ON GRADED CENTERS AND BLOCK COHOMOLOGY

Markus Linckelmann

Abstract. We extend the group theoretic notions of transfer and stable elements to graded centers of triangulated categories. When applied to the center Z^{∗}(D^{b}(B)) of the derived bounded category of a block algebra B we show that the block cohomology H^{∗}(B) is isomorphic to a quotient of a certain subalgebra of stable elements of Z^{∗}(D^{b}(B)) by some nilpotent ideal, and that a quotient of Z^{∗}(D^{b}(B)) by some nilpotent ideal is Noetherian over H^{∗}(B).

§1 Introduction

The graded center of a triangulated category C with shift functor Σ consists of all natural transformations Id_{C} → Σ^{n} which commute modulo the sign (−1)^{n} with Σ, where n runs over the integers. For B a source algebra or a block algebra of a p-block of a finite group, there are canonical graded maps from the block cohomology H^{∗}(B) and its Tate analogue Ĥ^{∗}(B) to the centers Z^{∗}(D^{b}(B)) and Z^{∗}(\overline{mod}(B)) of the bounded derived and stable module categories which factor through Hochschild cohomology HH^{∗}(B) and its Tate analogue ĤH^{∗}(B), respectively. Analogously to questions raised in [13] it is natural to ask when the canonical maps H^{∗}(B) → Z^{∗}(D^{b}(B)) or H^{∗}(B) → Z^{∗}(\overline{mod}(B)) are isomorphisms modulo nilpotent ideals. As in [13], the relevance of this type of question lies in the fact that H^{∗}(B) is an invariant of the local structure of the block while Z^{∗}(D^{b}(B)) and Z^{∗}(\overline{mod}(B)) are invariants of the bounded derived and stable module category, respectively, and it is not known to what extent these categories determine the local structure. The following weaker version of this question has an affirmative answer: the image of the canonical map H^{∗}(B) → Z^{∗}(D^{b}(B)) lies in a certain subalgebra of stable elements, defined in §5, and the induced map is then indeed an isomorphism modulo nilpotent ideals; more precisely:

Theorem 1.1. Let k be an algebraically closed field of prime characteristic p. Let G be a finite group and B be a source algebra of a block of kG with a defect group P. Denote by G : D^{b}(B) → D^{b}(kP) the restriction functor. The canonical map H^{∗}(B) → Z^{∗}(D^{b}(B)) sends H^{∗}(B) to the subalgebra of G-stable elements Z_{G}^{∗}(D^{b}(B)) and there is a nilpotent ideal N in Z_{G}^{∗}(D^{b}(B)) such that this map induces an isomorphism H^{∗}(B) ≅ Z_{G}^{∗}(D^{b}(B))/N.

This is proved in section 8. For completeness, we include several straightforward observations, with proofs at the end of section 2:

Proposition 1.2. Let k be an algebraically closed field of prime characteristic p, let G be a finite group and let B be a block algebra of kG. There is a nilpotent ideal N in Z^{∗}(D^{b}(B)) such that Z^{∗}(D^{b}(B))/N is Noetherian as H^{∗}(B)-module. In particular, the graded algebra Z^{∗}(D^{b}(B))/N is finitely generated.

Typeset by AMS-TEX

Proposition 1.3. Let k be a field of characteristic p and let P be a finite p-group. Let N be the kernel of the evaluation map Z^{∗}(D^{b}(kP)) → H^{∗}(P, k). The ideal N is nilpotent and the canonical map H^{∗}(P, k) → Z^{∗}(D^{b}(kP)) induces an isomorphism H^{∗}(P, k) → Z^{∗}(D^{b}(kP))/N.

This implies in particular that if B is a nilpotent block then the canonical map H^{∗}(B) → Z^{∗}(D^{b}(B)) is an isomorphism modulo nilpotent ideals. Graded centers of stable module categories seem to be more elusive; neither is it clear whether the degree zero component Z^{0}(\overline{mod}(B)) is finitedimensional, nor even what is the kernel of the canonical map Z(B) → Z^{0}(\overline{mod}(B)). For B a Brauer tree algebra, Z^{∗}(\overline{mod}(B)) is calculated in [8]. Rickard noted that almost vanishing morphisms in the sense of Happel [7] can be used to construct elements in Z^{0}(C) for C any triangulated category. Auslander-Reiten sequences for modules over a symmetric algebra A give rise to almost vanishing homomorphisms in \overline{mod}(A) of degree −1, and as a consequence we get that Z^{−1}(\overline{mod}(A)) is not finite-dimensional, in general:

Proposition 1.4. Let A be a finite-dimensional symmetric algebra over an algebraically closed field k and let U be a finitely generated indecomposable non-projective A-module. Suppose that U does not have odd Ω-period, or that char(k) = 2. There is an element ζ ∈ Z^{−1}(\overline{mod}(A)) such that ζ(U) : U → Ω(U) represents an almost split sequence ending in U and such that ζ(V) = 0 for any finitely generated indecomposable non-projective A-module V which is not isomorphic to Ω^{n}(U), for any integer n. In particular, if A has infinitely many Ω-orbits of isomorphism classes of finitely generated non projective modules whose Ω-period is not odd then Z^{−1}(\overline{mod}(A)) is not finite-dimensional.

