}$. 2. (2)

$1_{B}^{(0)}\otimes_{k}1_{B}^{(1)}\otimes_{k}1_{B}^{(2)}=1_{B}^{(0)}\otimes_{k}1^{(1)}1_{B}^{(1)}\otimes_{k}1^{(2)}$. 3. (3)

$b^{(0)}\otimes_{k}\overline{\Pi}^{R}(b^{(1)})=b1_{B}^{(0)}\otimes_{k}1_{B}^{(1)}$ for all $b\in B$. 4. (4)

$b^{(0)}\otimes_{k}\Pi^{L}(b^{(1)})=1_{B}^{(0)}b\otimes_{k}1_{B}^{(1)}$ for all $b\in B$. 5. (5)

$1_{B}^{(0)}\otimes_{k}\overline{\Pi}^{R}(1_{B}^{(1)})=1_{B}^{(0)}\otimes_{k}1_{B}^{(1)}$. 6. (6)

$1_{B}^{(0)}\otimes_{k}\Pi^{L}(1_{B}^{(1)})=1_{B}^{(0)}\otimes_{k}1_{B}^{(1)}$.

Proposition 1.7** (Comodule algebra structure on $E$).**

Assume that the hypotheses of Theorem 1.5 are satisfied. Then $E$ is a weak $H$-comodule algebra via the map $\delta_{E}\colon E\to E\otimes_{k}H$ defined by

[TABLE]

Proof.

See [GGV1]*Proposition 2.27. ∎

1.4   Weak crossed products of weak module algebras

Let $H$ be a weak bialgebra, $A$ a $k$-algebra and $\rho\colon H\otimes_{k}A\to A$ a map. In this subsection we study the weak crossed products of $A$ with $H$ in which $A$ is a left weak $H$-module algebra.

Definition 1.8**.**

We say that $A$ is a left weak $H$-module algebra via $\rho$, if

1. (1)

$1\cdot a=a$ for all $a\in A$, 2. (2)

$h\cdot(aa^{\prime})=(h^{(1)}\cdot a)(h^{(2)}\cdot a^{\prime})$ for all $h\in H$ and $a,a^{\prime}\in A$, 3. (3)

$h\cdot(l\cdot 1_{A})=(hl)\cdot 1_{A}$ for all $h,l\in H$ and $a\in A$,

In this case we say that $\rho$ is a weak left action of $H$ on $A$. If we also have

[TABLE]

then we say that $A$ is a left $H$-module algebra.

Proposition 1.9**.**

For each weak $H$-module algebra $A$ the following assertions hold:

1. (1)

$\Pi^{L}(h)\cdot a=(h\cdot 1_{A})a$* for all $h\in H$ and $a\in A$.* 2. (2)

$\overline{\Pi}^{L}(h)\cdot a=a(h\cdot 1_{A})$* for all $h\in H$ and $a\in A$.* 3. (3)

$\Pi^{L}(h)\cdot 1_{A}=h\cdot 1_{A}$* for all $h\in H$.* 4. (4)

$\overline{\Pi}^{L}(h)\cdot 1_{A}=h\cdot 1_{A}$* for all $h\in H$.* 5. (5)

$h\cdot(l\cdot 1_{A})=(h^{(1)}\cdot 1_{A})\epsilon(h^{(2)}l)$* for all $h,l\in H$.* 6. (6)

$h\cdot(l\cdot 1_{A})=(h^{(2)}\cdot 1_{A})\epsilon(h^{(1)}l)$* for all $h,l\in H$.*

Proof.

In [CDG] it was proven that these items are equivalent and the proof of item (3) it was kindly communicated to us by José Nicanor Alonso Álvarez and Ramón González Rodríguez (see [GGV1]*Proposition 4.2). ∎

Example 1.10**.**

The algebra $H^{\!L}$ is a left $H$-module algebra via $h\cdot l\coloneqq\Pi^{L}(hl)$. In fact item (1) of Definition 1.8 is trivial. By [BNS1]*equality (2.7a) we have

[TABLE]

Using this, equality [BNS1]*equality (2.5a) and the fact that $\Pi^{L}*\operatorname{id}=\operatorname{id}$, we obtain

[TABLE]

$h\in H$ and $l,m\in H^{\!L}$. This proves item (2). Finally equality (1.9) follows from [BNS1]*equality (2.5a). This example is known as the trivial representation.

Example 1.11**.**

Let $A$ be a left $H$-module algebra. The map $f(h\otimes_{k}l)\coloneqq hl\cdot 1_{A}$, named the trivial cocycle of $A$, satisfies the hypotheses of Theorem 1.5.

Definition 1.12**.**

A subalgebra $K$ of $A$ is stable under $\rho$ if $h\cdot\lambda\in K$ for all $h\in H$ and $\lambda\in K$.

Remark 1.13*.*

A subalgebra $K$ of $A$ is stable under $\rho$ if and only if $\chi_{\rho}(H\otimes_{k}K)\subseteq K\otimes_{k}H$.

Example 1.14**.**

Let $K\coloneqq\{h\cdot 1_{A}:h\in H\}$. By Definition 1.8 and Proposition 1.9, we know that $K$ is a subalgebra of $A$, which is stable under $\rho$, and $K=\{h\cdot 1_{A}:h\in H^{\!L}\}=\{h\cdot 1_{A}:h\in H^{\!R}\}$. Moreover, the map $\pi^{L}\colon H^{\!L}\to K$, defined by $\pi^{L}(h)\coloneqq h\cdot 1_{A}$, is a surjective morphism of algebras, and the map $\pi^{R}\colon H^{\!R}\to K$, defined by the same formula, is a surjective anti-morphism of algebras. By [BNS1]*Proposition 2.11, this implies that $K$ is separable.

Remark 1.15*.*

Each subalgebra of $A$, which is stable under $\rho$, includes $\{h\cdot 1_{A}:h\in H\}$.

From here until the end of this subsection $A$ is a left weak $H$-module algebra and $E$ is the unitary crossed product of $A$ by $H$ associated with $\rho$ and a map $f\colon H\otimes_{k}H\to A$. Thus, we assume that the hypotheses of Theorem 1.5 are fulfilled. In particular $\nu$, $\jmath_{\nu}$ and $\gamma$ are as at the beginning of Subsection 1.3. By [BNS1]*Proposition 2.4 we know that $H^{\!L}H^{\!R}$ is a subalgebra $H$.

Proposition 1.16**.**

The map $f$ factorizes throughout $H\otimes_{H^{\!L}H^{\!R}}H$.

Proof.

It suffices to prove that $f(hl\otimes_{k}m)=f(hl\otimes_{k}m)$ for all $h,m\in H$ and $l\in H^{\!L}\cup H^{\!R}$. For $l\in H^{\!R}$ this follows from [GGV1]*Proposition 2.7. Suppose now that $l\in H^{\!L}$. By Definition 1.8(3), the maps $u_{2},v_{2}\colon H\otimes_{k}H\to A$, defined by $u_{2}(h\otimes_{k}l)\coloneqq hl\cdot 1_{A}$ and $v_{2}(h\otimes_{k}l)\coloneqq h\cdot(l\cdot 1_{A})$ coincide. Consequently $f(hl\otimes_{k}m)=f(hl\otimes_{k}m)$ by [GGV1]*Proposition 2.8 and Remarks 2.16 and 2.17. ∎

1.4.1   Invertible cocycles and cleft extensions

Let $u_{2}\colon H\otimes_{k}H\to A$ be the map defined by $u_{2}(h\otimes_{k}l)\coloneqq hl\cdot 1_{A}$. We say that the cocycle $f$ is invertible if there exists a (unique) map $f^{-1}\colon H\otimes_{k}H\to A$ such that $u_{2}*f^{-1}=f^{-1}*u_{2}=f^{-1}$ and $f^{-1}*f=f*f^{-1}=u_{2}$.

Remark 1.17*.*

Condition (1) in Theorem 1.5 says that $f*u_{2}=f$. By [GGV1]*Remark 2.17 and the comment above [GGV1]*Proposition 4.4, this implies that $u_{2}*f=f$.

Example 1.18**.**

Assume that $A$ is a left $H$-module algebra and that $f$ is the trivial cocycle. By the previous remark and Definition 1.8(2), the cocycle $f$ is invertible and $f^{-1}=f$.

Definition 1.19**.**

A map $g\colon H\otimes_{k}H\to A$ is normal if $g(1\otimes_{k}h)=g(h\otimes_{k}1)=h\cdot 1_{A}$ for all $h$.

By [GGV1]*Proposition 2.17 and Remark 2.19 we know that the cocycle $f$ is normal if and only if the equalities in items (2) and (3) of Theorem 1.5 hold. When $f$ is invertible, we define $\gamma^{-1}\colon H\to E$ by

[TABLE]

Assume for instance that $A$ is a left $H$-module algebra and that $f$ is the trivial cocycle. By Example 1.18 and the fact that $S*\operatorname{id}=\Pi^{R}$, we have

[TABLE]

Using this, Proposition 1.9(3) and the fact that $\Pi^{L}\circ\Pi^{R}=\Pi^{L}\circ\overline{\Pi}^{L}\circ S=\Pi^{L}\circ S$, we obtain

[TABLE]

where the second equality holds by [GGV1]*Propositions 2.22 and 4.5; and the last one, since $S$ is anticomultiplicative and $\Pi^{L}*\operatorname{id}=\operatorname{id}$.

