HOMOLOGY OF FULLY GRADED ALGEBRAS,

MORITA AND DERIVED EQUIVALENCES

Andrei Marcus “Babeş-Bolyai” University Faculty of Mathematics and Computer Science Str. Mihail Kogălniceanu nr. 1 RO-3400 Cluj-Napoca Romania

1. Introduction

Let k be a commutative ring, G a group and R a fully G graded k - algebra. By definition, R has a decomposition R = ⨁_{x∈G} R_{g} such that R_{g}R_{h} = R_{gh} for all g, h ∈ G. These algebras are also called “strongly graded” in literature, and we refer to [D] for a dicussion on terminology. There are spectral sequences converging to the Hochschild and the cyclic (co)homology of R (see for instance [L], [C], [S], [GJ] and the references given there, and also [St] for the more general case of Hopf-Galois extensions).

It is also well-known that the Hochschild and the cyclic homology are invariant under Morita equivalence. Dealing with group-graded algebras, one can consider Morita equivalences induced by graded bimodules (see [Bo] and [M, Section 3]). The principal aim of this paper is to show that if the fully G-graded k -algebras R and S are graded Morita equivalent, then the isomorphisms between their Hochschild and cyclic homologies are induced by isomorphisms between the second pages of the corresponding spectral sequences.

The paper is organized as follows. In Section 2 we recall some facts concerning modules graded by G-sets and G-graded bimodules. The main result of Section 3 is the invariance of the above mentioned spectral sequences. Also, our treatment of these spectral sequences differs from those of [L] and [S].

J. Rickard has proved in [R] that two derived equivalent k -algebras have isomorphic Hochschild cohomology. We have studied in [M] derived equivalences induced by complexes of graded bimodules, and in Section 4 below we show that Rickard’s result can be refined as in the case of Morita equivalences. Finally, in Section 5 we discuss the graded Morita invariance of cyclic homology.

In this paper, rings will be always associative with unity, modules are left and unitary, and an (R, S)-bimodule is a module over R ⊗_{k} S^{op}. We refer to [D] and [NRV] for general results on group-graded algebras and their modules, and to [Be] for basic facts concerning Hochschild and cyclic homology. We denote by R-Mod ( Mod-R ) the category of R-modules (right R-modules) and by R-Mod-S the category of (R, S)-bimodules.

2. Graded modules and graded Morita equivalences

Let R = ⨁_{x∈G} R_{g} be a fully G-graded k -algebra as in the introduction. If H is a subgroup of G, R_{H} = ⨁_{h∈H} R_{h} is a subring of R and we denote by (G/H, R)-Gr the category of G/H -graded R-modules and grade-preserving R-homomorphisms. It is well-known that the functors R⊗_{R_{H}} −: R_{H}-Mod → (G/H, R)-Gr and (−)_{H}: (G/H, R)-Gr → R_{H}-Mod are inverse equivalences of categories.

If M and N are R-modules, then Hom_{R_{1}}(M, N) is a kG-module with

gf(m) = ∑_{i=1}^{λ} r_{i}f(r′_{i}m),

where f ∈ Hom_{R_{1}}(M, N), g ∈ G, m ∈ M and r_{i} ∈ R_{g}, r′_{i} ∈ R_{g^{−1}} are such that ∑_{i=1}^{λ} r_{i}r′_{i} = 1. Also, Hom_{R_{1}}(M, N)^{H} = Hom_{R_{H}} (M, N) for every subgroup H of G. This action is natural, that is, Hom_{R_{1}}(−, −) is a bifunctor from R-Mod × R-Mod to kG-Mod.

Similarly, we have a functor − ⊗_{R_{1}} −: Mod-R × R-Mod → kG-Mod. If M is a right R-module and N ∈ R-Mod, then

g(m ⊗_{R_{1}} n) = ∑_{j=1}^{ν} mρ′_{j} ⊗_{R_{1}} ρ_{j}n,

where ρ′_{j} ∈ R_{g^{−1}} and ρ_{j} ∈ R_{g} are such that ∑_{j=1}^{ν} ρ′_{j}ρ_{j} = 1. It can be easily seen that taking the coinvariants, we have

(M ⊗_{R_{1}} N)_{H} ≃ k ⊗_{kH} (M ⊗_{R_{1}} N) ≃ M ⊗_{R_{H}} N

for each subgroup H of G.