Notation. Throughout this note, R is a commutative ring with unit element. Let C, D, E be categories, let F, F′ : C → D and G, G′ : D → E be covariant functors. For any natural transformation ψ : G → G′ we denote by ψF : GF → G′F the natural transformation given by (ψF)(X) = ψ(F(X)) : G(F(X)) → G′(F(X)) for any object X in C. Similarly, for any natural transformation φ : F → F′ we denote by Gφ : GF → GF′ the natural transformation given by (Gφ)(X) = G(φ(X)) : G(F(X)) → G(F′(X)) for any object X in C.

2 Graded centers of graded categories

Graded centers are being considered by a growing number of authors; see for instance [2], [5]. We review the terminology and basic properties.

Definition 2.1. Let C be an R-linear category and let Σ : C ⟶ C be an R-linear equivalence. We define a graded R−module Z^{∗}(C) as follows. For any integer n we denote by Z^{n}(C, Σ) the R-module of natural transformations φ : Id_{C} → Σ^{n} satisfying Σφ = (−1)^{n}φΣ. We call Z^{∗}(C, Σ) the graded center of the graded category (C, Σ). If Σ is clear from the context we write Z^{∗}(C) instead of Z^{∗}(C, Σ).

Remark 2.2. In particular, Z^{0}(C, Σ) consists of all natural transformations φ : C → C satisfying Σφ = φΣ. This condition is void if Σ = Id_{C}. In that case, Z^{0}(C, Id_{C}) = Z(C) consists of all natural transformations φ : Id_{C} → Id_{C}, traditionally called the center of the category C. It is easy to see that Z(C) is commutative. If C = mod(A), where A is any ring, we have a canonical isomorphism Z(A) ≅ Z(C) mapping z ∈ Z(A) to the natural transformation consisting, for any A−module U, of the endomorphism given by multiplication with z on U. We ignore in this paper set theoretic issues since the results stated in section 1 involve only finitely generated module categories and their derived and stable categories.

Remark 2.3. The equality Σφ = (−1)^{n}φΣ in 2.1 is an equality of natural transformations from Σ to Σ^{n+1}. The sign is motivated by the “turning triangles” axiom: if (C, Σ) is a triangulated

category and if \xymatrix{U \ar[r]^{f} & V \ar[r]^{g} & W \ar[r]^{h} & Σ(U)} is an exact triangle in C then the triangle

\xymatrix{Σ^{n}(U) \ar[r]^{(−1)^{n}Σ^{n}(f)} & Σ^{n}(V) \ar[r]^{(−1)^{n}Σ^{n}(g)} & Σ^{n}(W) \ar[r]^{(−1)^{n}Σ^{n}(h)} & Σ^{n+1}(U)}

is exact. Hence, if φ : Id_{C} → Σ^{n} is any natural transformation, we have a commutative diagram

\xymatrix{U \ar[r]^{f} \ar[d]_{φ(U)} & V \ar[r]^{g} \ar[d]_{(−1)^{n}φ(V)} & W \ar[r]^{h} \ar[d]^{φ(W)} & Σ(U) \ar[d]^{(−1)^{n}φ(Σ(U))} \\ Σ^{n}(U) \ar[r]_{(−1)^{n}Σ^{n}(f)} & Σ^{n}(V) \ar[r]_{(−1)^{n}Σ^{n}(g)} & Σ^{n}(W) \ar[r]_{(−1)^{n}Σ^{n}(h)} & Σ^{n+1}(U)}

Thus this diagram defines a morphism of triangles provided that (−1)^{n}φ(Σ(U)) = Σ(φ(U)).

The graded R-module Z^{∗}(C) becomes an associative graded R-algebra via the R-bilinear product mapping a pair (ψ, φ) ∈ Z^{n}(C) × Z^{m}(C) to the composition (Σ^{m}ψ) ∘ φ ∈ Z^{m+n}(C) for any two integers m, n. Strictly speaking, for this to make sense in negative degrees, we have to choose an inverse equivalence Σ^{−1} and natural isomorphisms Σ∘Σ^{−1} ≅ Id_{C} ≅ Σ^{−1} ∘Σ, so that we can identify the functors Σ^{n} ∘ Σ^{m} ≅ Σ^{n+m} for any two integers n, m, positive or negative. In practice, this may not be an issue because Σ is frequently not only an equivalence, but an automorphism; that is, Σ ∘ Σ^{−1} = Id_{C} = Σ^{−1} ∘ Σ. This is, for instance, the case if Σ is a shift functor in some category of complexes. The sign convention has the following immediate consequence:

Proposition 2.4. Let C be an R-linear category and let Σ : C ⟶ C be an R-linear equivalence. The R-algebra Z^{∗}(C) is graded commutative.