1.5   Mixed complexes

In this subsection we recall briefly the notion of mixed complex. For more details about this concept we refer to [B] and [K].

A mixed complex $\mathcal{X}\coloneqq(X,b,B)$ is a graded $k$-module $(X_{n})_{n\geq 0}$, endowed with morphisms

[TABLE]

such that $b\circ b=0$, $B\circ B=0$ and $B\circ b+b\circ B=0$. A morphism of mixed complexes $g\colon(X,b,B)\longrightarrow(Y,d,D)$ is a family of maps $g\colon X_{n}\to Y_{n}$, such that $d\circ g=g\circ b$ and $D\circ g=g\circ B$. Let $u$ be a degree $2$ variable. A mixed complex $\mathcal{X}\coloneqq(X,b,B)$ determines a double complex

[TABLE]

where $b(\mathbf{x}u^{i})\coloneqq b(\mathbf{x})u^{i}$ and $B(\mathbf{x}u^{i})\coloneqq B(\mathbf{x})u^{i-1}$. By deleting the positively numbered columns we obtain a subcomplex $\operatorname{BN}(\mathcal{X})$ of $\operatorname{BP}(\mathcal{X})$. Let $\operatorname{BN}^{\prime}(\mathcal{X})$ be the kernel of the canonical surjection from $\operatorname{BN}(\mathcal{X})$ to $(X,b)$. The quotient double complex $\operatorname{BP}(\mathcal{X})/\operatorname{BN}^{\prime}(\mathcal{X})$ is denoted by $\operatorname{BC}(\mathcal{X})$. The homology groups $\operatorname{HC}_{*}(\mathcal{X})$, $\operatorname{HN}_{*}(\mathcal{X})$ and $\operatorname{HP}_{*}(\mathcal{X})$, of the total complexes of $\operatorname{BC}(\mathcal{X})$, $\operatorname{BN}(\mathcal{X})$ and $\operatorname{BP}(\mathcal{X})$, respectively, are called the cyclic, negative and periodic homology groups of $\mathcal{X}$. The homology $\operatorname{HH}_{*}(\mathcal{X})$, of $(X,b)$, is called the Hochschild homology of $\mathcal{X}$. Finally, it is clear that a morphism $f\colon\mathcal{X}\to\mathcal{Y}$ of mixed complexes induces a morphism from the double complex $\operatorname{BP}(\mathcal{X})$ to the double complex $\operatorname{BP}(\mathcal{Y})$.

Notation 1.20**.**

Let $C$ be an algebra. If $K$ is a subalgebra of $C$ we will say that $C$ is a $K$-algebra. Given a $K$-bimodule $M$, we let $M\otimes_{K}$ denote the quotient $M/[M,K]$, where $[M,K]$ is the $k$-submodule of $M$ generated by all the commutators $m\lambda-\lambda m$, with $m\in M$ and $\lambda\in K$. Moreover, for $m\in M$, we let $[m]$ denote the class of $m$ in $M\otimes_{K}$.

By definition, the normalized mixed complex of the $K$-algebra $C$ is $(C\otimes_{K}\overline{C}^{\otimes_{K}^{*}}\otimes_{K},b_{*},B_{*})$, where $\overline{C}\coloneqq C/K$, $b_{*}$ is the canonical Hochschild boundary map and the Connes operator $B_{*}$ is given by

[TABLE]

The cyclic, negative, periodic and Hochschild homology groups $\operatorname{HC}^{K}_{*}(C)$, $\operatorname{HN}^{K}_{*}(C)$, $\operatorname{HP}^{K}_{*}(C)$ and $\operatorname{HH}^{K}_{*}(C)$ of $C$ are the respective homology groups of $(C\otimes_{K}\overline{C}^{\otimes_{K}^{*}}\otimes_{K},b_{*},B_{*})$.

2   Hochschild homology of cleft extensions

Let $H$ be a weak Hopf algebra, $A$ an algebra, $\rho\colon H\to A$ a weak left action and $f\colon H\otimes_{k}H\to A$ a linear map. Let $\chi_{\rho}$, $\gamma$, $\nu$, $\jmath_{\nu}$ and $\mathcal{F}_{f}$ be as at the beginning of Subsection 1.3. Assume that the hypotheses of Theorem 1.5 are fulfilled. Let $E$ be the crossed product associated with $\rho$ and $f$ and let $K$ be a subalgebra of $A$, which is stable under $\rho$. For instance we can take $K$ as the minimal subalgebra of $A$ that is stable under $\rho$ (see Example 1.14). By Theorem 1.5, the tuple $(A,\chi_{\rho},\mathcal{F}_{f},\nu)$ is a crossed product system with preunit, $\mathcal{F}_{f}$ is a cocycle that satisfies the twisted module condition and $E$ is an associative algebra with unit $1_{E}\coloneqq 1_{A}\times 1$. Moreover, by Remark 1.15 we know that $1_{E}\in K\otimes_{k}H$. So, the hypotheses of [GGV2]*Section 3 are satisfied. For the sake of simplicity in the sequel we will write $\otimes$ instead of $\otimes_{K}$. Let $M$ be an $E$-bimodule. By definition the Hochschild homology $\operatorname{H}^{K}_{*}(E,M)$, of the $K$-algebra $E$ with coefficients in $M$, is the homology of the normalized Hochschild chain complex \cramped{\bigl{(}M\otimes\overline{E}^{\otimes^{*}}\otimes,b_{*}\bigr{)}}, where $b_{*}$ is the canonical Hochschild boundary map. In [GGV2]*Section 3 a chain complex $(\widehat{X}_{*}(M),\widehat{d}_{*})$ was obtained, simpler than the canonical one, that gives the Hochschild homology $\operatorname{H}^{K}_{*}(E,M)$, of the $K$-algebra $E$ with coefficients in $M$. From now on we assume that the cocycle $f$ is convolution invertible. In this Section we prove that $(\widehat{X}_{*}(M),\widehat{d}_{*})$ is isomorphic to a simpler complex $(\overline{X}_{*}(M),\overline{d}_{*})$. If $K$ is separable (for instance when $K\coloneqq H\cdot 1_{A}$), then $(\overline{X}_{*}(M),\overline{d}_{*})$ gives the absolute Hochschild homology of $E$ with coefficients in $M$. Recall that $M$ is an $A$-bimodule via the map $\jmath_{\nu}\colon A\to E$. Let $\overline{A}\coloneqq A/K$, $\overline{E}\coloneqq E/\jmath_{\nu}(K)$ and $\widetilde{E}\coloneqq E/\jmath_{\nu}(A)$. We recall from [GGV2]*Section 3 that

[TABLE]

and that there exist maps $\widehat{d}^{l}_{rs}\colon\widehat{X}_{rs}(M)\to\widehat{X}_{r-l,s+l-1}(M)$ such that

[TABLE]

Notations 2.1**.**

We will use the following notations:

1. (1)

For each $a\in A$ we let $\overline{a}$ denote its class in $\overline{A}$. For each $x\in E$, we let $\overline{x}$ and $\widetilde{x}$ denote its class in $\overline{E}$ and $\widetilde{E}$, respectively. 2. (2)

Given $a_{1},\dots,a_{r}\in A$ and $1\leq i\leq j\leq r$, we set $\mathbf{a}_{ij}\coloneqq a_{i}\otimes\cdots\otimes a_{j}$, and we let $\overline{\mathbf{a}}_{ij}$ denote the class of $\mathbf{a}_{ij}$ in $\overline{A}^{\otimes^{j-i+1}}$. 3. (3)

Given $h_{1},\dots,h_{s}\in H$ we set $\mathfrak{h}_{1s}\coloneqq h_{1}\cdots h_{s}$ and $\mathrm{h}_{1s}\coloneqq h_{1}\otimes_{k}\cdots\otimes_{k}h_{s}$. 4. (4)

Given $h_{1},\dots,h_{s}\in H$ we set

[TABLE] 5. (5)

Given $h\in H$ and $a_{1},\dots,a_{r}\in A$ we set $h\cdot\overline{\mathbf{a}}_{1r}\coloneqq\overline{h^{(1)}\cdot a_{1}}\otimes\cdots\otimes\overline{h^{(r)}\cdot a_{r}}$. 6. (6)

We let $\overline{\gamma}\colon H\to\overline{E}$ and $\widetilde{\gamma}\colon H\to\widetilde{E}$ denote the maps induced by $\gamma$. 7. (7)

Given $h_{1},\dots,h_{s}\in H$ we set $\widetilde{\gamma}_{A}(\mathrm{h}_{1s})\coloneqq\widetilde{\gamma}(h_{1})\otimes_{A}\cdots\otimes_{A}\widetilde{\gamma}(h_{s})$. 8. (8)

Given $h_{1},\dots,h_{s}\in H$ we set $\gamma_{{}_{\!\times}}(\mathrm{h}_{1s})\coloneqq\gamma(h_{1})\cdots\gamma(h_{s})$. 9. (9)

Given $h_{1},\dots,h_{s}\in H$ we set $\gamma_{{}_{\!\times}}^{-1}(\mathrm{h}_{1s})\coloneqq\gamma^{-1}(h_{s})\cdots\gamma^{-1}(h_{1}),$ where $\gamma^{-1}$ is as in (1.11).