If S is another fully G-graded k -algebra, then S^{op} is also G-graded with components S_{g}^{op} = S_{g^{−1}} , and A(R, S) = R ⊗_{k} S^{op} is G × G-graded with A(R, S)_{(g,h)} = R_{g} ⊗_{k} S_{h}^{op}. Let δ(G) = {(g, g) | g ∈ G} and Δ(R, S) = A(R, S)_{δ(G)} = ⨁_{x∈G}(R_{g} ⊗_{k} S_{g^{−1}}), which is again a fully G-graded k -algebra with Δ(R, S)_{g} = R_{g}⊗_{k}S_{g}^{op}; we denote Δ(R) = Δ(R, R) and R^{e} = A(R, R) = R ⊗_{k} R^{op}.

An (R, S)-bimodule M is G-graded if M = ⨁_{x∈G} M_{g} and R_{g}M_{x}S_{h} = M_{gxh} for all g, h, x ∈ G. Then M can be obviously regarded as a G × G/δ(G)-graded A(R, S)-module, where the δ(G)-component of M is M_{1} . Therefore, the category R-Gr-S of G-graded (R, S)-bimodules is equivalent with the category of Δ(R, S)-modules via the functors (−)_{1} = (−)_{δ(G)} and A(R, S) ⊗_{Δ(R,S)} − = Ind_{δ(G)}^{G×G}(−) (see [M, Section 2]). In particular, we have that R_{1} is a Δ(R)-module and R ≃ R^{e} ⊗_{Δ(R)} R_{1} as R^{e} -modules.

Morita equivalences induced by graded bimodules have the feature that they induce Morita equivalences for each subgroup of G. More precisely, we have:

2.1. Theorem. ([M, Theorem 3.4]) The following statements are equivalent:

(i) There is a G-graded (R, S)-bimodule P, a G-graded (S, R)-bimodule Q, an isomorphism α : P ⊗_{S} Q → R of G-graded (R, R)-bimodules and an isomorphism β: Q ⊗_{R} P → S of G-graded (S, S)-bimodules.

(i) There is a Δ(R, S)-module P_{1} , a Δ(S, R)-module Q_{1} , an isomorphism α_{1} : P_{1} ⊗_{S_{1}} Q_{1} → R_{1} of Δ(R)-modules and an isomorphism β_{1} : Q_{1} ⊗_{R_{1}} P_{1} → S_{1} of Δ(S)-modules.

2.2. If these conditions hold, we say that R and S are graded Morita equivalent, and then for each subgroup H of G, the categories (G/H, R)-Gr and (G/H, S)-Gr are equivalent. Also, the algebras R_{H} and S_{H} are graded Morita equivalent. Moreover, if M, N ∈ R-Mod, then Hom_{R_{1}}(M, N) and Hom_{S_{1}}(Q ⊗_{R} M, Q ⊗_{R} N) are isomorphic kG-modules.

For tensor products, we similarly have that if M ∈ Mod-R and N ∈ R-Mod, then M ⊗_{R_{1}} N and (M ⊗_{R} P) ⊗_{S_{1}} (Q ⊗_{R} N) are isomorphic kGmodules.

Indeed, by [M, Lemma 2.6], P ≃ R⊗_{R_{1}}P_{1} and Q ≃ Q_{1}⊗_{R_{1}}R as bimodules, hence (M ⊗_{R} P) ⊗_{S_{1}} (Q ⊗_{R} N) ≃ (M ⊗_{R_{1}} P_{1}) ⊗_{S_{1}} (Q_{1} ⊗_{R_{1}} N). Let

(M ⊗_{R_{1}} P_{1}) ⊗_{S_{1}} (Q_{1} ⊗_{R_{1}} N) → M ⊗_{R_{1}} N,
(m ⊗_{R_{1}} p) ⊗_{S_{1}} (q ⊗_{R_{1}} n) ↦ mα_{1}(p ⊗_{S_{1}} q) ⊗_{R_{1}} n.

