Auto-equivalences of derived categories acting on cohomology

Alexander Zimmermann^{∗}

Faculté de Mathématiques et CNRS (FRE 2270); Université de Picardie; 33 rue St Leu; 80039 Amiens Cedex; France

                  electronic mail: Alexander.Zimmermann@u-picardie.fr

To Idun Reiten on the occasion of her 60th birthday

Abstract

Let A be a k-algebra which is projective as a k-module, let M be an A-module whose endomorphisms are given by multiplication by central elements of A, and let T rP ic_{k}(A) be the group of standard self-equivalences of the derived category of bounded complexes of A-modules. Then we define an action of the stabilizer of M in T rP ic_{k}(A) on the Ext-algebra of M. In case M is the trivial module for the group algebra kG = A, this defines an action on the cohomology ring of G which extends the well-known action of the automorphism group of G on the cohomology group.

Introduction

Let A and B be R-algebras over the commutative ring R so that A is projective as an R-module. If there is an equivalence between the derived categories of bounded complexes of A-modules D^{b}(A) and the derived category D^{b}(B), Rickard and Keller proved that there is a complex X in D^{b}(A ⊗_{R} B^{op}) so that F_{X} := X ⊗_{B}^{𝕃} − induces such an equivalence. In case B is projective as an R-module as well, there is also an object Y in D^{b}(B ⊗_{R} A^{op}) so that F_{Y} := Y ⊗_{B}^{𝕃} − is a quasi-inverse to F_{X}. In case A = B, equivalences of this type form a group which, in an earlier work with R. Rouquier [8], is called T rP ic_{R}(A).

Let M be an A-module. Then, it is reasonable to expect that the set HD_{M}(A) of elements in T rP ic_{R}(A), which fix M up to isomorphism, acts on the Ext-algebra Ext_{A}^{∗}(M, M) of M as ring automorphisms since Ext_{A}^{n}(M, M) = Hom_{D^{b}(A)}(M, M[n]) for any integer n. To get an actual action one has to be a bit more careful. We prove the above statement if any automorphism of M is induced by multiplication by an invertible element of the centre of A. For other modules with more complicated automorphism groups an extension H̃D_{M}(A) of HD_{M}(A) by some quotient of the automorphism group of M acts on Ext_{A}^{∗}(M, M).

The above defined action is well behaved with respect to change of base rings. In case A is the group ring RG of a group G, the action of HD_{R}(RG) extends the well known action of the outer automorphism group of G on the cohomology ring H^{∗}(G, R). This action of HD_{R}(RG) is functorial with respect to the second variable. In further work [10, 11] we study the functoriality with respect to the first variable. There, the situation is more complicate and we only have partial answers.

Acknowledgement: I thank the referee for his careful reading of the manuscript and for pointing out an error in an earlier version of the paper.

∗Mathematics Subject Classification: 16E30, 18E30, 20C05; Secondary: 16W20, 20J06

1 Brief review on derived equivalences

We shall present briefly what we will need concerning Rickard’s tilting theory. Our notation closely follows [4]. Let A and B be R-algebras over a commutative ring R and assume that A and B are projective as modules over R.

Rickard [6], and under weaker hypotheses Keller [2], show that if there is an equivalence of triangulated categories D^{b}(A) ≃ D^{b}(B) then there is a bounded complex X in D^{b}(A⊗_{R}B^{op}) so that X ⊗_{B}^{𝕃} − is an equivalence. Moreover, the inverse functor is given by the left derived tensor product with a complex Y in D^{b}(B ⊗_{R} A^{op}). Equivalences given by (left derived) tensor product by a bounded complex of bimodules are called of standard type. The complex X is called a twosided tilting complex.

We define [8] the group

T rP ic_{R}(A) := { isomorphism classes of two-sided tilting complexes in D^{b}(A ⊗_{R} A^{op})}

It is clear by the definition that in case A and B are R-algebras which are projective as R-modules and D^{b}(A) ≃ D^{b}(B), then T rP ic_{R}(A) ≃ T rP ic_{R}(B) .

2 Operation on Ext-algebras

We are interested in the action of T rP ic_{R}(A) on Ext-algebras and in particular on group cohomology. Let us recall the definitions. For an R-algebra A and two A-modules M and N let ⊕_{n∈ℕ}Ext_{A}^{n}(M, N) =: Ext_{A}^{∗}(M, N). Then, Ext_{A}^{∗}(M, N) is a module over the R-algebra Ext^{∗}(M, M). This follows easily by the observation Ext_{A}^{n}(M, N) = Hom_{D^{b}(A)}(M, N[n]) for any n ∈ ℕ. Then, the ring structure of Ext_{A}^{∗}(M, M) is given by composition of mappings and the Ext_{A}^{∗}(M, M)-module structure of Ext_{A}^{∗}(M, N) as well is given by composing mappings.