2.1   Technical results

Lemma 2.2**.**

Let $a,a^{\prime}\in A$ and $h\in H$. The following equalities hold:

1. (1)

\delta_{E}\bigl{(}\jmath_{\nu}(a)\gamma(l)\jmath_{\nu}(a^{\prime})\gamma(h)\bigr{)}=\jmath_{\nu}(a)\gamma(l)\jmath_{\nu}(a^{\prime})\gamma(h^{(1)})\otimes_{k}h^{(2)}, for all $l\in H^{\!L}$. 2. (2)

\delta_{E}\bigl{(}\jmath_{\nu}(a^{\prime})\gamma(h)\jmath_{\nu}(a)\gamma(l)\bigr{)}=\jmath_{\nu}(a^{\prime})\gamma(h^{(1)})\jmath_{\nu}(a)\gamma(l)\otimes_{k}h^{(2)}, for all $l\in H^{\!L}$. 3. (3)

\delta_{E}\bigl{(}\jmath_{\nu}(A)\gamma(H^{\!L})\bigr{)}\subseteq E\otimes_{k}H^{\!L}. 4. (4)

$\jmath_{\nu}(a)\gamma^{-1}(h)=\gamma^{-1}(h^{(1)})\jmath_{\nu}(h^{(2)}\cdot a)$.

Proof.

1) We have

[TABLE]

where the first and the last equality hold by the first identity in (1.8) and Theorem 1.5(d); the second one, by [BNS1]*(2.6a) and [GGV1]*Proposition 2.22; the third one because, by equality (1.7), the definition of $\delta_{E}$, and the fact that $\jmath_{\nu}(A)\gamma(H)\subseteq E$, for all $a\in A$ and $h\in H$, we have

[TABLE]

the fourth one, by [BNS1](2.6a) and the fact that by [BNS1](2.4), (2.7a) and (2.10)

[TABLE]

and the fifth one, by [GGV1]*Proposition 2.22.

2) Mimic the proof of item (1).

3) This follows from equality (2.1) and [BNS1]*(2.6a).

4) This is [GGV1]*Proposition 5.22. ∎

Remark 2.3*.*

Since $\jmath_{\nu}(1_{A})=1_{E}$, from equality (2.1) it follows that $\gamma$ is $H$-colinear.

Lemma 2.4**.**

For all $h\in H$ and $l\in H^{\!L}$, we have \gamma^{-1}(hl)=\gamma\bigl{(}S(l)\bigr{)}\gamma^{-1}(h) and \gamma^{-1}(lh)=\gamma^{-1}(h)\gamma\bigl{(}S(l)\bigr{)}.

Proof.

We prove the first equality and leave the second one, which is similar, to the reader. We have

[TABLE]

where the first and last equality hold by the definition of $\gamma^{-1}$; the second one, by [BNS1]*(2.6a) and identity (1.10); the third one, by [GGV1]*Proposition 2.22 and the fact that $S$ is antimultiplicative; and the fourth one, by [GGV1]*Proposition 4.6. ∎

Lemma 2.5**.**

Let $h_{1},\dots,h_{s}\in H$ and $a\in A$. The following equality holds:

[TABLE]

Proof.

We proceed by induction on $s$. By the first identity in (1.8) and Lemma 2.2(4), for $s=1$ we have

[TABLE]

Assume $s>1$ and that the result is valid for $s-1$. Let $T\coloneqq\gamma_{{}_{\!\times}}^{-1}(\mathrm{h}_{1s}^{(1)})\otimes_{A}\widetilde{\gamma}_{A}(\mathrm{h}_{1s}^{(2)})\cdot\jmath_{\nu}(a)$. By the first identity in (1.8),

[TABLE]

Consequently, by the inductive hypothesis and Lemma 2.2(4),

[TABLE]

as desired. ∎

Lemma 2.6**.**

Let $s\geq 1$. for all $h_{1},\dots,h_{s}\in H$, we have $\gamma_{{}_{\!\times}}(\mathrm{h}_{1s}^{(1)})\gamma_{{}_{\!\times}}^{-1}(\mathrm{h}_{1s}^{(2)})\otimes_{A}\widetilde{\gamma}_{A}(\mathrm{h}_{1s}^{(3)})=1_{E}\otimes_{A}\widetilde{\gamma}_{A}(\mathrm{h}_{1s})$.

Proof.

We proceed by induction on $s$. Using the first equality in (1.3) and arguing as in [BNS1]*equality (2.8b), we obtain that

[TABLE]

Using this, [GGV1]*Lemma 2.20 and Propositions 4.5 and 5.19, we obtain

[TABLE]

This proves the case $s=1$. Assume that the result is true for $s$ and set $T\coloneqq\gamma_{{}_{\!\times}}(\mathrm{h}_{1,s+1}^{(1)})\gamma_{{}_{\!\times}}^{-1}(\mathrm{h}_{1,s+1}^{(2)})\otimes_{A}\widetilde{\gamma}_{A}(\mathrm{h}_{1,s+1}^{(3)})$. By equality (2.3), [GGV1]*Propositions 4.5 and 5.19,

[TABLE]

Hence

[TABLE]

Using now the first identity in (1.8) and Definition 1.8(3) again and again, we obtain

[TABLE]

Consequently, by the inductive hypothesis,

[TABLE]

where the last equality follows using Definition 1.8(3) and the first identity in (1.8) again and again. Hence, by [GGV1]*Lemma 2.20 and Proposition 4.5, we have $T=1_{E}\otimes_{A}\widetilde{\gamma}_{A}(\mathrm{h}_{1,s+1})$, as desired. ∎

Lemma 2.7**.**

Let $s\geq 1$. For all $z\in H^{\!R}$ and $h_{1},\dots,h_{s}\in H$, we have

[TABLE]

Proof.

Set $T\coloneqq\gamma_{{}_{\!\times}}^{-1}(\mathrm{h}_{1s}^{(1)})\gamma(z)\gamma_{{}_{\!\times}}(\mathrm{h}_{1s}^{(2)})\otimes_{k}\overline{\mathbf{h}}_{1s}^{(3)}$. We proceed by induction on $s$. Let $s=1$. By [BNS1]*equality (2.7b), [GGV1]*Propositions 2.22 and 5.19, we have

[TABLE]

Consequently, by the definition of $\Pi^{R}$, [BNS1]*(2.6a) and identities (1.2) and (1.10),

[TABLE]

So, by the definition of $\overline{\Pi}^{R}$,

[TABLE]

Assume now that $s>1$ and the result holds by $s-1$. By the inductive hypothesis, we have

[TABLE]

Thus, by the case $s=1$, [BNS1]*(2.4) and [GGV1]*Proposition 2.22, we have

[TABLE]

as desired. ∎

2.2   Main results

Let $r,s\geq 0$. By [GGV1]*Propositions 2.22 and 4.6, we know that $M\otimes\overline{A}^{\otimes^{r}}\otimes$ is a left $H^{\!L}$-module via

[TABLE]

where $[m\otimes\overline{\mathbf{a}}_{1r}]$ denotes the class of $m\otimes\overline{\mathbf{a}}_{1r}$ in $M\otimes\overline{A}^{\otimes^{r}}\otimes$ (see Notation 1.20 and remember that $\otimes$ stands for $\otimes_{K}$). Write

[TABLE]

Since $\overline{H}^{\otimes_{\!H^{\!L}}^{0}}=H^{\!L}$ and $\overline{A}^{\otimes^{0}}=K$, we have

[TABLE]

Let $\Theta^{\prime}_{rs}\colon M\otimes_{k}E^{\otimes_{k}^{s}}\otimes_{k}\overline{A}^{\otimes^{r}}\longrightarrow\overline{X}_{rs}(M)$ and $\Lambda^{\prime}_{rs}\colon H^{\otimes_{k}^{s}}\otimes_{k}M\otimes_{k}\overline{A}^{\otimes^{r}}\longrightarrow\widehat{X}_{rs}(M)$ be the maps defined by

[TABLE]

where $\mathbf{x}\coloneqq m\otimes_{k}\jmath_{\nu}(a_{1})\gamma(h_{1})\otimes_{k}\cdots\otimes_{k}\jmath_{\nu}(a_{s})\gamma(h_{s})\otimes_{k}\overline{\mathbf{a}}_{s+1,s+r}$ and $\mathbf{y}\coloneqq\mathrm{h}_{1s}\otimes_{k}m\otimes_{k}\overline{\mathbf{a}}_{1r}$.

Proposition 2.8**.**

For each $r,s\geq 0$ the maps $\Theta^{\prime}_{rs}$ and $\Lambda^{\prime}_{rs}$ induce morphisms

[TABLE]

which are inverse one of each other.

Proof.

First we show that $\Theta_{rs}$ is well defined. Let $\mathbf{x}$ be as in the definition of $\Theta^{\prime}_{rs}$. We must prove that

1. (1)

If there exists $i$ such that $h_{i}\in H^{\!L}$, then $\Theta^{\prime}(\mathbf{x})=0$, 2. (2)

$\Theta^{\prime}$ is $A$-balanced in the first $s$ tensors of $M\otimes_{k}E^{\otimes_{k}^{s}}\otimes_{k}A^{\otimes^{r}}$, 3. (3)

$\Theta^{\prime}$ is $K$-balanced in the $(s+1)$-tensor of $M\otimes_{k}E^{\otimes_{k}^{s}}\otimes_{k}\overline{A}^{\otimes^{r}}$, 4. (4)

$\Theta^{\prime}(\jmath_{\nu}(\lambda)\cdot\mathbf{x})=\Theta^{\prime}(\mathbf{x}\cdot\jmath_{\nu}(\lambda))$ for all $\lambda\in K$.