Since α_{1} is a Δ(R)-isomorphism, one can calculate using the definitions given in [M, Section 2], that ϕ is a kG-isomorphism. 2.3. Let P and Q be G-graded bimodules as in the theorem. Then P ⊗_{k} Q is a G × G-graded (R^{e}, S^{e})-bimodule (with components (P ⊗_{k} Q)_{(g,h)} = P_{g} ⊗_{k} Q_{h}) and Q⊗_{k} P is a G×G-graded (S^{e}, R^{e})-bimodule. It can be easily seen that (P ⊗_{k} Q, Q ⊗_{k} P) induce a graded Morita equivalence between R^{e} and S^{e}, and that R and S correspond under this equivalence.

3. Hochschild homology of fully graded algebras

In this section we deduce the spectral sequences of [L] and [S] by a slightly different approach and we show that these spectral sequences are graded Morita invariants.

Let R be a fully G-graded k -algebra. For simplicity, we assume that R_{1} , and hence R is a projective k -module. Let also M be an (R, R)-bimodule, that is, a module over R^{e} = R ⊗_{k} R^{op}. 3.1. Cohomology. Since both R_{1} and M can be regarded as Δ(R)- modules, we have that Hom_{R_{1}^{e}}(R_{1}, M) = Hom_{Δ(R)_{1}}(R_{1}, M) is a kG-module. Further

Hom_{kG}(k, Hom_{R_{1}^{e}}(R_{1}, M)) = Hom_{R_{1}^{e}}(R_{1}, M)^{G}

= Hom_{Δ(R)}(R_{1}, Res_{δ(G)}^{G×G}(M))

= Hom_{R^{e}}(Ind_{δ(G)}^{G×G}(R_{1}), M) = Hom_{R^{e}}(R, M).

If we consider the functors

F_{1} = Hom_{kG}(k, −): kG-Mod → k-Mod,
F_{2} = Hom_{R_{1}^{e}}(R_{1}, −): R^{e}-Mod → kG-Mod

and

F = Hom_{R^{e}}(R, −): R-Mod → k-Mod,

it follows that F = F_{1} ∘ F_{2} .

To obtain an action of G on H^{n}(R_{1}, M) = Ext_{R_{1}^{e}}^{n}(R_{1}, M), let P_{•} be a projective resolution of R_{1} regarded as a Δ(R)-module. (Note that the terms of the acyclic Hochschild complex are not, in general, Δ(R)-modules.) Then P_{•} is also a projective resolution of R_{1} regarded as an R_{1}^{e} -module, and Hom_{R_{1}^{e}}(P_{•}, M) is a complex of kG-modules; its cohomology is the ℤgraded kG-module H^{•}(R_{1}, M). (The same could be obtained using an injective resolution of M regarded as a Δ(R)-module.) Moreover, the argument of [St, Proposition 3.2] shows that if M is an injective R^{e} -module, then Ext_{kG}^{q}(k, F_{2}(M)) = 0 for q ≥ 1. We conclude that there is a Grothendieck spectral sequence

H^{p}(G, H^{q}(R_{1}, M)) ⇒ H^{p+q}(R, M).

3.2. Decomposition. Assume in addition that M = ⨁_{x∈G} M_{g} is a Ggraded (R, R)-bimodule, so M ≃ R^{e} ⊗_{Δ(R)} M_{1} = Ind_{δ(G)}^{G×G}(M_{1}). For g ∈ G, denote by [g] the conjugacy class of g, and let T (G) = {[g] | g ∈ G} be the set of conjugacy classes.

Since Mackey’s formula holds for fully graded algebras (see [D, Section 3]) we have a decomposition of Hom_{R^{e}}(R, M) as follows:

Hom_{R^{e}}(R, M) = Hom_{R^{e}}(Ind_{δ(G)}^{G×G}(R_{1}), Ind_{δ(G)}^{G×G}(M_{1}))

≃ Hom_{Δ(R)}(R_{1}, Res_{δ(G)}^{G×G} Ind_{δ(G)}^{G×G}(M_{1}))

≃ ⨁_{[g]∈T (G)} Hom_{Δ(R)}(R_{1}, Ind_{δC_{G}(g)}^{δ(G)}(M_{g}))

≃ ⨁_{[g]∈T (G)} Hom_{Δ(R)_{C_{G}(g)}}(R_{1}, M_{g}),

where Δ(R)_{C_{G}(g)} = ⨁_{x∈C_{G}(g)}(R_{x} ⊗_{k} R_{x}^{op}). We denote

Hom_{R^{e}}(R, M)_{[g]} = Hom_{Δ(R)_{C_{G}(g)}}(R_{1}, M)

and

H^{n}(R, M)_{[g]} = Ext_{Δ(R)_{C_{G}(g)}}^{n}(R_{1}, M_{g}).