Proof. Let m, n be two integers, let φ ∈ Z^{m}(C) and ψ ∈ Z^{n}(C). Since φΣ = (−1)^{m}Σφ we have φΣ^{n} = (−1)^{mn}Σ^{n}φ. Using that φ is a natural transformation from Id_{C} to Σ^{m} we get a commutative diagram

Id_{C} ⟶^{φ} Σ^{m}
ψ↓  ↓^{Σ^{m}ψ}
Σ^{n} ⟶_{φΣ^{n}} Σ^{m+n}

Together we get ψφ = (Σ^{m}ψ) ∘ φ = (φΣ^{n}) ∘ ψ = (−1)^{mn}(Σ^{n}φ) ∘ ψ = (−1)^{mn}φψ. □

Definition 2.5. Let C be an R-linear category and let Σ : C ⟶ C be an R-linear equivalence. For any two objects U, V in C we define a graded R-module Ext_{(C,Σ)}^{∗}(U, V) by setting Ext_{(C,Σ)}^{n}(U, V) = Hom_{C}(U, Σ^{n}(V)) for any integer n. As before, if Σ is clear from the context, we write Ext_{C}^{n}(U, V) instead of Ext_{(C,Σ)}^{n}(U, V).

Remark 2.6. If U, V, W are three objects in an R-linear category C endowed with an R-linear equivalence Σ : C → C, there is a graded composition Ext_{C}^{∗}(V, W) × Ext_{C}^{∗}(U, V) ⟶ Ext_{C}^{∗}(U, W) mapping a pair (ψ, φ) ∈ Ext_{C}^{n}(V, W) × Ext_{C}^{m}(U, V) to the composition Σ^{m}(ψ) ∘ φ ∈ Ext_{C}^{m+n}(U, W). This defines an associative graded R-algebra structure on Ext_{C}^{∗}(U, U), and Ext_{C}^{∗}(U, V) becomes an Ext_{C}^{∗}(V, V)-Ext_{C}^{∗}(U, U)-bimodule in this way. Moreover, for any object U in C there is a canonical graded algebra homomorphism Z^{∗}(C) ⟶ Ext_{C}^{∗}(U, U) mapping a natural transformation φ : Id_{C} → Σ^{n} to the morphism φ(U) : U → Σ^{n}(U), where n is any integer.

Example 2.7. Let A be an R-algebra. The derived bounded category of finitely generated Amodules D^{b}(A) together with the shift functor is triangulated, and for any bounded complexes of A-modules U, V we have Ext_{D^{b}(A)}^{∗}(U, V) = Ext_{A}^{∗}(U, V). If moreover A is finitely generated projective as R-module then HH^{∗}(A) = Ext_{A⊗_{R}A^{op}}(A, A); thus an element in HH^{n}(A) can be

represented by a morphism ζ : A ⟶ A[n] in the bounded derived category of A-A-bimodules

D^{b}(A ⊗_{R} A^{op}). For any bounded complex U of A-modules, the functor − ⊗^{𝕃} U applied to ζ yields

a morphism φ(U) : U → U[n] in D^{b}(A). This family of morphisms defines clearly a natural transformation from the identity functor on D^{b}(A) to [n]. By chasing signs one verifies that this is in fact an element φ ∈ Z^{∗}(D^{b}(A)). Thus the map ζ ↦ φ defines a homomorphism of graded algebras HH^{∗}(A) ⟶ Z^{∗}(D^{b}(A)) . This is not surjective, in general, not even in degree zero; see [9] for an example. For any bounded complex of A-modules U, evaluation at U defines a canonical graded algebra homomorphism (cf. 2.6) Z^{∗}(D^{b}(A)) → Ext_{A}^{∗}(U, U), and the composition of the two graded algebra homomorphisms HH^{∗}(A) ⟶ Z^{∗}(D^{b}(A)) ⟶ Ext_{A}^{∗}(U, U) is the canonical map

induced by the functor − ⊗_{A}^{𝕃} U from D^{b}(A ⊗_{R} A^{op}) to D^{b}(A).

Example 2.8. Let A be an R-algebra which is finitely generated projective as R-module and suppose that A is relatively R-injective as left A-module; this is the case, for instance, if A is symmetric and hence in particular if A = RG for some finite group G. Let \overline{mod}(A) be the R-stable category of the category mod(A) of finitely generated A-modules. That is, \overline{mod}(A) is the R-linear quotient category of mod(A) obtained from identifying to zero all relatively R-projective A-modules. Taking cokernels of relatively R-injective envelopes defines an equivalence Σ on \overline{mod}(A). Together with triangles induced by R-split exact sequences, \overline{mod}(A) endowed with Σ becomes a triangulated category (see e.g. [7]). Applied to (C, Σ) = (\overline{mod}(A), Σ) and A-modules, we get Tate cohomology, Ext_{C}^{∗}(U, V) = Êxt_{A}^{∗}(U, V). As in 2.7, for any A-module U we get graded algebra homomorphisms ĤH^{∗}(A) ⟶ Z^{∗}(\overline{mod}(A)) ⟶ Êxt_{A}^{∗}(U, U) whose composition is induced by the functor − ⊗_{A} U.

This map is neither surjective nor injective, in general; see [8] for an example.