Condition (1) follows from Lemma 2.2(3). Next we prove that Condition (2) is satisfied at the first tensor. Let $a\in A$. By Lemma 2.2(1), we have

[TABLE]

A similar argument using items (1) and (2) of Lemma 2.2 proves Condition (2) at the $i$-th tensor with $2\leq i\leq s$. We now prove Condition (3). Let $\lambda\in K$. By Lemma 2.2(2), we have

[TABLE]

Finally, when $r\geq 1$ the fourth assertion is trivial, while, when $r=0$ it follows from Lemma 2.2(2).

We next show that $\Lambda_{rs}$ is well defined. Let $\mathbf{y}$ be as in the definition of $\Lambda^{\prime}_{rs}$. We must prove that

1. (1)

If some $h_{i}\in H^{\!L}$, then $\Lambda^{\prime}(\mathbf{y})=0$, 2. (2)

$\Lambda^{\prime}$ is $H^{\!L}$-balanced in the first $(s-1)$-th tensors of $\cramped{H^{\otimes_{k}^{s}}\otimes_{k}M\otimes_{k}\overline{A}^{\otimes^{r}}}$, 3. (3)

If $r>0$, then $\Lambda^{\prime}$ is $K$-balanced in the $(s+1)$-th tensor of $\cramped{H^{\otimes_{k}^{s}}\otimes_{k}M\otimes_{k}\overline{A}^{\otimes^{r}}}$, 4. (4)

$\Lambda^{\prime}(\mathrm{h}_{1s}\otimes_{k}\jmath_{\nu}(\lambda)\cdot m\otimes_{k}\overline{\mathbf{a}}_{1r})=\Lambda^{\prime}_{rs}(\mathrm{h}_{1s}\otimes_{k}m\otimes_{k}\overline{\mathbf{a}}_{1r}\cdot\lambda)$ for all $\lambda\in K$, 5. (5)

If $s>0$, then $\Lambda^{\prime}(\mathrm{h}_{1s}\otimes_{k}m\cdot\gamma(S(l))\otimes_{k}\overline{a}_{1r})=\Lambda^{\prime}(\mathrm{h}_{1s}\cdot l\otimes_{k}m\otimes_{k}\overline{\mathbf{a}}_{1r})$ for all $l\in H^{\!L}$,

Item (1) holds since, by [BNS1]*(2.6a) and [GGV1]*Proposition 4.5, $\gamma^{-1}(l^{(1)})\otimes_{k}\gamma(l^{(2)})\in E\otimes_{k}\jmath_{\nu}(A)$ for all $l\in H^{\!L}$. In order to prove item (2) we must check that $\Lambda^{\prime}(\mathrm{h}_{1i}\cdot l\otimes_{k}h_{i+1,s}\otimes_{k}m\otimes_{k}\overline{\mathbf{a}}_{1r})=\Lambda^{\prime}(\mathrm{h}_{1i}\otimes_{k}l\cdot h_{i+1,s}\otimes_{k}m\otimes_{k}\overline{\mathbf{a}}_{1r})$, for all $i\!<\!s$ and $l\in H^{\!L}$. But this follows from Lemma 2.4 and identities (1.10) and (2.2). Item (3) holds since, by Lemma 2.5,

[TABLE]

When $r\geq 1$, Item (4) is trivial, while, when $r=0$, it holds since, by Lemma 2.5,

[TABLE]

Finally, for Item (5) we have

[TABLE]

where the second equality holds by Lemma 2.4; and the third one, by identity (1.10).

We next prove that $\Theta_{rs}$ and $\Lambda_{rs}$ are inverse one of each other. To begin with, note that under the first identifications in [GGV2]*(3.1) and (2.5),

[TABLE]

which proves the case $s=0$. Assume $s\geq 1$ and let $m\in M$, $h_{1},\dots,h_{s}\in H$ and $a_{1},\dots,a_{r}\in A$. Set

[TABLE]

By Lemma 2.6

[TABLE]

as desired. For the other composition, by Lemma 2.7 and [GGV1]*Lemma 2.20, we have

[TABLE]

which finishes the proof. ∎

For each $0\leq l\leq s$ and $r\geq 0$ such that $r+l\geq 1$, let $\overline{d}^{l}_{rs}\colon\overline{X}_{rs}(M)\longrightarrow\overline{X}_{r+l-1,s-l}(M)$ be the map defined by $\overline{d}^{l}_{rs}\coloneqq\Theta_{r+l-1,s-l}\circ\widehat{d}^{l}_{rs}\circ\Lambda_{rs}$.

Theorem 2.9**.**

The Hochschild homology of the $K$-algebra $E$ with coefficients in $M$ is the homology of the chain complex $(\overline{X}_{*}(M),\overline{d}_{*})$, where

[TABLE]

Proof.

By Proposition 2.8 and the definition of $(\overline{X}_{*}(M),\overline{d}_{*})$, the maps

[TABLE]

given by $\Theta_{n}\coloneqq\bigoplus_{r+s=n}\Theta_{rs}$ and $\Lambda_{n}\coloneqq\bigoplus_{r+s=n}\Lambda_{rs}$, are inverse one of each other. ∎

By [GGV2]*Remark 3.2 if $f$ takes its values in $K$, then $(\overline{X}_{*}(M),\overline{d}_{*})$ is the total chain complex of the double complex $(\overline{X}_{**}(M),\overline{d}^{0}_{**},\overline{d}^{1}_{**})$; while if $K=A$, then $(\overline{X}_{*}(M),\overline{d}_{*})=(\overline{X}_{0*}(M),\overline{d}^{1}_{0*})$.

Lemma 2.10**.**

Let $m\in M$, $a,a_{1}\dots,a_{r}\in A$, $h_{1},\dots,h_{s}\in H$ and $z\in H^{\!R}$.

1. (1)

For \mathbf{x}\coloneqq\bigl{[}m\cdot\gamma_{{}_{\!\times}}^{-1}(\mathrm{h}_{1s}^{(1)})\otimes_{A}\widetilde{\gamma}_{A}(\mathrm{h}_{1,s-1}^{(2)})\otimes_{A}\stackon[-8pt]{\gamma(h_{s}^{(2)})\jmath_{\nu}(a)}{\vstretch{1.5}{\hstretch{3.3}{\widetilde{\phantom{\;\;\;\;\;\;}}}}}\otimes\overline{\mathbf{a}}_{1r}\bigr{]}, we have

[TABLE] 2. (2)

For \mathbf{x}\coloneqq\bigl{[}m\cdot\gamma_{{}_{\!\times}}^{-1}(\mathrm{h}_{1s}^{(1)})\otimes_{A}\widetilde{\gamma}_{A}(\mathrm{h}_{1,i-1}^{(2)})\otimes_{A}\stackon[-8pt]{\gamma(h_{i}^{(2)})\gamma(h_{i+1}^{(2)})}{\vstretch{1.5}{\hstretch{3.2}{\widetilde{\phantom{\;\;\;\;\;\;\;\,}}}}}\otimes_{A}\widetilde{\gamma}_{A}(\mathrm{h}_{i+2,s}^{(2)})\otimes\overline{\mathbf{a}}_{1r}\bigr{]}, we have

[TABLE]

(of course, we are assuming that $s\geq 2$ and $1\leq i<s$). 3. (3)

For \mathbf{x}\coloneqq\bigl{[}m\cdot\gamma_{{}_{\!\times}}^{-1}(\mathrm{h}_{1s}^{(1)})\gamma(z)\otimes_{A}\widetilde{\gamma}_{A}(\mathrm{h}_{1s}^{(2)})\otimes\overline{\mathbf{a}}_{1r}\bigr{]}, we have $\Theta(\mathbf{x})=(-1)^{rs}\,\overline{\Pi}^{R}(z)\cdot\overline{\mathbf{h}}_{1s}\otimes_{H^{\!L}}[m\otimes\overline{\mathbf{a}}_{1r}]$.

Proof.

1) Under the first identifications in [GGV2]*(3.1) and (2.5), for $s=0$ the equality in item (1) becomes $\Theta([m\cdot\jmath_{\nu}(a)\otimes\overline{\mathbf{a}}_{1r}])=[m\cdot\jmath_{\nu}(a)\otimes\overline{\mathbf{a}}_{1r}]$, which follows immediately from equality (2.6). Assume now that $s\geq 1$. By [GGV1]*Proposition 2.21, we know that

[TABLE]

By this, the first identity in (1.8), Lemma 2.7, [BNS1]*(2.4) and the definition of the action in (2.4),

[TABLE]

as desired.

2) By the second identity in (1.8), Lemma 2.7, [BNS1]*(2.4), the definition of the action in (2.4) and equality (2.7),

[TABLE]

as desired.