Since R_{1} and M_{g} are Δ(R)_{C_{G}(g)} -modules, Hom_{Δ(R)_{1}}(R_{1}, M_{g}) is a kC_{G}(g)- module, and

Hom_{kC_{G}(g)}(k, Hom_{Δ(R)_{1}}(R_{1}, M_{g})) = Hom_{Δ(R)_{1}}(R_{1}, M_{g})^{C_{G}(g)}
≃ Hom_{Δ(R)_{C_{G}(g)}}(R_{1}, M_{g})
≃ Hom_{R^{e}}(R, M)_{[g]}.

As above, we have a spectral sequence

H^{p}(C_{G}(g), H^{q}(R_{1}, M_{g})) ⇒ H^{p+q}(R, M)_{[g]}.

3.3. Homology. Let M be an (R, R)-bimodule regarded as a right R⊗_{k}R^{op} - module. Regarding R_{1} as a Δ(R)-module, we see that M ⊗_{R_{1}^{e}} R_{1} = M ⊗_{Δ(R_{1})} R_{1} is a kG-module as in Section 2, and

k ⊗_{kG} (M ⊗_{Δ(R)_{1}} R_{1}) = M ⊗_{Δ(R)} R_{1} = Res_{δ(G)}^{G×G}(M) ⊗_{Δ(R)} R_{1}
≃ M ⊗_{R^{e}} Ind_{δ(G)}^{G×G}(R_{1}) ≃ M ⊗_{R^{e}} R.

Similarly to 3.1, we have a spectral sequence

H_{p}(G, H_{q}(R_{1}, M)) ⇒ H_{p+q}(R, M).

Moreover, if M is a G-graded bimodule, then

M ⊗_{R^{e}} R ≃ Ind_{δ(G)}^{G×G}(M_{1}) ⊗_{R^{e}} Ind_{δ(G)}^{G×G}(R_{1})

≃ Res_{δ(G)}^{G×G} Ind_{δ(G)}^{G×G}(M_{1}) ⊗_{Δ(R)} R_{1}

≃ ⨁_{[g]∈T (G)} Ind_{δC_{G}(g)}^{δ(G)}(M_{g}) ⊗_{Δ(R)} R_{1}

≃ ⨁_{[g]∈T (G)} M_{g} ⊗_{Δ(R)_{C_{G}(g)}} R_{1}.

Denote (M ⊗_{R^{e}} R)_{[g]} = M_{g} ⊗_{Δ(R)_{C_{G}(g)}} R_{1} and

H_{n}(R, M)_{[g]} = Tor_{n}^{Δ(R)_{C_{G}(g)}}(M_{g}, R_{1}).

As in 3.3, we have a spectral sequence

H_{p}(C_{G}(g), H_{q}(R_{1}, M_{g})) ⇒ H_{p+q}(R, M)_{[g]}.

3.4. Morita invariance. Let S be another G-graded k -algebra, projective over k, and assume that (P, Q) induce a graded Morita equivalence between R and S, where P is a G-graded (R, S)-bimodule and Q is a G-graded (S, R)-bimodule. Then the functors

F = Q ⊗_{R} (−) ⊗_{R} P: R-Mod-R → S-Mod-S

and

G = P ⊗_{S} (−) ⊗_{S} Q: S-Mod-S → R-Mod-R

are inverse equivalences; in fact, the G × G-graded k -algebras R^{e} and S^{e} are graded Morita equivalent and F(R) = S. We shall prove that the above spectral sequences are isomorphic. More precisely, we have

3.5. Theorem. a) If M is an (R, R)-bimodule, then

H^{p}(G, H^{q}(R_{1}, M)) ≃ H^{p}(G, H^{q}(S_{1}, F(M)))

and

H_{p}(G, H_{q}(R_{1}, M)) ≃ H_{p}(G, H_{q}(S_{1}, F(M))).