If R = k is an algebraically closed field and C a triangulated category such that Z^{∗}(C) is finitely generated as k-algebra, one can associate with any object U in C a variety V_{C}(U) in the spirit of Carlson’s cohomology varieties for modules over group algebras, by taking for V_{C}(U) the maximal ideal spectrum of the quotient of Z^{∗}(C) by the kernel of the algebra homomorphism Z^{∗}(C) → Ext_{C}^{∗}(U, U). In fact, for this to make sense it would suffice if the quotient of Z^{∗}(C) by some nilpotent ideal is finitely generated as k-algebra. This motivates the next Proposition, which requires the following concept from Rouquier [14]. Let (C, Σ) be a triangulated category and let M be an object in C. We denote by ⟨M⟩_{1} the full additive subcategory of C consisting of all objects isomorphic to finite direct sums of summands of the objects Σ^{n}(M), with n ∈ ℤ. For i ≥ 2 we define inductively ⟨X⟩_{i} as the full additive subcategory of C consisting of all objects isomorphic to direct summands of objects Z for which there exists an exact triangle X → Y → Z → Σ(X) with X in ⟨M⟩_{i−1} and Y in ⟨M⟩_{1}. Following [14, 3.6], the dimension of C, denoted dim(C), is the smallest positive integer d for which there exists an object M in C such that ⟨M⟩_{d+1} = C.

Proposition 2.9. Let (C, Σ) be a triangulated category and let M be an object in C for which there exists a positive integer d satisfying ⟨M⟩_{d+1} = C. Let N be the kernel of the canonical graded algebra homomorphism Z^{∗}(C) → Ext_{C}^{∗}(M, M). We have N^{2^{d}} = {0}.

Proof. Let m, n be integers and let φ ∈ Z^{m}(C) and ψ ∈ Z^{n}(C). If φ ∈ N then φ(M) = 0 and hence φ(X) = 0 for all X in ⟨M⟩_{1}. What we show is that if φ(X) = 0 = ψ(X) for all objects in X in ⟨M⟩_{i} then (ψφ)(Z) = 0 for all objects in ⟨M⟩_{i+1}; the result follows then by induction. This is a standard argument: consider an exact commutative diagram in C of the form

\xymatrix{X \ar[r] \ar[d]_{0} & Y \ar[r] \ar[d]_{0} & Z \ar[r] \ar[d]_{φ(Z)} \ar@{-->}[dl] & Σ(X) \ar[d]^{0} \\ Σ^{m}(X) \ar[r] \ar[d]_{0} & Σ^{m}(Y) \ar[r] \ar[d]_{0} & Σ^{m}(Z) \ar[r] \ar[d]_{ψ(Σ^{m}(Z))} & Σ^{m+1}(X) \ar[d]^{0} \ar@{-->}[dl] \\ Σ^{m+n}(X) \ar[r] & Σ^{m+n}(Y) \ar[r] & Σ^{m+n}(Z) \ar[r] & Σ^{m+n+1}(X)}

with X in ⟨M⟩_{i} and Y in ⟨M⟩_{1}. Since φ(Z) composed with the morphism Σ^{m}(Z) → Σ^{m+1}(X) is zero, φ(Z) factors through Σ^{m}(Y). For similar reasons, ψ(Σ^{m}(Z)) factors through Σ^{m+1}(X). But then ψ(Σ^{m}(Z)) ∘ φ(Z) factors through two consecutive morphisms in an exact triangle, hence is zero. □

Proof of Proposition 1.2. Let B be a block algebra over an algebraically closed field k of prime characteristic p. We refer to [10, §5] for the definition of block cohomology. By [10, 5.6.(iii)] there is a canonical graded algebra homomorphism H^{∗}(B) ⟶ HH^{∗}(B). Thus, for any B-module U, we have a sequence of canonical algebra homomorphisms

H^{∗}(B) → HH^{∗}(B) → Z^{∗}(D^{b}(B)) → Ext_{B}^{∗}(U, U)

By 2.9 there exists a finitely generated B-module U such that N = ker(Z^{∗}(D^{b}(B)) → Ext_{A}^{∗}(U, U)) is a nilpotent ideal (for instance U = B/J(B) would have this property because then ⟨U⟩_{d+1} = D^{b}(B) for some integer d bounded by the Loewy length of B by [14, 3.7]). Since Ext_{B}^{∗}(U, U) is Noetherian over H^{∗}(G, k) (cf. [1, 4.2.4]) it follows from the commutative diagram in [11, 4.2.(iii)] that Ext_{A}^{∗}(U, U) is Noetherian as module over H^{∗}(B). Thus its submodule isomorphic to Z^{∗}(D^{b}(B))/N is Noetherian over H^{∗}(B). □