3) Under the first identifications in [GGV2]*(3.1) and (2.5), for $s=0$ item (3) becomes

[TABLE]

which follows immediately from identities (1.3) and (2.6). Assume now that $s\geq 1$. We have

[TABLE]

where the first equality holds by the definition of $\Theta$ and Lemma 2.7; the second one, by [BNS1]*(2.4) and the definition of the action in (2.4); and the last one, by equality (2.7). ∎

Notation 2.11**.**

Given a $k$-subalgebra $R$ of $A$ and $0\!\leq\!u\!\leq\!r$, we let $\overline{X}^{u}_{rs}(R,M)$ denote the $k$-submodule of $\overline{X}_{rs}(M)$ generated by all the elements $\overline{\mathbf{h}}_{1s}\otimes_{H^{\!L}}[m\otimes\overline{\mathbf{a}}_{1r}]$ with $m\in M$, $a_{1},\dots,a_{r}\in A$, $h_{1},\dots,h_{s}\in H$, and at least $u$ of the $a_{j}$’s in $R$.

Theorem 2.12**.**

Let $\mathbf{y}\coloneqq\overline{\mathbf{h}}_{1s}\otimes_{H^{\!L}}[m\otimes\overline{\mathbf{a}}_{1r}]\in\overline{X}_{rs}(M)$, where $m\in M$, $a_{1},\dots,a_{r}\in A$ and $h_{1},\dots,h_{s}\in H$. The following assertions hold:

1. (1)

For $r\geq 1$ and $s\geq 0$, we have

[TABLE] 2. (2)

For $r\geq 0$ and $s=1$, we have

[TABLE] 3. (3)

For $r\geq 0$ and $s\geq 2$, we have

[TABLE]

where

[TABLE] 4. (4)

Let $R$ be a $k$-subalgebra of $A$. If $R$ is stable under $\rho$ and $f$ takes its values in $R$, then

[TABLE]

for each $r\geq 0$ and $1<l\leq s$.

Proof.

1) By the definition of $\Lambda$ and [GGV2]*Theorem 3.5(1), we have

[TABLE]

The formula for $\overline{d}^{0}$ follows from this using Lemma 2.10(1).

2) By the definition of $\Lambda$, [GGV1]*Proposition 5.19 and [GGV2]*Theorem 3.5(2), we have

[TABLE]

The formula for $\overline{d}^{1}$ follows from this using Lemma 2.10 and the fact that $\overline{\Pi}^{R}\circ\Pi^{R}=\overline{\Pi}^{R}$.

3) By the definition of $\Lambda$ and [GGV2]*Theorem 3.6, we have

[TABLE]

The formula for $\overline{d}^{2}$ follows from this using Lemma 2.10(1).

4) Let $\widehat{X}^{l-1}_{r+l-1,s-l}(R,M)$ be as in [GGV2]*Notation 3.4. By Remark 1.13 and [GGV2]*Theorem 3.5(3) this item follows from the fact that \overline{X}^{l-1}_{r+l-1,s-l}(R,M)=\Theta\bigl{(}\widehat{X}^{l-1}_{r+l-1,s-l}(R,M)\bigr{)}. ∎

Remark 2.13*.*

By the second equality in (1.3), we have

[TABLE]

This gives an alternative formula for $\overline{d}^{1}(\mathbf{y})$ in (2.8). We will use this fact in the proof of Proposition 2.16.

Proposition 2.14**.**

For each $h\in H$, the map \cramped{F_{*}^{h}\colon\bigl{(}M\otimes\overline{A}^{\otimes^{*}}\otimes,b_{*}\bigr{)}\longrightarrow\bigl{(}M\otimes\overline{A}^{\otimes^{*}}\otimes,b_{*}\bigr{)}}, defined by

[TABLE]

is a morphism of complexes. Moreover the following facts hold:

1. (1)

For each $h,l\in H$, the endomorphisms of $\operatorname{H}^{K}_{*}(A,M)$ induced by $F_{*}^{h}\circ F_{*}^{l}$ and $F_{*}^{hl}$ coincide. 2. (2)

$F_{r}^{l}([m\otimes\overline{\mathbf{a}}_{1r}])=l\cdot[m\otimes\overline{\mathbf{a}}_{1r}]$, for all $l\in H^{\!L}$ (see (2.4)). In particular $F_{*}^{1}$ is the identity map.

Consequently, $\operatorname{H}^{K}_{*}(A,M)$ is a left $H$-module.

Proof.

Using the first equality in Proposition 1.8 and Lemma 2.2(4), it is easy to see that the maps $F_{*}^{h}$ are well defined. Moreover by Definition 1.8(2), the first identity in (1.8) and Lemma 2.2(4), they are morphisms of complexes.

For $h,l\in H$, let \cramped{\bigl{(}\mathfrak{h}_{r}\colon M\otimes\overline{A}^{\otimes^{r}}\otimes\longrightarrow M\otimes\overline{A}^{\otimes^{r+1}}\otimes\bigr{)}_{r\geq 0}} be the family of maps, defined by

[TABLE]

where $\mathfrak{T}(h,l,\overline{\mathbf{a}}_{1r})$ is as in Theorem 2.12(3). In order to prove item (1) it suffices to show that \bigl{(}\mathfrak{h}_{r}\bigr{)}_{r\geq 0} is a homotopy from $F^{hl}_{*}$ to $F^{h}_{*}\circ F^{l}_{*}$. For this we must check that

[TABLE]

Let \mathbf{y}\coloneqq(\overline{h}\otimes_{H^{\!L}}\overline{l})\otimes_{H^{\!L}}\bigl{[}m\otimes\overline{\mathbf{a}}_{1r}\bigr{]}\in\overline{X}_{r2}(M). Since $(\overline{X}_{*}(M),\overline{d}_{*})$ is a chain complex,

[TABLE]

By this fact and Theorem 2.12, to prove equality (2.9) it suffices to show that

[TABLE]

Now, a direct computation shows that

[TABLE]

Using the second equality in (1.3), and the fact that, by Lemma 2.4 and identity (2.2),

[TABLE]

we obtain that equality (2.10) holds.

We next prove item (2). Let $l\in H^{\!L}$. For $r=1$, we have

[TABLE]

where the second equality holds by [BNS1]*(2.6a), [GGV1]*Proposition 4.5 and Remark 1.15; the third one, by [GGV1]*Proposition 5.19; and the fourth one, by the second identity in (1.3) and the fact that $\overline{\Pi}^{R}(l)=l$. Assume now $r\geq 1$. To begin note that, by Remark 1.15,  [GGV1]*Proposition 4.5 and Definition 1.8(2),

[TABLE]

We claim that

[TABLE]

By [BNS1]*(2.6a) and Proposition 1.9(1),

[TABLE]

If $r=1$ this ends the proof of the claim. Assume that $r>1$. In this case, by Remark 1.15 and Definition 1.8(2),

[TABLE]

and the claim follows from an evident inductive argument. Thus, by [GGV1]*Proposition 4.5 and Remark 1.15,

[TABLE]

and so, by [GGV1]*Propositions 4.5 and 5.19, the second identity in (1.3) and the fact that $\overline{\Pi}^{R}(l)=l$,

[TABLE]

as desired. ∎

Example 2.15**.**

If $A=K$, then \operatorname{H}^{K}_{*}(E,M)=\operatorname{H}_{*}\bigl{(}H,M\otimes\bigr{)}, where $M\otimes$ is considered as a left $H$-module via the action given by $h\cdot[m]\coloneqq[\gamma(h^{(2)})\cdot m\cdot\gamma^{-1}(h^{(1)})]$.

In the following proposition, for each $r\geq 0$, we consider $M\otimes\overline{A}^{\otimes^{*}}\otimes$ as a left $H$-module via the action introduced in Proposition 2.14, and we consider $M\otimes\overline{A}^{\otimes^{*}}\otimes$ as a left $H^{\!L}$-module via the canonical inclusion of $H^{\!L}$ into $H$. In Proposition 2.14(2) we prove that these structures coincide with the ones introduced in (2.4) (which are the ones used in Theorem 2.12).

Proposition 2.16**.**

The spectral sequence of [GGV2](3.3) satisfies*

[TABLE]

Proof.

For each $i,n\geq 0$, let $F^{i}(\overline{X}_{n}(M))\coloneqq\bigoplus_{s=0}^{i}\overline{X}_{n-s,s}(M)$ . The chain complex $(\overline{X}_{*}(M),\overline{d}_{*})$ is filtrated by

[TABLE]

Moreover the isomorphism $\Theta_{*}\colon(\widehat{X}_{*}(M),\widehat{d}_{*})\to(\overline{X}_{*}(M),\overline{d}_{*})$ preserves filtrations, where we consider the chain complex $(\widehat{X}_{*}(M),\widehat{d}_{*})$ endowed with the filtration introduced at the beginning of [GGV2]Subsection 3.2. So, the spectral sequence of [GGV2](3.3) coincides with the spectral sequence determined by the filtration (2.12). Thus, the formula for $E^{1}_{rs}$ follows from Theorem 2.12(1) and the formula for $E^{2}_{rs}$ follows from Theorem 2.12(2), Remark 2.13 and Proposition 1.4(1). ∎