b) If M is a G-graded (R, R)-bimodule, then for all g ∈ G,

H^{p}(C_{G}(g), H^{q}(R_{1}, M_{g})) ≃ H^{p}(C_{G}(g), H^{q}(S_{1}, F_{1}(M_{g})))

and

H_{p}(C_{G}(g), H_{q}(R_{1}, M_{g})) ≃ H_{p}(C_{G}(g), H_{q}(S_{1}, F_{1}(M_{g}))),

where F_{1} = Q_{1} ⊗_{R_{1}} (−) ⊗_{R_{1}} P_{1} : R_{1}-Mod-R_{1} → S_{1}-Mod-S_{1} is the equivalence induced by F.

Proof. a) It is enough to show that H^{q}(R_{1}, M) and H^{q}(S_{1}, F(M)) are isomorphic kG-modules. Since H^{q}(R_{1}, M) = Ext_{R_{1}^{e}}^{q}(R_{1}, M), it is in fact enough to show that

Hom_{R_{1}^{e}}(R_{1}, M) ≃ Hom_{R_{1}^{e}}(S_{1}, F(M))

as kG-modules. Recall from 2.2 and 2.3 that F and G induce a graded Morita equivalence between Δ(R)-Mod and Δ(S)-Mod. Moreover, a graded Morita equivalence commutes with scalar restriction, so the image of Res_{δ(G)}^{G×G} is Res_{δ(G)}^{G×G} F(M). It follows by 2.2 that Hom_{R_{1}^{e}}(R_{1}, M) ≃ Hom_{R_{1}^{e}}(S_{1}, F(M)) in kG-Mod.

The same argument works in the case of homology.

b) Again it’s enough to see that Hom_{R_{1}^{e}}(R_{1}, M_{g}) and Hom_{S_{1}^{e}}(S_{1}, F_{1}(M_{g})) are isomorphic kC_{G}(g)-modules. By 2.2 and 2.3 we know that F and G induce a graded Morita equivalence between R_{C_{G}(g)}^{e} and S_{C_{G}(g)}^{e}, and then we only need to use the last part of 2.2 with C_{G}(g) instead of G and Δ(R)_{C_{G}(g)} instead of R.

3.6. Remark. It is not difficult to see that the isomorphism

H_{•}(R, M) ≃ H_{•}(S, Q ⊗_{R} M ⊗_{R} P)

induced by the Morita equivalence between R and S is the same as the isomorphism coming from the spectral sequence and the isomorphism

H_{•}(G, H_{•}(R_{1}, M)) ≃ H_{•}(G, H_{•}(S_{1}, Q_{1} ⊗_{R_{1}} M ⊗_{R_{1}} P_{1})).

Actually. one can argue in the general situation of the Grothendieck spectral sequence. Assume that we have the commutative diagram of abelian categories

A ⟶^{F} B ⟶^{G} C
S↓  ↓^{T}  ↓^{U}
A′ ⟶^{F′} B′ ⟶^{G′} C′

where F, F′, G, G′ are right exact functors and S, T , U are equivalences. If A, A′, B, B′ have enough projectives then the following diagrams are also commutative

\begin{matrix}A & ⟶^{L_{q}F} & B & ⟶^{L_{p}G} & C \\ ^{S}↓ & & ↓^{T} & & ↓^{U} \\ A′ & ⟶^{L_{q}F′} & B′ & ⟶^{L_{p}G′} & C′ \end{matrix}  \begin{matrix}A & ⟶^{L_{n}(G∘F)} & C \\ ^{S}↓ & & ↓^{U} \\ A′ & ⟶^{L_{n}(G′∘F′)} & C′ \end{matrix}

If F (respectively F′) send projectives to G -acyclic (respectively G′ -acyclic) objects, then for A ∈ A and A′ ∈ A′ there are spectral sequences

E_{pq}^{2}(A) = ((L_{p}G) ∘ (L_{q}F))(A) ⇒ L_{p+q}(G ∘ F)(A),
E_{pq}^{2}(A′) = ((L_{p}G′) ∘ (L_{q}F′))(A′) ⇒ L_{p+q}(G′ ∘ F′)(A′),

The isomorphism between U(E_{pq}^{2}(A)) and E′_{pq}^{2}(S(A)) induces the isomorphism U(E_{pq}^{∞}(A)) ≃ E′_{pq}^{∞}S(A)) which in turn, induces an isomorphism between U(L_{n}(G∘F)(A)) and L_{n}(G′∘F′)(S(A)). By inspecting the construction ofthe above spectral sequences, one can observe that this isomorphism is the same as the isomorphism induced by the commutativity of the third diagram.