Proof of Proposition 1.3. Set ΔP = {(u, u) | u ∈ P } ⊆ P × P. The “diagonal” induction functor Ind_{ΔP}^{P ×P} sends the trivial kΔP -module k to kP viewed as k(P × P)-module via (u, v) · y = uyv^{−1} for u, v, y ∈ P. This functor induces hence a graded homomorphism δ_{P} : H^{∗}(P, k) → HH^{∗}(kP) (this is well-known; see e.g. [10, §4] for some more details). When composed with the canonical homomorphisms HH^{∗}(kP) → Z^{∗}(D^{b}(kP)) → Ext_{kP}^{∗} (k, k) = H^{∗}(P, k) this yields the identity on H^{∗}(P, k); in particular, the canonical map Z^{∗}(D^{b}(kP)) → H^{∗}(P, k) is surjective. Since k is the unique simple kP -module, up to isomorphism, we have D^{b}(kP) = ⟨k⟩_{d+1} for some positive integer d, and hence the kernel N of this map is nilpotent by 2.9. The result follows. □

Proof of Proposition 1.4. Let ζ(U) ∈ \overline{Hom}_{A}(U, Ω(U)) such that ζ(U) represents an Auslander-Reiten sequence ending in U. Then ζ(U) is almost vanishing in the sense of Happel [7, I.4.1]; that is, for any morphism β : W → U in \overline{mod}(A) which is not a split epimorphism, we have ζ(U)∘β = 0, and, for any morphism γ : Ω(U) → W in \overline{mod}(A) which is not a split monomorphism, we have γ ∘ ζ(U) = 0. For any integer n set ζ(Ω^{n}(U)) = (−1)^{n}Ω^{n}(ζ(U)); this is possible even if U has an odd period thanks to the assumption char(k) = 2 in that case. For any indecomposable non projective A-module V not isomorphic to Ω^{n}(U) for any integer n set ζ(V) = 0 in \overline{Hom}_{A}(V, Ω(V)). Note that Ω^{n}(ζ(U)) represents an Auslander-Reiten sequence ending in Ω^{n}(U). In order to show that this family of morphisms defines an element in the graded center it suffices to show that for any two indecomposable non-projective A-modules X, Y and any α ∈ \overline{Hom}_{A}(X, Y) the diagram in \overline{mod}(A)

\xymatrix{X \ar[r]^{ζ(X)} \ar[d]_{α} & Ω(X) \ar[d]^{Ω(α)} \\ Y \ar[r]_{ζ(Y)} & Ω(Y)}

is commutative. It suffices in fact to do this for X and Y running over a set of representatives of the isomorphism classes of indecomposable non projective modules. If none of X, Y is isomorphic to Ω^{n}(U) for some integer n this holds trivially as then both ζ(X) and ζ(Y) are zero. Thus we may assume that one of X or Y is equal to Ω^{n}(U) for some integer n. By applying Ω^{−n} to such a diagram we may in fact assume that on of X or Y is equal to U. Suppose that both X = Y = U. Since k is algebraically closed and since U is indecomposable non projective, the algebra \overline{End}_{A}(U) is split local; that is, we can write α = λĪd_{U} + ρ for some λ ∈ k and some ρ ∈ J(\overline{End}_{A}(U)). If ρ = 0 then α is a multiple of the identity on U , hence Ω(α) is the same multiple of the identity on Ω(U), and the commutativity of the above diagram is clear. If λ = 0 we have α = ρ, hence neither α nor Ω(α) is a split epimorphism or split monomorphism in \overline{mod}(A) and hence ζ(U) ∘ α = 0 = Ω(α) ∘ ζ(U) in that case. For general α ∈ \overline{End}_{A}(U) the required commutativity follows from adding the two previous cases.

If X = U but Y ≅ U̸ then Ω(X) ≅ Ω̸(Y), hence Ω(α) ∘ ζ(X) = 0. We need to show that then also ζ(Y) ∘ α = 0. If Y ≅ Ω̸^{n}(U) for any integer n then ζ(Y) = 0, so this is clear. Suppose that Y ≅ Ω^{n}(U) for some integer n. Since Y ≅ X̸, the morphism α is not a split epimorphism and therefore ζ(Y)∘α = 0 because ζ(Y) represents an Auslander-Reiten sequence ending in Y. Similarly, if Y = U but X ≅ U̸, then α is not a split epimorphism, hence ζ(Y) ∘ α = 0. To see that also Ω(α) ∘ ζ(X) = 0 we distinguish as before two cases: if X ≅ Ω̸^{n}(U) for any integer n then ζ(X) = 0, so the commutativity is clear. If X ≅ Ω^{n}(U) for some integer n then ζ(X) represents an Auslander-Reiten sequence ending in Y and Ω(α) is not a split monomorphism because Ω(X) ≅ Ω̸(Y). Hence Ω(α) ∘ ζ(X) = 0 as required. □

3 Compatible adjunctions for functors between graded categories

Definition 3.1 Let C, D be R−linear categories, endowed with R-linear equivalences Σ : C → C and Δ : D → D. Let F : C → D and G : D → C be R-linear functors such that G is left adjoint to F, and let f : Id_{D} → FG be the unit and g : GF → Id_{C} the counit of an adjunction. Suppose that there are isomorphisms of functors a : ΔF ≅ FΣ and b : GΔ ≅ ΣG.