3   Hochschild cohomology of cleft extensions

Let $H$, $A$, $\rho$, $\chi_{\rho}$, $f$, $\mathcal{F}_{f}$, $E$, $K$, $M$, $\nu$, $\jmath_{\nu}$ and $\gamma$ be as in the previous section. Assume that the hypotheses of that section are fulfilled. In particular $H$ is a weak Hopf algebra, $A$ is a weak module algebra and $f$ is convolution invertible. Let $\gamma^{-1}$ be as in equality (1.11). By definition the Hochschild cohomology $\operatorname{H}_{K}^{*}(E,M)$, of the $K$-algebra $E$ with coefficients in an $E$-bimodule $M$, is the cohomology of the normalized Hochschild cochain complex \cramped{\bigl{(}\operatorname{Hom}_{K^{e}}\bigl{(}\overline{E}^{\otimes*},M\bigr{)},b^{*}\bigr{)}}, where $b^{*}$ is the canonical Hochschild boundary map. In [GGV2]*Section 4 a cochain complex $(\widehat{X}^{*}(M),\widehat{d}^{*})$ was obtained, simpler than the canonical one, that gives the Hochschild cohomology of $E$ with coefficients in $M$. In this section we prove that $(\widehat{X}^{*}(M),\widehat{d}^{*})$ is isomorphic to a simpler complex $(\overline{X}^{*}(M),\overline{d}^{*})$. When $K$ is separable, the complex $(\overline{X}^{*}(M),\overline{d}^{*})$ gives the absolute Hochschild cohomology of $E$ with coefficients in $M$. We recall from [GGV2]*Section 4 that

[TABLE]

and that there exist maps $\widehat{d}_{l}^{rs}\colon\widehat{X}^{r+l-1,s-l}(M)\longrightarrow\widehat{X}^{rs}(M)$ such that

[TABLE]

By [GGV1]*Propositions 2.22 and 4.6, we know that $M$ is a $(K,K\otimes_{k}H^{\!L})$-bimodule via

[TABLE]

For each $r,s\geq 0$, we set \overline{X}^{rs}(M)\coloneqq\operatorname{Hom}_{(K,K\otimes_{k}H^{\!L})}\bigl{(}\overline{H}^{\otimes_{H^{\!L}}^{s}}\otimes_{k}\overline{A}^{\otimes^{r}},M\bigr{)} , where we consider $\overline{H}^{\otimes_{\!H^{\!L}}^{s}}\otimes_{k}\overline{A}^{\otimes^{r}}$ as a $(K,K\otimes_{k}H^{\!L})$-bimodule via

[TABLE]

Let M^{\!K}\coloneqq\{m\in M:\lambda\cdot m=m\cdot\lambda\text{ for all \lambda\in K}\}. Since $\overline{H}^{\otimes_{\!H^{\!L}}^{0}}=H^{\!L}$ and $\overline{A}^{\otimes^{0}}=K$, we have

[TABLE]

where $M^{\!K}$ is considered as a right $H^{\!L}$-module via $m\cdot l\coloneqq\gamma(S(l))\cdot m$.

Remark 3.1*.*

For each $r,s\geq 0$, we have \overline{X}^{rs}(M)\simeq\operatorname{Hom}_{\!H^{\!L}}\bigl{(}\overline{H}^{\otimes_{\!H^{\!L}}^{s}},\operatorname{Hom}_{K^{e}}\bigl{(}\overline{A}^{\otimes^{r}},M\bigr{)}\bigr{)}, where \operatorname{Hom}_{K^{e}}\bigl{(}\overline{A}^{\otimes^{r}},M\bigr{)} is considered as right $H^{\!L}$-module via $(\beta\cdot l)(\overline{\mathbf{a}}_{1r})\coloneqq\gamma(S(l))\cdot\beta(\overline{\mathbf{a}}_{1r})$.

Proposition 3.2**.**

For each $r,s\geq 0$ there exist maps

[TABLE]

such that for

[TABLE]

we have

[TABLE]

Moreover the maps $\Theta^{rs}$ and $\Lambda^{rs}$ are inverse one of each other.

Proof.

Mimic the proof of Proposition 2.8 . ∎

For each $0\leq l\leq s$ and $r\geq 0$ such that $r+l\geq 1$, let $\overline{d}_{l}^{rs}\colon\overline{X}^{r+l-1,s-l}(M)\to\overline{X}^{rs}(M)$ be the map $\overline{d}_{l}^{rs}\coloneqq\Lambda^{r+l-1,s-l}\circ\widehat{d}_{l}^{rs}\circ\Theta^{rs}$.

Theorem 3.3**.**

The Hochschild cohomology of the $K$-algebra $E$ is the cohomology of $(\overline{X}^{*}(M),\overline{d}^{*})$, where

[TABLE]

Proof.

By Proposition 3.2 and the definition of $(\overline{X}_{*}(M),\overline{d}_{*})$, the map $\Theta^{*}\colon(\overline{X}^{*}(M),\overline{d}^{*})\longrightarrow(\widehat{X}^{*}(M),\widehat{d}^{*})$, given by $\Theta^{n}\coloneqq\bigoplus_{r+s=n}\Theta^{rs}$, is an isomorphism of complexes. ∎

By [GGV2]*Remark 4.2, if $f$ takes its values in $K$, then $(\overline{X}^{*}(M),\overline{d}^{*})$ is the total cochain complex of the double complex $(\overline{X}^{**}(M),\overline{d}_{0}^{**},\overline{d}_{1}^{**})$; while, if $K=A$, then $(\overline{X}^{*}(M),\overline{d}^{*})=(\overline{X}^{*}(M),\overline{d}_{1}^{0*})$.

Lemma 3.4**.**

Let $\beta\in\overline{X}^{rs}(M)$, $a,a_{1}\dots,a_{r}\in A$, $h_{1},\dots,h_{s}\in H$ and $z\in H^{\!R}$.

1. (1)

We have \gamma_{{}_{\!\times}}^{-1}(\mathrm{h}_{1s}^{(1)})\cdot\Theta(\beta)\bigl{(}\widetilde{\gamma}_{A}(\mathrm{h}_{1s}^{(2)})\cdot\jmath_{\nu}(a)\otimes\overline{\mathbf{a}}_{1r}\bigr{)}=(-1)^{rs}\jmath_{\nu}(a)\cdot\beta\bigl{(}\overline{\mathbf{h}}_{1s}\otimes_{k}\overline{\mathbf{a}}_{1r}\bigr{)}. 2. (2)

For $s\geq 2$ and $1\leq i<s$, we have

[TABLE] 3. (3)

We have \gamma_{{}_{\!\times}}^{-1}(\mathrm{h}_{1s}^{(1)})\gamma(z)\cdot\Theta(\beta)\bigl{(}\widetilde{\gamma}_{A}(\mathrm{h}_{1s}^{(2)})\otimes\overline{\mathbf{a}}_{1r}\bigr{)}=(-1)^{rs}\beta\bigl{(}\overline{\Pi}^{R}(z)\cdot\overline{\mathbf{h}}_{1s}\otimes_{k}\overline{\mathbf{a}}_{1r}\bigr{)}.

Proof.

Mimic the proof of Lemma 2.10. ∎

Notation 3.5**.**

For each $k$-subalgebra $R$ of $A$ and $0\!\leq\!u\!\leq\!r$, we set \overline{X}_{u}^{rs}(R,M)\coloneqq\Lambda\bigl{(}\widehat{X}_{u}^{rs}(R,M)\bigr{)}, where $\widehat{X}_{u}^{rs}(R,M)$ is as in [GGV2]*Notation 4.4

Theorem 3.6**.**

Let $\mathbf{y}\coloneqq\overline{\mathbf{h}}_{1s}\otimes_{k}\overline{\mathbf{a}}_{1r}$, where $h_{1},\dots,h_{s}\in H$ and $a_{1},\dots,a_{r}\in A$. The following assertions hold:

1. (1)

For $r\geq 1$ and $s\geq 0$, we have

[TABLE] 2. (2)

For $r\geq 0$ and $s=1$, we have

[TABLE] 3. (3)

For $r\geq 0$ and $s\geq 2$, we have

[TABLE]

where $\mathfrak{T}(h_{s-1},h_{s},\overline{\mathbf{a}}_{1r})$ is as in Theorem 2.12(3). 4. (4)

Let $R$ be a $k$-subalgebra of $A$. If $R$ is stable under $\rho$ and $f$ takes its values in $R$, then

[TABLE]

for each $r\geq 0$ and $1<l\leq s$.

Proof.

Mimic the proof of Theorem 2.12. ∎

Remark 3.7*.*

By the second equality in (1.3), we have

[TABLE]

This gives an alternative formula for $\overline{d}^{1}(\beta)(\mathbf{y})$ in (3.2). We will use this fact in the proof of Proposition 3.10.

Proposition 3.8**.**

For each $h\in H$ the map F^{*}_{h}\colon\bigl{(}\operatorname{Hom}_{K^{e}}\bigl{(}\overline{A}^{\otimes^{r}},M\bigr{)},b^{*}\bigr{)}\longrightarrow\bigl{(}\operatorname{Hom}_{K^{e}}\bigl{(}\overline{A}^{\otimes^{r}},M\bigr{)},b^{*}\bigr{)}, defined by

[TABLE]

is a morphism of complexes. Moreover the following facts hold

1. (1)

For each $h,l\in H$, the endomorphisms of $\operatorname{H}_{K}^{*}(A,M)$ induced by $F^{*}_{l}\circ F^{*}_{h}$ and $F^{*}_{hl}$ coincide. 2. (2)

F^{r}_{l}(\beta)(\overline{\mathbf{a}}_{1r})=(\beta\cdot l)\bigl{(}\overline{\mathbf{a}}_{1r}\bigr{)}, for all $l\in H^{\!L}$ (see Remark 3.1). In particular $F_{1}^{*}$ is the identity map.