4. Derived equivalences

Let again R and S be fully G-graded algebras, projective over k, and denote by D^{b}(A) the derived category of bounded complexes of objects of the abelian category A. By [R, Proposition 2.5], if R and S are derived equivalent, then HH^{•}(R) ≃ HH^{•}(S). Considering now derived equivalences induced by G-graded bimodules, we have

4.1. Theorem. ([M, Theorem 4.7]) The following statements are equivalent:

i) There are objects C ∈ D^{b}(R-Gr-S) and D ∈ D^{b}(S-Gr-R) inducing a derived equivalence between R and S such that α: C ⊗^{L}_{S} D → R and β: D⊗^{L}_{R} C → S are isomorphisms in D^{b}(R-Gr-R), respectively in D^{b}(S-Gr-S).

ii) There are objects C_{1} ∈ D^{b}(Δ(R, S)-Mod), D_{1} ∈ D^{b}(Δ(S, R)-Mod), and isomorphisms α_{1} : C_{1} ⊗^{L}_{S_{1}} D_{1} → R_{1} in D^{b}(Δ(R)-Mod) and β_{1} : D_{1} ⊗^{L}_{R_{1}} C_{1} → S_{1} in D^{b}(Δ(S)-Mod).

In this case we say that R and S are graded derived equivalent, and it follows that for each subgroup H of G, the categories D^{b}((G/H, R)-Gr) and D^{b}((G/H, S)-Gr), respectively D^{b}(R_{H}-Mod) and D^{b}(S_{H}-Mod) are equivalent.

Moreover, if X, Y ∈ D^{b}(R-Mod) then

RHom_{R_{1}}(X, Y) ≃ RHom_{S_{1}}(C_{1} ⊗^{L}_{R_{1}} X, C_{1} ⊗^{L}_{R_{1}} Y)

in D^{b}(kG-Mod).

Again by [R, Proposition 2.5], C ⊗_{k} D and D ⊗_{k} C are tilting complexes inducing an equivalence between D^{b}(R^{e}-Mod) and D^{b}(S^{e}-Mod) such that RR_{R} corresponds to _{S}S_{S}.

Further, C ⊗_{k} D and D ⊗_{k} C are complexes of G × G-graded bimodules,

(C_{1} ⊗_{k} D_{1}) ⊗_{S_{1}} (D_{1} ⊗_{k} C_{1}) ≃ R_{1}^{e} in D^{b}(Δ(R)-Mod)

and

(D_{1} ⊗_{k} C_{1}) ⊗_{S_{1}} (C_{1} ⊗_{k} D_{1}) ≃ S_{1}^{e} in D^{b}(Δ(S)-Mod)

Since by [H, Theorem 6.4],

HH^{n}(R_{1}) = Ext_{R_{1}^{e}}^{n}(R_{1}, R_{1}) = H^{n}(RHom(R_{1}, R_{1})),

as in Theorem 3.6 we may conclude

4.2. Corollary. If R and S are graded derived equivalent, then

H^{p}(G, HH^{q}(R_{1})) ≃ H^{p}(G, HH^{q}(S_{1})).

5. Cyclic homology of graded algebras

We continue to use the notations introduced in Section 3. By [C, Theorem 3.8], there is a spectral sequence

(E_{p,q}^{2})_{[1]} = H_{p}(G, HC_{q}(R_{1})) ⇒ HC_{n}(R)_{[1]}.

The decomposition of HC_{n}(R) is deduced in this case from a decomposition of the acyclic Hochschild complex. Our result of this section is

5.1. Theorem. If R and S are graded Morita equivalent, then

H_{p}(G, HC_{q}(R_{1})) ≃ H_{p}(G, HC_{q}(S_{1})).