In particular if G is a group and R is a commutative ring, then for any RG-module M we have

H^{n}(G, M) ≃ Ext_{RG}^{n}(R, M) ≃ Hom_{D^{b}(RG)}(R, M[n]) .

Let X be a complex with isomorphism class in T rP ic_{R}(A) and let

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

be the corresponding self-equivalence. Then, F_{X} induces a mapping

Hom_{D^{b}(A)}(M, M[n]) ⟶ Hom_{D^{b}(A)}(F_{X}(M), F_{X}(M[n])) = Hom_{D^{b}(A)}(F_{X}(M), F_{X}(M)[n]) .

Assume now that we have an isomorphism F_{X}(M) ⟶^{α_{X}} M. Then, composing by α_{X} and its inverse from the left and from the right, the pair (α_{X}, X) induces automorphisms F_{X}^{(n)}

F_{X}^{(n)} : Hom_{D^{b}(A)}(M, M[n]) ⟶^{α_{X}[n]_{∗}(α_{X}^{−1})^{∗}F_{X}} Hom_{D^{b}(A)}(M, M[n])
λ  ⟶  α_{X}[n] · F_{X}(λ) · α_{X}^{−1}

for any positive integer n.

We shall discuss what happens if we have two isomorphic functors F_{X} and F_{Y}.

Lemma 2.1 Let F_{X} and F_{Y} be two functors D^{b}(A) ⟶ D^{b}(A) and suppose that F_{Y} ⟶^{η} F_{X} is an isomorphism of functors. If there exists an isomorphism α_{X} : F_{X}(M) ⟶ M. Then, the action of (α_{X}η_{M}, F_{Y}) and the action of (α_{X}, F_{X}) on Hom_{D^{b}(RG)}(M, M[n]) coincide.

Proof. Let λ ∈ Hom_{D^{b}(A)}(M, M[n]). Then, setting α_{Y} := α_{X}η_{R} we get a commutative diagram

M[−n] ⟶^{α_{X}^{−1}[−n]}  F_{X}(M)[−n]  ⟶^{F_{X}(λ)} F_{X}(M) ⟶^{α_{X}} M
∥  ↑ η_{M}[−n]  ↑ η_{M}  ∥
M[−n] ⟶^{α_{Y}^{−1}[−n]}  F_{Y} (M)[−n]  ⟶^{F_{Y} (λ)} F_{Y} (M) ⟶^{α_{Y}} M

This proves the lemma.

Let F_{X_{1}} : D^{b}(A) ⟶ D^{b}(A) and F_{X_{2}} : D^{b}(A) ⟶ D^{b}(A) be equivalences of triangulated categories and suppose there are isomorphisms α_{X_{i}} : F_{X_{i}}(M) ⟶ M for i = 1, 2. We define the composition of two objects (α_{X_{1}}, F_{X_{1}}) and (α_{X_{2}}, F_{X_{2}}) by:

(α_{X_{2}}, F_{X_{2}}) · (α_{X_{1}}, F_{X_{1}}) = (α_{X_{2}}F_{X_{2}}(α_{X_{1}}), F_{X_{2}}F_{X_{1}})

Definition 2.2 Let ĤD_{M}(A) be the class of pairs (α_{X}, X) where F_{X} = X ⊗_{A}^{𝕃} − is a selfequivalence of D^{b}(A) of standard type so that X ⊗_{A}^{𝕃} M ≃ M and α_{X} : X ⊗_{A}^{𝕃} M ⟶ M is an isomorphism of complexes of A-modules. Two of these pairs (α_{X}, X) and (α_{Y} , Y) are called isomorphic if there is an isomorphism η : Y ⟶ X so that α_{Y} = α_{X} ∘ (η ⊗ id_{M}). Let H̃D_{M}(A) be the set of isomorphism classes in ĤD_{M}(A).

In case A = RG is a group ring and M = R the trivial RG-module, then we denote ĤD_{R}(RG) =: ĤD_{R}(G) and H̃D_{R}(RG) =: H̃D_{R}(G) for short.

With this definition we get

Lemma 2.3 For any λ ∈ Hom_{D^{b}(A)}(M[−n], M) and (α_{X_{2}}, X_{2}) and (α_{X_{1}}, X_{1}) in ĤD_{M}(A) we get

(id_{R}, Id_{D^{b}(A)}) (λ) = λ
((α_{X_{2}}, X_{2}) · (α_{X_{1}}, X_{1})) (λ) = (α_{X_{2}}, X_{2}) ((α_{X_{1}}, X_{1}) (λ))

Proof. The first assertion is immediate. For the second assertion we compute

(α_{X_{2}}, X_{2}) ((α_{X_{1}}, X_{1}) (λ)) = (α_{X_{2}}, X_{2}) (α_{X_{1}}F_{X_{1}}(λ)α_{X_{1}}^{−1}[−n])
= α_{X_{2}}(F_{X_{2}}(α_{X_{1}})F_{X_{2}}F_{X_{1}}(λ)F_{X_{2}}(α_{X_{1}})^{−1}[−n])α_{X_{2}}^{−1}[−n]
= ((α_{X_{2}}, X_{2}) · (α_{X_{1}}, X_{1})) (λ)

This proves the second assertion.