Consequently $\operatorname{H}_{K}^{*}(A,M)$ is a right $H$-module.

Proof.

Mimic the proof of Proposition 2.14. ∎

Example 3.9**.**

If $A=K$, then \operatorname{H}_{K}^{*}(E,M)=\operatorname{H}^{*}\bigl{(}H,M^{\!K}), where $M^{\!K}$ is considered as a right $H$-module via $m\cdot h\coloneqq\gamma(h^{(2)})\cdot m\cdot\gamma^{-1}(h^{(1)})$ (for the well definition use the first identity in (1.8) and Lemma 2.2(4)).

Proposition 3.10**.**

The spectral sequence of [GGV2](4.3) satisfies*

[TABLE]

Proof.

For each $s,n\geq 0$, let $F_{i}(\overline{X}^{n}(M))\coloneqq\bigoplus_{s=i}^{n}\overline{X}^{n-s,s}(M)$. The cochain complex $(\overline{X}^{*}(M),\overline{d}^{*})$ is filtrated by

[TABLE]

Since the isomorphism $\Theta^{*}\colon(\overline{X}^{*}(M),\overline{d}^{*})\to(\widehat{X}^{*}(M),\widehat{d}^{*})$ preserves filtrations, the spectral sequence of [GGV2]*(4.3) coincides with the one determined by the filtration (3.3). Thus, the formula for $E_{1}^{rs}$ follows from Theorem 3.6(1) and the formula for $E_{2}^{rs}$ follows from Theorem 3.6(2), Remark 3.7 and Proposition 1.4(2). ∎

4   The cup and cap products for cleft extensions

Let $H$, $A$, $\rho$, $f$, $E$, $K$, $M$, $\nu$, $\gamma$ and $\gamma^{-1}$ be as in Section 2. Thus $H$ is a weak Hopf algebra, $A$ is a weak module algebra and $f$ is convolution invertible. In this section we obtain formulas involving the complexes $(\overline{X}^{*}(E),\overline{d}^{*})$ and $(\overline{X}_{*}(M),\overline{d}_{*})$ that induce the cup product of $\operatorname{HH}_{K}^{*}(E)$ and the cap product of $\operatorname{H}^{K}_{*}(E,M)$. We will use freely the operators $\bullet$ and $\diamond$ introduced in [GGV2]*Section 5.

Notation 4.1**.**

Given $h_{1},\dots,h_{s}\in H$ and $a_{1},\dots,a_{r}\in A$ we set $\mathrm{h}_{1s}\cdot\overline{\mathbf{a}}_{1r}\coloneqq h_{1}\cdot(h_{2}\cdot(\dots\cdot(h_{s}\cdot\overline{\mathbf{a}}_{1r}))\dots)$.

Definition 4.2**.**

Let $\beta\in\overline{X}^{rs}(E)$ and $\beta^{\prime}\in\overline{X}^{r^{\prime}s^{\prime}}(E)$. We define $\beta\centerdot\beta^{\prime}\in\overline{X}^{r^{\prime\prime},s^{\prime\prime}}(E)$ by

[TABLE]

where $r^{\prime\prime}\coloneqq r+r^{\prime}$, $s^{\prime\prime}\coloneqq s+s^{\prime}$, $h_{1},\dots,h_{s^{\prime\prime}}\in H$, $a_{1},\dots,a_{r^{\prime\prime}}\in A$ and $\mathbf{y}\coloneqq\overline{\mathbf{h}}_{1s^{\prime\prime}}\otimes_{k}\overline{\mathbf{a}}_{1r^{\prime\prime}}$.

Proposition 4.3**.**

If $f$ takes its values in $K$, then the cup product in $\operatorname{HH}_{K}^{*}(E)$ is induced by $\centerdot$.

Proof.

By [GGV2]*Corollary 5.3 it suffices to prove that

[TABLE]

For this, take \mathbf{x}\!\coloneqq\!\widetilde{\gamma}_{A}\bigl{(}\mathrm{h}_{1s^{\prime\prime}})\otimes\overline{\mathbf{a}}_{1,r^{\prime\prime}}, where $r^{\prime\prime}\!\coloneqq\!r+r^{\prime}$, $s^{\prime\prime}\!\coloneqq\!s+s^{\prime}$, $h_{1},\dots,h_{s^{\prime\prime}}\!\in\!H$ and $a_{1},\dots,a_{r^{\prime\prime}}\!\in\!A$. By Lemma 2.6 and the definitions of $\Theta$, $\bullet$ and $\centerdot$,

[TABLE]

as desired. ∎

Definition 4.4**.**

Let $\beta\in\overline{X}^{r^{\prime}s^{\prime}}(E)$ and let $\mathbf{y}\coloneqq\overline{\mathbf{h}}_{1s}\otimes_{H^{\!L}}[m\otimes\overline{\mathbf{a}}_{1r}]\in\overline{X}_{rs}(M)$, where $m\in M$, $a_{1},\dots,a_{r}\in A$ and $h_{1},\dots,h_{s}\in H$. Assume that $r\geq r^{\prime}$ and $s\geq s^{\prime}$. We define $\mathbf{y}\ast\beta$ by

[TABLE]

If $r^{\prime}>r$ or $s^{\prime}>s$, then we set $\mathbf{y}\ast\beta=0$.

Proposition 4.5**.**

If $f$ takes its values in $K$, then in terms of the complexes $(\overline{X}_{*}(M),\overline{d}_{*})$ and $(\overline{X}^{*}(E),\overline{d}^{*})$, the cap product is induced by the operation $\ast$.

Proof.

By [GGV2]*Corollary 5.7 it suffices to prove that

[TABLE]

If $r^{\prime}>r$ or $s^{\prime}>s$, then both sides of this equality are zero. Then we can assume that $r^{\prime}\leq r$, $s^{\prime}\leq s$ and $\mathbf{y}$ is as in Definition 4.4. We have

[TABLE]

as desired. Here the first equality holds by the definition of $\Lambda$; the second one, by the definition of $\diamond$; the third one, by the definition of $\Theta(\beta)$; the fourth one, by Lemma 2.7; the fifth one, since $\beta$ is a right $H^{\!L}$-module morphism and $\gamma(1^{(1)})\gamma(S(1^{(2)}))=1_{E}$; the fifth one, by the definition of $\Lambda$ and Lemma 2.6; and the last one, by the definition of $\ast$. ∎

5   Cyclic homology of cleft extensions

Let $H$, $A$, $\rho$, $f$, $E$, $K$, $M$, $\nu$, $\gamma$ and $\gamma^{-1}$ be as in Section 2. Thus $H$ is a weak Hopf algebra, $A$ is a weak module algebra and $f$ is convolution invertible. In this section we prove that the mixed complex $(\widehat{X}_{*}(E),\widehat{d}_{*},\widehat{D}_{*})$ of [GGV2]*Section 6 is isomorphic to a simpler mixed complex $(\overline{X}_{*}(E),\overline{d}_{*},\overline{D}_{*})$. Let

[TABLE]

be the maps introduced in the proof of Theorem 2.9. For each $n\!\geq\!0$, let $\overline{D}_{n}\colon\overline{X}_{n}(E)\to\overline{X}_{n+1}(E)$ be the map defined by $\overline{D}_{n}\coloneqq\Theta_{n+1}\circ\widehat{D}_{n}\circ\Lambda_{n}$.

Theorem 5.1**.**

The triple \bigl{(}\overline{X}_{*}(E),\overline{d}_{*},\overline{D}_{*}\bigr{)} is a mixed complex that gives the Hochschild and that cyclic type homologies of the $K$-algebra $E$. More precisely, the mixed complexes \bigl{(}\overline{X}_{*}(E),\overline{d}_{*},\overline{D}_{*}\bigr{)} and \cramped{\bigl{(}E\otimes\overline{E}^{\otimes^{*}},b_{*},B_{*}\bigr{)}} are homotopically equivalent.

Proof.

Since $\Theta_{*}$ and $\Lambda_{*}$ are inverse one of each other it is clear that \bigl{(}\overline{X}_{*}(E),\overline{d}_{*},\overline{D}_{*}\bigr{)} is a mixed complex and that \Theta_{*}\colon\bigl{(}\widehat{X}_{*}(E),\widehat{d}_{*},\widehat{D}_{*}\bigr{)}\to\bigl{(}\overline{X}_{*}(E),\overline{d}_{*},\overline{D}_{*}\bigr{)} is an isomorphism of mixed complexes. Consequently the result follows from [GGV2]*Theorem 6.3. ∎

Definition 5.2**.**

For each $r,s\geq 0$, let $\widehat{D}^{0}_{rs}\colon\widehat{X}_{rs}(E)\to\widehat{X}_{r,s+1}(E)$ and $\widehat{D}^{1}_{rs}\colon\widehat{X}_{rs}(E)\to\widehat{X}_{r+1,s}(E)$ be the maps introduced in [GGV2]*Definition 6.5. We define $\overline{D}^{0}_{rs}\colon\overline{X}_{rs}(E)\to\overline{X}_{r,s+1}(E)$ and $\overline{D}^{1}_{rs}\colon\overline{X}_{rs}(E)\to\overline{X}_{r+1,s}(E)$ by $\overline{D}^{0}_{rs}\coloneqq\Theta_{r,s+1}\circ\widehat{D}^{0}_{rs}\circ\Lambda_{rs}$ and $\overline{D}^{1}_{rs}\coloneqq\Theta_{r+1,s}\circ\widehat{D}^{1}_{rs}\circ\Lambda_{rs}$, respectively.