We say that the left adjunction given by f and g is compatible with Σ and Δ with respect to a and b if the following diagrams of natural transformations are commutative:

\xymatrix{Δ \ar[r]^{Δf} \ar[d]^{fΔ} & ΔFG \ar[d]_{aG} & GΔF \ar[r]^{bF} \ar[d]^{Ga} & ΣGF \ar[d]_{Σg} \\ FGΔ \ar[r]_{Fb} & FΣG & GFΣ \ar[r]_{gΣ} & Σ}

If G is also right adjoint to F and f′ : Id_{C} → GF the unit, g′ : FG → Id_{D} the counit of a right adjunction, we say that the adjunction of F and G given by the units and counints f, g, f′, g′ is compatible with Σ and Δ with respect to a and b, if the left adjunction of G to F is compatible with Σ and Δ with respect to a, b, and if the left adjunction of F to G is compatible with Σ and Δ with respect to a^{−1}, b^{−1}; in other words, if in addition to the two previous diagrams the two following diagrams are commutative as well:

\xymatrix{Σ \ar[r]^{Σf′} \ar[d]^{f′Σ} & ΣGF \ar[d]_{b^{−1}F} & FΣG \ar[r]^{a^{−1}G} \ar[d]^{Fb^{−1}} & ΔFG \ar[d]_{Δg′} \\ GFΣ \ar[r]_{Ga^{−1}} & GΔF & FGΔ \ar[r]_{g′Δ} & Δ}

Remark 3.2. Especially when dealing with derived categories, the natural transformations a and b will typically be equalities ΔF = FΣ and GΔ = ΣG, in which case the four compatibility diagrams reduce to equalities of natural transformations Δf = fΔ, Σg = gΣ, Σf′ = f′Σ, Δg′ = g′Δ . Verdier [15] defined an exact functor of triangulated categories (C, Σ) → (D, Δ) to be an additive functor F : C → D satisfying FΣ = ΔF and mapping exact triangles to exact triangles. Other authors allow more flexibility in that they require only a natural isomorphism FΣ ≅ ΔF and 3.1 is intended to accomodate this extra degree of generality; the price to pay in this context is that the notion of an adjoint pair of functors between triangulated categories as in 3.1 becomes more formally involved.

The above definition is redundant in various ways - the following Lemmas describe this in detail. Lemma 3.3 says that the commutativity of the first two diagrams in 3.1 is equivalent:

Lemma 3.3. With the notation of 3.1, if one of the following two diagrams is commutative, so is the other:

\xymatrix{Δ \ar[r]^{Δf} \ar[d]^{fΔ} & ΔFG \ar[d]_{aG} & GΔF \ar[r]^{bF} \ar[d]^{Ga} & ΣGF \ar[d]_{Σg} \\ FGΔ \ar[r]_{Fb} & FΣG & GFΣ \ar[r]_{gΣ} & Σ}

Proof. If the first diagram is commutative, there is a commutative diagram

\xymatrix{GΔF \ar[r]^{GΔfF} \ar[d]_{GfGΔF} & GΔFGF \ar[r]^{GΔFg} \ar[d]_{GaGF} & GΔF \ar[d]^{Ga} \\ GFGΔF \ar[r]^{GFbF} \ar[d]_{gGΔF} & GFΣGF \ar[r]^{GFΣg} \ar[d]_{gΣGF} & GFΣ \ar[d]^{gΣ} \\ GΔF \ar[r]_{bF} & ΣGF \ar[r]_{Σg} & Σ}

The upper left square in this diagram is obtained by composing the first diagram in 3.1 with G on the left and F on the right. The other squares are commutative because f and g are natural transformations. Moreover, the first row and the first column of this diagram are the identity transformation. Thus by considering the four outer corners of the above 3 by 3 diagram it follows that the second diagram in 3.3 is commutative. A dual argument shows that the commutativity of the second diagram in 3.3 implies the commutativity of the first one. □

By the same argument, the commutativity of the third and fourth diagram in 3.1 is equivalent. When specialised to the case where a and b are equalities, this says that the equality Δf = fΔ holds if and only if Σg = gΣ, and similarly Σf′ = f′Σ if and only if Δg′ = g′Δ. If the first diagram in 3.1 commutes, then the natural isomorphisms a and b determine each other:

Lemma 3.4. With the notation of 3.1, suppose the following diagram is commutative:

\xymatrix{Δ \ar[r]^{Δf} \ar[d]^{fΔ} & ΔFG \ar[d]_{aG} \\ FGΔ \ar[r]_{Fb} & FΣG}

Then a = (FΣg) ∘ (FbF) ∘ (fΔF) and b = (gΣG) ∘ (GaG) ∘ (GΔf).