Lemma 2.4 The set H̃D_{M}(A) is a group with the above defined multiplication. The identity element is (nat, A) and the inverse of (α_{X}, X) is (nat ∘ (η_{X} ⊗ id_{M})F_{X}^{−1}(α_{X}^{−1}), F_{X}^{−1}) where η_{X} : X^{−1} ⊗_{A} X ⟶ A is an isomorphism in the derived category of A ⊗_{R} A^{op}-bimodules, and nat : A ⊗_{A} M ⟶ M is the natural isomorphism.

Moreover, projection onto the second component gives a group homomorphism

H̃D_{M}(A) ⟶ T rP ic_{R}(A)

whose image is the fix point stabilizer of the trivial module

HD_{M}(A) := {[X] ∈ T rP ic_{R}(A) | X ⊗_{A}^{𝕃} M ≃ M}

Proof. Let (α_{X}, X), (α_{Y} , Y) and (α_{Z}, Z) be elements of ĤD_{M}(A). It is clear that (nat, A) · (α_{X}, X) = (α_{X}, X) in H̃D_{M}(A). Now,

(nat ∘ (η_{X} ⊗ id_{M}) ∘ F_{X}^{−1}(α_{X}^{−1}), F_{X}^{−1}) · (α_{X}, F_{X})
= (nat ∘ (η_{X} ⊗ id_{M}) ∘ F_{X}^{−1}(α_{X}^{−1}) ∘ F_{X}^{−1}(α_{X}), F_{X}^{−1}F_{X})
= (nat ∘ (η_{X} ⊗ id_{M}) ∘ F_{X}^{−1}(id), F_{X}^{−1}F_{X})
= (nat, A)

Associativity is the following computation:

((α_{X}, X)(α_{Y} , Y))(α_{Z}, Z) = (α_{X}F_{X}(α_{Y}), X ⊗_{A} Y)(α_{Z}, Z)
= (α_{X}F_{X}(α_{Y})F_{X⊗_{A}Y} (α_{Z}), X ⊗_{A} Y ⊗_{A} Z)
= (α_{X}F_{X}(α_{Y} F_{Y} (α_{Z})), X ⊗_{A} Y ⊗_{A} Z)
= (α_{X}, X)(α_{Y} F_{Y} (α_{Z}), Y ⊗_{A} Z)
= (α_{X}, X)((α_{Y} , Y)(α_{Z}, Z))

The fact that the projection onto the second component is a group homomorphism with image HD_{R}(G) described above is immediate.

Observe that ĤD_{M}(A) is not a group in general. Moreover, we use the fact that we deal with standard equivalences and we use the fact that we work in algebras, which are projective as a module over the base ring. So, composition of functors is associative since we may replace a complex by its projective resolution and there we may use the ordinary tensor product which is associative.

Lemma 2.5 Let A be an R-algebra and let M be an A-module so that any A-linear automorphism of M is induced by multiplication by an invertible element of the centre Z(A) of A. Then, H̃D_{M}(A) ≃ HD_{M}(A). In particular, for any group G and any commutative ring R we have H̃D_{R}(G) ≃ HD_{R}(G).

Proof. The kernel of the canonical surjection H̃D_{M}(A) ⟶ HD_{M}(A) is formed by the set of A-linear automorphisms of M modulo the group of automorphisms nat ∘ (η ⊗ id_{M}) ∘ nat^{−1} for automorphisms η of A as bimodule. Automorphisms of A as bimodules are precisely the multiplications by central invertible elements of A. Now, Aut(M) is generated by multiplication by invertible elements of Z(A). Therefore, the kernel of the surjection H̃D_{M}(A) ⟶ HD_{M}(A) is trivial and this surjection induces the isomorphism as stated.

For a group ring RG, any automorphism of the trivial module is multiplication by a unit in R. It is clear that the group of units R^{∗} of R is in Z(RG).

Remark 2.6 Bass observed that there is a monomorphism of the outer automorphism group Out_{R}(A) of the R-algebra A to P ic_{R}(A). This monomorphism maps an automorphism f of the R-algebra A to the bimodule _{f}A_{1} on which a ∈ A acts by multiplication by a on the right and by multiplication by f(a) on the left. P ic_{R}(A) in turn is a subgroup of T rP ic_{R}(A) and we get a group homomorphism

{ϕ ∈ Aut_{R}(A) | ^{ϕ}M ≃ M} ⟶ HD_{M}(A).