Proposition 5.3**.**

let $R$ be a subalgebra of $A$ and let $\mathbf{y}\coloneqq\overline{\mathbf{h}}_{1s}\otimes_{H^{\!L}}[a_{0}\cdot\gamma(h_{0})\otimes\overline{\mathbf{a}}_{1r}]\in\overline{X}_{rs}(E)$. If $R$ is stable under $\rho$ and $f$ takes its values in $R$, then $\overline{D}(\mathbf{y})=\overline{D}^{0}(\mathbf{y})+\overline{D}^{1}(\mathbf{y})$ module $\bigoplus_{s=0}^{i}\overline{X}^{1}_{s,n+1-s}(R,M)$.

Proof.

This follows by Remark 1.13 and [GGV2]*Proposition 6.6(1). ∎

Corollary 5.4**.**

If $f$ takes its values in $K$, then $\overline{D}=\overline{D}^{0}+\overline{D}^{1}$.

We next compute the maps $\overline{D}^{0}$ and $\overline{D}^{1}$. We will need the following proposition.

Notation 5.5**.**

Given $h_{1},\dots,h_{s}\in S$ and $1\leq j\leq s$, we set $S_{{}_{\!\times}}(\mathrm{h}_{1s})\coloneqq S(h_{s})S(h_{s-1})\cdots S(h_{1})$.

Proposition 5.6**.**

For each $s\in\mathds{N}$ there exists a map $T_{s}\colon H^{\otimes_{k}^{s+1}}\to A$ such that

[TABLE]

Proof.

We will proceed by an inductive argument. Assume first that $s=1$. By equality (1.11), the first identity in (1.8) and Theorem 1.5(d), we have

[TABLE]

So, we have T_{1}(h_{01})=\bigl{(}h_{0}^{(1)}\!\cdot\!f^{-1}\bigl{(}S(h_{1}^{(2)})\otimes_{k}h_{1}^{(3)}\bigr{)}\bigr{)}f\bigl{(}h_{0}^{(2)}\otimes_{k}S(h_{1}^{(1)})\bigr{)}. The first equality in the statement follows immediately from the equality in Theorem 1.5(1). Assume now that $s\geq 1$ and there exists $T_{s}$ as in the statement. Then

[TABLE]

which provides the recursive step. The same argument as above proves the second equality in the statement. ∎

Remark 5.7*.*

By Theorem 1.5(d), Propositions 1.9(3) and 5.6, and [GGV1]*Propositions 4.5 and 5.19,

[TABLE]

Since $(A\otimes\epsilon)\circ\jmath_{\nu}=\operatorname{id}_{A}$, this implies that

[TABLE]

Proposition 5.8**.**

For $\mathbf{y}\coloneqq\overline{\mathbf{h}}_{1s}\otimes_{H^{\!L}}[\jmath_{\nu}(a_{0})\gamma(h_{0})\otimes\overline{\mathbf{a}}_{1r}]\in\overline{X}_{rs}(E)$, we have

[TABLE]

where $\alpha_{jrs}\coloneqq(-1)^{js+r+s}$ and $\beta_{jr}\coloneqq(-1)^{jr+r}$.

Proof.

By the definition of $\Lambda$ and the first equality in Proposition 5.6,

[TABLE]

So

[TABLE]

where $\alpha^{\prime}_{jrs}\coloneqq(-1)^{rs+js+s}$ and $\beta^{\prime}_{jrs}\coloneqq(-1)^{rs+jr+r+s}$. Now, we have

[TABLE]

where the first equality holds by the definition on $\Theta$ and the first identity in Proposition 5.6; the second one, by Lemma 2.7; the third one, by the definition of the action in (2.4) and [GGV1]*Proposition 2.22(2); and the fourth one, since $\gamma(1^{(1)}S(1^{(2)}))=\gamma(1)=1$. Similarly

[TABLE]

From these facts it follows that the formulas in the statement are true. ∎

Index of symbols

• $\overline{A}$ §2

• $\overline{a}$ item (1)

• $\mathbf{a}_{ij}$ item (2)

• $\overline{\mathbf{a}}_{ij}$ item (2)

• $b$ §1.5

• $B$ §1.5

• $\widehat{d}^{l}_{rs}$ §2

• $\widehat{d}_{n}$ §2

• $\overline{d}^{l}_{rs}$ §2.2

• $\overline{d}_{n}$ Theorem 2.9

• $\widehat{d}_{l}^{rs}$ §3

• $\widehat{d}^{n}$ §3

• $\overline{d}_{l}^{rs}$ §3

• $\overline{d}^{n}$ Theorem 3.3

• $\widehat{D}_{*}$ §5

• $\overline{D}_{n}$ §5

• $\widehat{D}^{0}_{rs}$ Definition 5.2

• $\overline{D}^{0}_{rs}$ Definition 5.2

• $\widehat{D}^{1}_{rs}$ Definition 5.2

• $\overline{D}^{1}_{rs}$ Definition 5.2

• $\overline{E}$ §2

• $\widetilde{E}$ §2

• $\overline{x}$ item (1)

• $\widetilde{x}$ item (1)

• $f$ §1.3

• $\mathcal{F}_{f}$ §1.3

• $f^{-1}$ §1.4.1

• $F_{*}^{h}$ Proposition 2.14

• $F^{i}(\overline{X}_{n}(M))$ §2.2

• $F_{i}(\overline{X}^{n}(M))$ §3

• $H^{L}$ §1.1

• $H^{R}$ §1.1

• $\overline{H}$ item (1)

• $\operatorname{HC}^{K}_{*}(C)$ §1.5

• $\operatorname{HN}^{K}_{*}(C)$ §1.5

• $\operatorname{HP}^{K}_{*}(C)$ §1.5

• §1.5

• $\operatorname{H}^{K}_{*}(E,M)$ §2

• §3

• $\overline{h}$ item (1)

• $\overline{\mathbf{h}}_{1s}$ item (2)

• $\mathfrak{h}_{1s}$ item (3)

• $\mathrm{h}_{1s}$ item (3)

• $\mathrm{h}_{1s}^{(1)}\otimes_{k}\mathrm{h}_{1s}^{(2)}$ item (4)

• $\overline{\mathbf{h}}_{1s}^{(1)}\otimes_{k}\overline{\mathbf{h}}_{1s}^{(2)}$ item (4)

• $h\cdot\overline{\mathbf{a}}_{1r}$ item (5)

• $\mathbf{h}_{1s}\cdot\overline{\mathbf{a}}_{1r}$ Notation 4.1

• $M\otimes$ Notation 1.20

• $[m]$ Notation 1.20

• $S$ §1.1

• $S_{{}_{\times}}(\mathrm{h}_{1s})$ Notation 5.5

• $\widehat{X}_{rs}(M)$ §2

• $\overline{X}_{rs}(M)$ §2.2

• $\widehat{X}_{n}(M)$ §2

• $\overline{X}_{n}(M)$ Theorem 2.9

• $\widehat{X}^{rs}(M)$ §3

• $\overline{X}^{rs}(M)$ §3

• $\widehat{X}^{n}(M)$ §3

• $\overline{X}^{n}(M)$ Theorem 3.3

• $\overline{X}^{u}_{rs}(R,M)$ Notation 2.11

• $\overline{X}_{u}^{rs}(R,M)$ Notation 3.5

• $\mathcal{X}$ §1.5

• $\operatorname{BC}(\mathcal{X})$ §1.5

• $\operatorname{BN}(\mathcal{X})$ §1.5

• $\operatorname{BN}^{\prime}(\mathcal{X})$ §1.5

• $\operatorname{BP}(\mathcal{X})$ §1.5

• $\operatorname{HP}_{*}(\mathcal{X})$ §1.5

• $\operatorname{HC}_{*}(\mathcal{X})$ §1.5

• $\operatorname{HH}_{*}(\mathcal{X})$ §1.5

• $\operatorname{HN}_{*}(\mathcal{X})$ §1.5

• $\delta_{E}$ Proposition 1.7

• $\gamma$ §1.3

• $\overline{\gamma}$ item (6)

• $\widetilde{\gamma}$ item (6)

• $\gamma^{-1}$ §1.4.1

• $\widetilde{\gamma}_{A}(\mathrm{h}_{1s})$ item (7)

• $\gamma_{{}_{\times}}(\mathrm{h}_{1s})$ item (8)

• $\gamma_{{}_{\times}}^{-1}(\mathrm{h}_{1s})$ item (9)

• $\Lambda_{rs}$ Proposition 2.8, Proposition 3.2

• $\nu$ §1.3

• $\jmath_{\nu}$ §1.3

• $\Pi^{L}$ §1.1

• $\Pi^{R}$ §1.1

• $\overline{\Pi}^{L}$ §1.1

• $\overline{\Pi}^{R}$ §1.1

• $\rho$ §1.3

• $\chi_{\rho}$ §1.3

• $\nabla_{\rho}$ §1.3

• $\Theta_{rs}$ Proposition 2.8

• $\Theta^{rs}$ Proposition 3.2

• $a\times h$ §1.3

• $\centerdot$ Definition 4.2

• $\ast$ Definition 4.4