Proof. We also need to use the acyclic Hochschild complex as in [L] in order to prove that HC_{q}(R_{1}) and HC_{q}(S_{1}) are isomorphic kG-modules. Let P and Q be as in Theorem 2.1 and choose dual bases {p_{i}} ⊂ P_{1} and {q_{i}} ⊂ Q_{1} , so that α_{1}(∑_{i=1}^{μ} p_{i} ⊗_{S_{1}} q_{i}) = 1. The one can define the chain map

ϕ_{n}: R_{1}^{⊗(n+1)} → S_{1}^{⊗(n+1)},

a_{0}⊗…⊗a_{n} ↦ ∑_{(i_{0},...,i_{n})}β_{1}(q_{i_{0}} ⊗_{R_{1}} a_{0}p_{i_{1}})⊗β_{1}(q_{i_{1}} ⊗_{R_{1}} a_{1}p_{i_{2}})⊗…⊗β_{1}(q_{i_{n}} ⊗_{R_{1}}

a_{n}p_{i_{0}});

ϕ_{n} commutes with the Hochschild boundary b and induces an isomorphism between HH_{•}(R_{1}) and HH_{•}(S_{1}); ϕ_{n} also commutes with the operations t_{n} and b′ and induces an isomorphism ϕ_{•}: HC_{•}(R_{1}) → HC_{•}(S_{1}), which is independent of the choice of the dual bases (see [Be, Theorem 2.13.2]).

For every g ∈ G, the (R_{1}, R_{1})-bimodules R_{g} and R_{g^{−1}} induce an autoequivalence of R_{1}-Mod. Let r_{j} ∈ R_{g}, r′_{j} ∈ R_{g^{−1}} be such that ∑_{j=1}^{λ} r_{j}r′_{j} = 1, and define the chain map

ϕ_{n}^{g}: R_{1}^{⊗(n+1)} → R_{1}^{⊗(n+1)},

a_{0} ⊗ … ⊗ a_{n} ↦ ∑_{(j_{0}...j_{n})}(r′_{j_{0}}a_{0}r_{j_{1}}) ⊗ (r′_{j_{1}}a_{1}r_{j_{2}}) ⊗ … ⊗ (r′_{j_{n}}a_{n}r_{j_{0}});

as above, ϕ_{n}^{g} induces an isomorphism ϕ_{•}^{g}: HC_{•}(R_{1}) → HC_{•}(R_{1}), and we obtain in this way the action of G on HC_{•}(R_{1}).

Similarly, for S_{1} we have the chain map

ψ_{n}^{g^{−1}} : S_{1}^{⊗(n+1)} → S_{1}^{⊗(n+1)},

b_{0} ⊗ … ⊗ b_{n} ↦ ∑_{(k_{0}...k_{n})}(s_{k_{0}}b_{0}s′_{k_{1}}) ⊗ (s_{k_{1}}b_{1}s′_{k_{2}}) ⊗ … ⊗ (s_{k_{n}}b_{n}s′_{k_{0}}),

where s_{k} ∈ S_{g}, s′_{k} ∈ S_{g^{−1}} are such that ∑_{k=1}^{ν} s′_{k}s_{k} = 1. It follows that ψ_{n}^{g^{−1}} induces the isomorphism ψ_{•}^{g^{−1}} : HC_{•}(S_{1}) → HC_{•}(S_{1}).

We must prove that ϕ_{•} = ψ_{•}^{g^{−1}} ∘ ϕ_{•} ∘ ϕ_{•}^{g}: HC_{•}(R_{1}) → HC_{•}(S_{1}). Remark that the composition ψ_{•}^{g^{−1}} ∘ ϕ_{•} ∘ ϕ_{•}^{g} is induced by the chain map