In particular, define HA_{R}(G) as the image of the automorphism group of G in HD_{R}(G) by mapping an automorphism f of G to the bimodule _{f}RG_{1} which is RG from the right and on which g ∈ G operates by multiplication by f(g) from the left.

Note that the question when an automorphism becomes inner in the group ring (and hence induces the identity in HD_{R}(G)) is far from trivial. The reader may consult for example M. Mazur [5] or Roggenkamp-Zimmermann [7].

We have obtained the following result.

Theorem 1 Let R be a commutative ring and let A be an R-algebra, projective as an Rmodule and let M be an A-module. Then, for any integer n the group Ext_{A}^{n}(M, M) is an R H̃D_{M}(A)-module.

If in addition any automorphism of M is induced by multiplication by an invertible element in Z(A), then for any integer n the group Ext_{A}^{n}(M, M) is an R HD_{M}(A)-module.

In particular, for any commutative ring R and any group G the cohomology group H^{n}(G, R) is an R HD_{R}(G)-module.

Proof. The results in Lemma 2.3, Lemma 2.4 and Lemma 2.5 imply the statement immediately.

Remark 2.7 The action of HD_{R}(G) extends the well known and well studied action of Out(G) on H^{n}(G, R). Our interest into the above defined action partly comes from this fact.

3 Properties of the action

We recall the trivial cases.

Proposition 3.1 [8] Let A be a commutative indecomposable R-algebra, or let A be a local algebra. Then T rP ic_{R}(A) = P ic_{R}(A)× < [1] > .

We shall study change of rings properties and functoriality with respect to the coefficient ring R.

Remark 3.2 What happens in the case of a group ring RG for a finite group G ? Since H^{0}(G, R) ≃ R is the trivial HD_{R}(G)-module, we may restrict our attention to H^{n}(G, R) for n ≥ 1. Moreover, assume that R is finitely generated over ℤ. Then, the universal coefficient theorem gives an exact sequence

0 ⟶ H^{n}(G, ℤ) ⊗_{ℤ} R ⟶ H^{n}(G, R) ⟶ T or_{1}^{ℤ}(H^{n+1}(G, ℤ), R) ⟶ 0

So, for certain coefficient rings R and for certain questions we may restrict attention to the coefficient ring ℤ. Moreover, for n ≥ 1, we get H^{n}(G, ℤ) ≃ ∏_{p∈Spec ℤ} H^{n}(G, ℤ̂_{p}). If p does not divide the order of the finite group G, one gets H^{n}(G, ℤ̂_{p}) = 0.

So, part of the problem is reduced to the case H^{n}(G, ℤ̂_{p}) for all prime numbers dividing the order of G.

Let S be a commutative R-algebra. Since we assumed that A is projective as an Rmodule, [6] proves that

S ⊗_{R} − : T rP ic_{R}(A) ⟶ T rP ic_{S}(S ⊗_{R} A)

is a homomorphism of groups. It is not true in general that this homomorphism is surjective or injective as the following example shows.

Example: Since ℤC_{3} is commutative indecomposable, Proposition 3.1 shows that T rP ic_{ℤ}(ℤC_{3}) =< [1] > while T rP icℤ̂_{7}(ℤ̂_{7}C_{3}) ≃ T rP icℤ̂_{7}(ℤ̂_{7} × ℤ̂_{7} × ℤ̂_{7}) ≃ C_{∞} ≀ 𝔖_{3} is a wreath product since ℤ̂_{7} contains a primitive third root of unity. So, the above mapping S ⊗_{R} − : T rP ic_{R}(A) ⟶ T rP ic_{S}(S ⊗_{R} A) is not surjective in general. We should remark that HDℤ̂_{7}(C_{3}) = C_{∞} ≀ C_{2} while HD_{ℤ}(C_{3}) = Gal(ℤ[ζ_{3}] : ℤ) = C_{2} is generated by the Galois automorphism of ℤ[ζ_{3}] over ℤ.

As is shown in [9] the kernel T rIℤ̂_{3}(ℤ̂_{3}𝔖_{3}) of the homomorphism T rP icℤ̂_{3}(ℤ̂_{3}𝔖_{3}) ⟶ T rP icℚ̂_{3}(ℚ̂_{3}𝔖_{3}) is a non abelian free group. So, the above mapping is not injective in general neither.

Theorem 2 Let ϕ : R ⟶ S be a homomorphism of commutative rings, let A be an Ralgebra which is projective as an R-module and let M be an A-module. Then, S ⊗_{R} − : T rP ic_{R}(A) ⟶ T rP ic_{S}(S ⊗_{R} A) lifts to a homomorphism

Hϕ : H̃D_{M}(A) ⟶ H̃D_{S⊗_{R}M}(S ⊗_{R} A) .