Proof. In order to prove the formula for a we observe that there is a commutative diagram

\xymatrix{ΔF \ar[r]^{ΔfF} \ar[d]^{fΔF} & ΔFGF \ar[r]^{ΔFg} \ar[d]^{aGF} & ΔF \ar[d]^{a} \\ FGΔF \ar[r]_{FbF} & FΣGF \ar[r]_{FΣg} & FΣ}

Indeed, the left square is obtained by composing the first diagram in 3.1 (which is commutative by the assumptions) with F on the right. The second square uses that g is a natural transformation from GF to Id_{C}. Moreover, the first row is the identity transformation on ΔF by the general properties of adjunctions. The formula for a follows. The formula for b is obtained by composing the first diagram in 3.1 with G on the left and then using g again, getting a commutative diagram of the form

\xymatrix{GΔ \ar[r]^{GΔf} \ar[d]^{GfΔ} & GΔFG \ar[d]_{GaG} \\ GFGΔ \ar[r]^{GFb} \ar[d]^{gGΔ} & GFΣG \ar[d]_{gΣG} \\ GΔ \ar[r]_{b} & ΣG}

Observing again that the left column is the identity transformation on GΔ yields the formula for b. □

If all four diagrams in 3.1 are commutative, then the left and right adjunctions are related:

Lemma 3.5. With the notation of 3.1, assume that all four diagrams in 3.1 are commutative. Then

(Δg′) ∘ (Δf) = (g′Δ) ∘ (fΔ)
(Σg) ∘ (Σf′) = (gΣ) ∘ (f′Σ)

Proof. To see the first equality (of natural transformations on Δ) combine the first diagram in 3.1 with the fourth diagram in 3.1 rotated by 90 degree counterclockwise, and for the second equality (of natural transformations on Σ) combine the second diagram in 3.1 with the third diagram rotated by 90 degree clockwise. □

Compatible adjunctions can be “added”:

Proposition 3.6. Let C, D be R-linear categories endowed with R-linear self-equivalences Σ, Δ, respectively. Let F, F′ : C → D and G, G′ : D → C be R-linear functors. Suppose that F, G and F′, G′ are two pairs of adjoint functors which are both compatible with Σ, Δ with respect to natural isomorphisms of functors a : Δ ∘ F ≅ F ∘ Σ, b : G ∘ Δ ≅ Σ ∘ G and a′ : Δ ∘ F′ ≅ F′ ∘ Σ b′ : G′ ∘ Δ ≅ Σ ∘ G′. Then the direct sum of of the adjunction isomorphisms of the adjoint pairs of functors F, G and F′, G′ is an adjunction between the functors F ⊕ F′ and G ⊕ G′ which is compatible with Σ, Δ with respect to a ⊕ a′, b ⊕ b′.

Proof. Let U be an object in C and let V be an object in D. The sum of the adjunction isomorphisms Hom_{C}(U, G(V)) ≅ Hom_{D}(F(U), V) and Hom_{C}(U, G′(V)) ≅ Hom_{D}(F′(U), V) yields obviously a natural isomorphism

Hom_{C}(U, G(V) ⊕ G′(V)) ≅ Hom_{D}(F(U) ⊕ F′(U), V)

showing that F ⊕ F′ is left adjoint to G ⊕ G′. Similarly one shows that F ⊕ F′ is right adjoint to G ⊕ G′. We determine the unit Id_{C} → (G ⊕ G′)(F ⊕ F′) of this adjunction in terms of the units f : Id_{C} → GF and f′ : Id_{C} → G′F′ as follows. The morphism f(U) : U → GF(U) is the image of Id_{F(U)} under the appropariate adjunction isomorphism; similarly for f′(U). Chasing Id_{F(U)⊕F′(U)} through the isomorphism

Hom_{D}(F(U) ⊕ F′(U), F(U) ⊕ F′(U)) ≅ Hom_{C}(U, (G ⊕ G′)(F(U) ⊕ F′(U))) =

= Hom_{C}(U, GF(U) ⊕ GF′(U) ⊕ G′F(U) ⊕ G′F′(U))

shows that the counit Id_{C} ⟶ (G ⊕ G′)(F ⊕ F′) is in fact simply obtained by taking the “sum” of f, f′ evaluated at the object U, that is,

( \begin{matrix}f(U) \\ f′(U) \end{matrix} ) : U ⟶ GF(U) ⊕ G′F′(U)

followed by the canonical inclusion

GF(U) ⊕ G′F′(U) ⟶ (G ⊕ G′)(F ⊕ F′)(U) = GF(U) ⊕ GF′(U) ⊕ G′F(U) ⊕ G′F′(U))

A similar statement holds for the counits. Since units and counits for F ⊕ F′, G ⊕ G′ are essentially obtained by adding those for the two pairs of adjoint functors, the result follows. □

Compatible adjunctions can be composed:

Proposition 3.7. Let C, D, E be R-linear categories endowed with R-linear self-equivalences Σ, Δ, Γ, respectively. Let F : C → D, G : D → C be adjoint R-linear functors, compatible with Σ, Δ with respect to isomorphisms a : FΣ ≅ ΔF, b : GΔ ≅ ΣG. Let F′ : D → E, G′ : E → D be adjoint R-linear functors, compatible with Δ, Γ with respect to isomorphisms a′ : F′Δ ≅ ΓF′, b′ : G′Γ ≅ ΔG′. The composition of adjunction isomorphisms of the adjoint pairs of functors F, G and F′, G′ is an adjunction between the compositions of functors F′ ∘ F, G ∘ G′ which is compatible with Σ, Γ with respect to the isomorphisms (a′F) ∘ (F′a) : F′FΣ ≅ ΓF′F and (bG′) ∘ (Gb′) : GG′Γ ≅ ΣGG′.