ψ_{n}^{g^{−1}} ∘ ϕ_{n} ∘ ϕ_{n}^{g}: R_{1}^{⊗(n+1)} → S_{1}^{⊗(n+1)},

a_{0} ⊗ … a_{n} ↦ ∑_{(k_{0},...,k_{n})} ∑_{(i_{0},...,i_{n})} ∑_{(j_{0},...,j_{n})}s_{k_{0}}β_{1}(q_{i_{0}} ⊗_{R_{1}} (r′_{j_{0}}a_{0}r_{j_{1}})p_{i_{1}})s′_{k_{1}}
⊗ s_{k_{1}}β_{1}(q_{i_{1}} ⊗_{R_{1}} (r′_{j_{1}}a_{1}r_{j_{1}})p_{i_{2}})s′_{k_{2}}
⊗ … ⊗ s_{k_{n}}β_{1}(q_{i_{n}} ⊗_{R_{1}} (r′_{j_{n}}a_{n}r_{j_{0}})p_{i_{0}})s′_{k_{0}}

= ∑_{(k_{0},...,k_{n})} ∑_{(i_{0},...,i_{n})} ∑_{(j_{0},...,j_{n})}β_{1}(s_{k_{0}}β_{1}(q_{i_{0}} ⊗_{R_{1}} (r′_{j_{0}}a_{0}r_{j_{1}})p_{i_{1}})s′_{k_{1}})

⊗ β_{1}(s_{k_{1}}(q_{i_{1}} ⊗_{R_{1}} (r′_{j_{1}}a_{1}r_{j_{1}})p_{i_{2}})s′_{k_{2}})

⊗ … ⊗ β_{1}(s_{k_{n}}(q_{i_{n}} ⊗_{R_{1}} (r′_{j_{n}}a_{n}r_{j_{0}})p_{i_{0}})s′_{k_{0}})

Here we have used [M, Lemma 2.9] and the fact that β_{1} is a Δ(S)- isomorphism. In the same way we compute that

∑_{i=1}^{n} ∑_{j=1}^{λ} ∑_{k=1}^{ν} α_{1}(r_{j}p_{i}s′_{k} ⊗_{S_{1}} s_{k}q_{i}r′_{j}) = ∑_{i=1}^{n} ∑_{j=1}^{λ} α_{1}(r_{j}(p_{i} ⊗_{S_{1}} q_{i})r′_{j})

= ∑_{i=1}^{n} ∑_{j=1}^{λ} r_{j}α_{1}(p_{i} ⊗_{S_{1}} q_{i})r′_{j} = 1.

It follows that {r_{j}p_{i}s′_{k}} ⊂ P_{1} and {s_{k}q_{i}r′_{j}} ⊂ Q_{1} are dual bases, and if we denote by ϕ̄_{n} the chain map induced by them, we immediately see that ϕ̄_{n} = ψ_{n}^{g^{−1}} ∘ ϕ_{n} ∘ ϕ_{n}^{g}. By [L, Lemma 2.1], the homotopy class of ϕ_{n} does not depend on the choice of the dual bases. Consequently, ϕ_{•} = ψ_{•}^{g^{−1}} ∘ ϕ_{•} ∘ ϕ_{•}^{g}, so the theorem is proved.

References

[Be] D. Benson, Representations and cohomology II, Cambridge University Press, Cambridge, 1991.

[Bo] P. Boisen, Graded Morita theory, J. Algebra 164 (1994), 1–25.

[C] J. Cornick, On the homology of group graded algebras, J. Algebra 174 (1995), 999– 1023.

[D] E.C. Dade, The equivalence of various generalizations of group rings and modules, Math. Z. 181 (1982), 335–344.

[GJ] E. Getzler and J.D.S. Jones, On the cyclic homology of crossed product algebras, J. Reine Angew. Math. 445 (1993), 161–174.

[H] R. Hartshorne, Residues and duality, LNM 20, Springer-Verlag, Berlin, 1966.

[L] M. Lorenz, On the homology of graded algebras, Comm. Algebra 20 (1992), 489–507.

[M] A. Marcus, Equivalences induced by graded bimodules, to appear.

[NRV] C Năstăsescu, Ş. Raianu and F. van Oystaeyen, Graded modules over G-sets, Math. Z. 203 (1990), 605–627.

[R] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc. 43 (1991), 37–48.

[S] K. Sanada, On the Hochschild cohomology of crossed products, Comm. Algebra 21 (1993), 2727–2748.

[St] D. Ştefan, Hochschild cohomology of Hopf-Galois extensions, J.P.A.A. 103 (1995), 221–233.