Moreover,

S ⊗_{R} − : Ext_{A}^{∗}(M, M) ⟶ Ext_{S⊗_{R}A}^{∗}(S ⊗_{R} M, S ⊗_{R} M)

is R H̃D_{M}(A)-linear.

Proof. Let X be a complex with isomorphism class in HD_{M}(A). Since A is assumed to be projective as an R-module, we may and will assume that the homogeneous components of X are projective as A-modules and projective as A^{op}-modules. Then, we can replace the left derived tensor product by the ordinary tensor product.

First we observe that there is an isomorphism

ν_{X} : (S ⊗_{R} X) ⊗_{S⊗_{R}A} (S ⊗_{R} M) ˜ ⟶S ⊗_{R} (X ⊗_{A} M)

given by

(s ⊗ x) ⊗ (t ⊗ m) = (st ⊗ x) ⊗ (1 ⊗ m) ↦ st ⊗ (x ⊗ m)

in each homogeneous component of X. We define

α_{S⊗_{R}X} : (S ⊗_{R} X) ⊗_{S⊗_{R}A} (S ⊗_{R} M) ˜ ⟶S ⊗_{R} M

by α_{S⊗_{R}X} := (id_{S} ⊗ α_{X}) ∘ ν_{X}.

In order to prove the first statement we define

H̃D_{M}(A) ⟶^{Hϕ} H̃D_{S⊗_{R}M}(S ⊗_{R} A)
(α_{X}, X) ↦ (α_{S⊗_{R}X}, (S ⊗_{R} X))

If η : X ⟶ Y is an isomorphism, then id_{S} ⊗_{R} η : S ⊗_{R} X ⟶ S ⊗_{R} Y is an isomorphism as well. It follows that Hϕ does not depend on the chosen representative in the isomorphism class in H̃D_{M}(A). Moreover, the above shows that

(α_{S⊗_{R}X}, (S ⊗_{R} X)) ∈ ĤD_{S⊗_{R}M}(S ⊗_{R} A) .

This proves the first statement.

We have to show that

S ⊗_{R} − : Hom_{D^{b}(A)}(M, M[n]) ⟶ Hom_{D^{b}(S⊗_{R}A)}(S ⊗_{R} M, S ⊗_{R} M[n])

is linear under the action of H̃D_{M}(A). This is equivalent to proving that the diagram

Hom_{D^{b}(A)}(M, M[n])  ⟶  Hom_{D^{b}(S⊗_{R}A)}(S ⊗_{R} M, S ⊗_{R} M[n])
↓  ↓
Hom_{D^{b}(A)}(X ⊗_{A} M, X ⊗_{A} M[n])  Hom_{D^{b}(S⊗_{R}A)}((SX) ⊗_{SA} (SM), ((SX) ⊗_{SA} (SM[n]))
↓ α_{X}[n]_{∗} · (α_{X}^{−1})^{∗}  ↓ α_{S⊗_{R}X}[n]_{∗} · (α_{S⊗_{R}X}^{−1})^{∗}
Hom_{D^{b}(A)}(M, M[n])  ⟶  Hom_{D^{b}(S⊗_{R}A)}(S ⊗_{R} M, S ⊗_{R} M[n])

is commutative. (We wrote SM for S ⊗_{R} M and likewise SX for S ⊗_{R} X.) The diagram is commutative if and only if

α_{S⊗_{R}X}[n] ∘ ((S ⊗_{R} X) ⊗ f) ∘ α_{S⊗_{R}X}^{−1} = id_{S} ⊗ (α_{X}[n] ∘ (X ⊗ f) ∘ α_{X}^{−1}) .

But,

α_{S⊗_{R}X}[n] ∘ ((S ⊗_{R} X) ⊗ f)∘α_{S⊗_{R}X}^{−1}

= (id_{S} ⊗_{R} α_{X}[n]) ∘ ν ∘ ((S ⊗_{R} X) ⊗ f) ∘ ν^{−1} ∘ (α_{X}^{−1} ⊗ id_{S})
= (id_{S} ⊗_{R} α_{X}[n]) ∘ ((S ⊗_{R} X) ⊗ f) ∘ (α_{X}^{−1} ⊗ id_{S})

since conjugation by ν acts trivially on the morphism (S ⊗_{R} X) ⊗_{A} f. Now, since (S ⊗_{R} X) ⊗_{A} f ≃ S ⊗_{R} (X ⊗_{A} f), we see that under this isomorphism we obtain

(id_{S} ⊗_{R} α_{X}[n]) ∘ ((S ⊗_{R} X) ⊗ f) ∘ (α_{X}^{−1} ⊗ id_{S}) = id_{S} ⊗ (α_{X}[n] ∘ (X ⊗ f) ∘ α_{X}^{−1}) .

This proves the second statement.

As a consequence, one might get a rather different group acting on H^{∗}(G, S) than on H^{∗}(G, R) even though it might happen that H^{≥1}(G, S) ≃ H^{≥1}(G, R). For example ℤ̂_{p} contains a primitive p − 1-th root of unity but ℤ_{p} does not. Nevertheless, H^{≥1}(G, ℤ̂_{p}) ≃ H^{≥1}(G, ℤ_{p}) for a finite group G.

Lemma 3.3 Let ϕ : R ⟶ S and ψ : S ⟶ T be homomorphisms of commutative rings and let A be an R-algebra which is projective as an R-module. Let M be an A-module. Then, Hψ ∘ Hϕ = H(ψ ∘ ϕ).

Proof. It is clear that there is an isomorphism T ⊗_{S} (S ⊗_{R} X) ≃ T ⊗_{R} X and so, the only thing to prove is that under this isomorphism (α_{T ⊗_{S}(S⊗_{R}X)}, T ⊗_{S} (S ⊗_{R} X)) becomes equivalent to (α_{T ⊗_{R}X}, T ⊗_{R} X). But, this is immediate.

Remark 3.4 Let R be any commutative ring, let A and B be R-algebras which are projective as R-modules and let M be an A-module. Assume now that we have an equivalence of triangulated categories D^{b}(A) ≃ D^{b}(B). Then, by [6], there is a complex Y in D^{b}(B ⊗_{R} A^{op}) so that the left derived tensor product Y ⊗_{A}^{𝕃} − is an equivalence and a (quasi-)inverse equivalence is given by left derived tensor product with a complex X in D^{b}(A⊗_{R} B^{op}). Therefore, the group Y ⊗_{A} HD_{M}(A) ⊗_{B} X ⊆ T rP ic_{R}(B) fixes Y ⊗_{A} M. It is now immediate that H̃D_{M}(A) ≃ H̃D_{Y ⊗_{A}M}(B) We should note however that one has to enlarge the definition of H̃D_{M}(A) in the obvious way in case Y ⊗_{A} M is not necessarily isomorphic to a module.

Proposition 3.5 Let R be a commutative ring, let A and B be R-algebras which are projective as R-modules and let M be an A-module. Let F_{U} := U ⊗_{A}^{𝕃} − : D^{b}(A) ⟶ D^{b}(B) be an equivalence of standard type of triangulated categories.

Then, F_{U} induces an isomorphism

HU : H̃D_{M}(A) ⟶^{≃} H̃D_{U⊗_{A}M}(B)

and an isomorphism

Ext_{A}^{∗}(M, M) ⟶^{≃} ^{HU}Ext_{B}^{∗}(U ⊗_{A} M, U ⊗_{A} M)

as H̃D_{M}(A)-modules.

Proof. We note by U^{−1} a complex of A ⊗_{R} B^{op}-modules so that U ⊗_{A}^{𝕃} U^{−1} ≃ B and U^{−1} ⊗_{B}^{𝕃} U ≃ A. Since A and B are projective as R-modules, we may and will assume that the homogeneous components of U are projective as A^{op}-modules and as B-modules and that the homogeneous components of U^{−1} are projective as B^{op}-modules and as A-modules. Let ρ_{T} : T ⊗_{A} M ⟶ M be an isomorphism fixed for each T which has isomorphism class in HD_{M}(A). Set

ρ_{UT U^{−1}} : U ⊗_{A} T ⊗_{A} U^{−1} ⊗_{B} U ⊗_{A} M ≃ U ⊗_{A} T ⊗_{A} M ⟶^{id_{U} ⊗ρ_{T}} U ⊗_{A} M

Then, define a mapping

HU : H̃D_{M}(A) ⟶ H̃D_{U⊗_{A}M}(B)
(ρ_{T} , T) ⟶ (ρ_{UT U^{−1}}, UT U^{−1})

The above is well defined since if there is an isomorphism η : T ⟶ T′, then η induces ρ_{T′} = ρ_{T} ∘ (η ⊗ id_{M}) and so ρ_{UT′U^{−1}} = ρ_{UT U^{−1}} ∘ (id_{U} ⊗ η ⊗ id_{M}) which gives

HU(ρ_{T} , T) = (ρ_{UT U^{−1}}, UT U^{−1}) = (ρ_{UT U^{−1}} ∘ (id_{U} ⊗ η ⊗ id_{M}), UT′U^{−1}) = HU(ρ_{T′}, T′) .

It is clear that the mapping HU is bijective since the inverse is given in the very same way, replacing U by U^{−1}. Moreover, for any (ρ_{T} , T) and (ρ_{S}, S) in H̃D_{M}(A), we get

HU(ρ_{T} ∘ F_{T} (ρ_{S}), T S) = (F_{U}(ρ_{T} ∘ F_{T} (ρ_{S})), UT SU^{−1})
= (F_{U}(ρ_{T}), UT U^{−1})(F_{U}(ρ_{S}), USU^{−1})
= HU(ρ_{T} , T) · HU(ρ_{S}, S)

and this proves that HU is multiplicative. We obtained the first statement.

Recall that the action of (ρ_{T} , T) with equivalence class in HD_{M}(A) on Ext_{A}^{n}(M, M) is given by the following construction. An α ∈ Hom_{D^{b}(A)}(M[−n], M) is mapped by (ρ_{T} , T) to ρ_{T} [−n]T (α)ρ_{T}^{−1}.

Take β ∈ Hom_{D^{b}(B)}(U ⊗_{A} M[−n], U ⊗_{A} M).

(HU(ρ_{T} , T)) · β = (F_{U}(ρ_{T}), UT U^{−1}) · β
= F_{U}(ρ_{T}) · F_{UT U^{−1}}(β) · F_{U}(ρ_{T})[−n]
= F_{U}(ρ_{T} · F_{T} F_{U^{−1}}(β) · ρ_{T} [−n])
= F_{U}((ρ_{T} , T) · F_{U^{−1}}(β))

This proves the second statement.

Now, we shall study the case of a group ring in more detail. In modular representation theory of finite groups it has proved useful to look at the stable module category. Most of what follows works equally well for a selfinjective R-algebra, which is projective as an R-module. We will not go into these details.

Let R be a Dedekind domain. Then, the R-stable module category \overline{mod}(RG) is the quotient category of RG − mod by the subcategory of R-projective modules.

Let StP ic(RG) be the group of isomorphism classes of RG ⊗_{R} RG^{op}-modules X up to isomorphism in \overline{mod}(RG ⊗_{R} RG^{op}) so that X ⊗_{RG} − induces an equivalence \overline{mod}(RG) ⟶ \overline{mod}(RG). Set

HSt_{R}(G) := \{[X] ∈ StP ic(RG) | X ⊗_{RG} R ≃ R in \overline{mod}(RG)\}

Analogously to H̃D_{R}(G) and ĤD_{R}(G) we define H̃St_{R}(G) and ĤSt_{R}(G).

Following [8] we have a group homomorphism

T rP ic_{R}(RG) ⟶^{σ} StP ic_{R}(RG) .

This is defined as follows: Take a two-sided tilting complex X with isomorphism class in T rP ic_{R}(RG). Then, choose any projective resolution (X′, d′) of X as complex of RG ⊗_{R} RG^{op} bimodules. Let n be the highest degree (differentials are of degree −1) so that X has non zero homology in degree n. Then, X_{∞} := ker d′_{n+1}. Since RG is a Gorenstein order (that is Ext_{RG}^{1}(M, RG) = 0 for any RG-lattice M), the functor Ω : \overline{mod}(RG) ⟶ \overline{mod}(RG) is a self-equivalence.

Define σ(X) := Ω^{−n}(X_{∞}). It can be shown that this does not depend on the chosen projective resolution of X.

Lemma 3.6 σ(HD_{R}(G)) ⊆ HSt_{R}(G).

Proof. Let X be a two-sided tilting complex with isomorphism class in HD_{R}(G). Replace X by an isomorphic copy which has projective homogeneous components, also denoted by X. Since now all components are projective as right-RG-modules, one can replace the left derived tensor product by the ordinary tensor product. Now, X is isomorphic to the complex Y whose homogeneous components of degree higher than n + 2 are 0, the component in degree n + 1 is X_{∞} = ker d_{n+1} and all the other homogeneous components of Y are identical to those of X. The differentials of Y are the obvious ones. The image of X in StP ic_{R}(RG) is Ω^{−n}(X_{∞}) where Ω is the syzygy operator. We have to prove that X_{∞} ⊗_{RG} R ≃ Ω^{n}(R) in \overline{mod}(RG).

If M is a projective RG ⊗_{R} RG^{op}-module, then M ⊗_{RG} R is a projective RG-module. In fact, this is true for free RG ⊗_{R} RG^{op}-modules, and hence it holds for projective modules as well. Since, X ⊗_{RG} R ≃ R, the complex X ⊗_{RG} R gives the first n terms of a projective resolution of the trivial module R. Since X ⊗_{RG} R ≃ Y ⊗_{RG} R ≃ R we get a commutative diagram with exact rows

0  → X_{∞} ⊗_{RG} R → X_{n} ⊗_{RG} R → … → X_{m} ⊗_{RG} R → R → 0
↑ ↑ ∥ ∥ ∥
X_{n+2} ⊗_{RG} R → X_{n+1} ⊗_{RG} R → X_{n} ⊗_{RG} R → … → X_{m} ⊗_{RG} R → R → 0

This means X_{∞} ⊗_{RG} R ≃ Ω^{n}(R).

Lemma 3.7 If RG has the Krull-Schmidt property on lattices, then H̃St(RG) acts on H^{∗}(G, R) and the action of HD_{R}(G) on H^{∗}(G, R) factors through the action of H̃St_{R}(G).

Proof. If RG has the Krull-Schmidt property on lattices one gets

H^{n}(G, R) = Ext_{RG}^{n}(R, R) = Hom_{\overline{mod}(RG)}(Ω^{n}(R), R)

It is now clear that HSt_{R}(G) acts on H^{∗}(G, R) and that the action of HD_{R}(G) factors through the action of HSt_{R}(G).

It should be noted that the question when RG has the Krull Schmidt property on lattices is a delicate one if R is only local and not complete. We refer to [1, § 36].

Let R be a complete discrete valuation ring of characteristic 0 with residue field of characteristic p and let G be a finite group with cyclic p-Sylow subgroup so that the principal block has no exceptional vertex. Lemma 3.7 enables us to prove that the action of HD_{R}(G) on H^{∗}(G, R) is trivial.

In fact, Linckelmann computed StP ic_{R}(B_{0}(RG)) for groups with cyclic p-Sylow subgroup P. He obtained in [4, Theorem 11.4.9] and the preceding remarks that in this case StP ic_{R}(B_{0}(RG)) ≃ C_{2e} × Aut(P)/E where E is the inertia quotient and e = |E| is the number of simple modules. In our case, in the absence of an exceptional vertex one gets StP ic_{R}(B_{0}(RG)) ≃ C_{2(p−1)} and this cyclic group correspond to the 2(p − 1) syzygies of the trivial module. So, HSt_{R}(G) = {1} . We note that in [9], using a deep result of Khovanov and Seidel [3], we prove that HD_{R}(G) contains a braid group on e = p − 1 strings.

Moreover, Linckelmann obtained in [4, Theorem 11.4.10] for R being a complete discrete valuation domain with residue field of characteristic 2 and 𝔄_{4} be the alternating group of order 12 that StP ic(R𝔄_{4}) ≃ C_{∞} × P ic_{R}(R𝔄_{4}). Here C_{∞} is the group consisting of taking syzygies, which is the image of the subgroup of T rP ic_{R}(R𝔄_{4}) generated by shift in degrees. Hence, the action of HD_{R}(𝔄_{4}) on H^{∗}(𝔄_{4}, R) is the action of Aut_{R}(R𝔄_{4}) on H^{∗}(𝔄_{4}, R). Again, HD_{R}(𝔄_{4}) contains a braid group on 3 strings as is shown in [12].

The above proof uses the functor T rP ic_{R}(RG) ⟶ StP ic_{R}(RG). The existence of this functor needs some hypotheses on the coefficient domain R. It should be noted that it is actually possible to compute the action of a self-equivalence on the cohomology explicitly without passing through the stable category. The result is hence valid in a more general context.

References

[1] Charles Curtis and Irving Reiner, “Methods of Representation Theory”; Vol 1, Wiley Interscience, 1981.

[2] Bernhard Keller, A remark on tilting theory and DG-algebras, manus. math. 79 (1993) 247-253.

[3] Mikhael Khovanov and Paul Seidel, Quivers, Floer Cohomology and Braid Group Actions, preprint (30 August 2000). http://xxx.lanl.gov/abs/math.QA/0006056

[4] Steffen König and Alexander Zimmermann, “Derived equivalences for group rings (with contributions by Bernhard Keller, Markus Linckelmann, Jeremy Rickard and Raphaël Rouquier)”, Lecture Notes in Mathematics 1685, Springer Verlag, Berlin 1998.

[5] Marcin Mazur, The normalizer of a group in the unit group of its group ring, J. of Algebra 212 (1999), 175-189.

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

[7] Klaus W. Roggenkamp and Alexander Zimmermann, Outer group automorphisms may become inner in the integral group ring, J. of Pure and Appl. Alg. 103 (1995), 91-99.

[8] Raphaël Rouquier and Alexander Zimmermann, A Picard group for derived module categories, accepted for publication in Proceedings of the London Mathematical Society.

[9] Alexander Zimmermann, Automorphisms of Green orders and their derived categories, preprint.

[10] Alexander Zimmermann, Cohomology of groups and splendid equivalences of derived categories, to appear in Math. Proc. Cambridge Phil. Soc.

[11] Alexander Zimmermann, Cohomology of groups, abelian Sylow subgroups and splendid equivalences, Algebra Montpellier Announcements 01-2000. http://www.emis.de/journals/AMA/index.html

[12] Alexander Zimmermann, Braid groups as self-equivalences of derived categories, preprint.

Amiens, November 2000; revised January 2001.