Proof. Straightforward verification. □

§4 Adjunction and transfer

In the context of group cohomology and Hochschild cohomology of symmetric algebras, transfer appears to be the translation of the adjunction principle to cohomology rings arising from triangulated categories. The following definition extends the pattern of various sources such as [3], [4], [10], [12].

Definition 4.1 Let C, D be R-linear categories endowed with R-linear equivalences Σ : C → C and Δ : D → D. Let F : C → D and G : D → C be R-linear functors which are adjoint to each other, compatible with Σ and Δ with respect to natural transformations a : Δ ∘ F ≅ F ∘ Σ and b : G ∘ Δ ≅ Σ ∘ G. Denote by f : Id_{D} → FG, g : GF → Id_{C} and f′ : Id_{C} → GF, g′ : FG → Id_{D}, the units and counints of adjunctions which are compatible with Σ and Δ, with respect to a, b. For any integer n and any natural transformation φ : Id_{C} → Σ^{n} we define the natural transformation tr_{F}(φ) : Id_{D} → Δ^{n} to be the natural transformation equal to the composition

Id_{D} ⟶^{f} FG = FId_{C}G ⟶^{FφG} FΣ^{n}G ≅ Δ^{n}FG ⟶^{Δ^{n}g′} Δ^{n} ,

where the isomorphism FΣ^{n}G ≅ Δ^{n}FG is given by applying repeatedly a. Similarly, for any two objects U, V in D, we define the relative transfer map tr_{F}(U, V) : Ext_{C}^{∗}(G(U), G(V)) ⟶ Ext_{D}^{∗}(U, V) to be the graded R-linear map sending a morphism α ∈ Hom_{C}(G(U), Σ^{n}(G(V))) to the morphism β ∈ Hom_{D}(U, Δ^{n}(V)), which is equal to the composition

U ⟶^{f(U)} FG(U) ⟶^{F(α)} FΣ^{n}G(V) ≅ Δ^{n}FG(V) ⟶^{Δ^{n}(g′)} Δ^{n}(V) ,

where the isomorphism FΣ^{n}G(V) ≅ Δ^{n}FG(V) is again given by evaluating the natural transformation a.

Remarks 4.2.

(a) The definition of tr_{F} makes use of the unit f and the counit g′; using the unit f′ and counit g instead, one gets analogously tr_{G}. The transfer map tr_{F} defined above depends on the choices of the adjunction isomorphisms as well as on the choices of the natural transformations a, b. Whenever we do not specify these choices in a statement, this means that we implicitly assert, that either the statement does not depend on this choice or the context determines such a choice canonically. It can in fact be useful to play two different choices off against each other. In practice, the natural isomorphisms a, b are frequently given canonically by the context; they may sometimes just be equalities of functors.

(b) Examples of pairs of biadjoint functors include induction/restriction between algebras of finite groups, Harish-Chandra induction/restriction, and more generally, functors between module categories of symmetric algebras induced by bimodules which are finitely generated projective on both sides; see §7 for details.

(c) With the notation of 4.1, the degree zero component tr_{F} sends a natural transformation φ : Id_{C} → Id_{C} to the natural transformation

tr_{F}(φ) : Id_{D} ⟶^{f} FG = FId_{C}G ⟶^{FφG} FId_{C}G = FG ⟶^{g′} Id_{D} .

Similarly, for any two objects U, V in D, the degree zero component of the relative transfer map tr_{F}(U, V) is the map t_{F}(U, V) : Hom_{C}(G(U), G(V)) ⟶ Hom_{D}(U, V) sending a morphism α : G(U) → G(V) to the morphism β = g′(V) ∘ F(α) ∘ f(U) : U → V.

Theorem 4.3. Let C, D be R-linear categories endowed with R-linear equivalences Σ : C → C and Δ : D → D. Let F : C → D and G : D → C be R-linear functors which are adjoint to each other, compatible with Σ and Δ with respect to natural transformations a : Δ ∘ F ≅ F ∘ Σ and b : G ∘ Δ ≅ Σ ∘ G. The map sending a natural transformation φ : Id_{C} → Σ^{n} to tr_{F}(φ) for any integer n induces a graded R-linear map tr_{F} : Z^{∗}(C) ⟶ Z^{∗}(D) such that, for any object V in D, the diagram

\xymatrix{Z^{∗}(C) \ar[r] \ar[d]_{_{tr_{F}}} & Ext_{C}^{∗}(G(V), G(V)) \ar[d]^{tr_{F} (V,V)} \\ Z^{∗}(D) \ar[r] & Ext_{D}^{∗}(V, V)}

is commutative, where the horizontal maps are the canonical graded algebra homomorphisms. More explicitly, for any integer n, any natural transformation φ : Id_{C} → Σ^{n} and any object V in D we have tr_{F}(φ)(V) = tr_{F}(V, V)(φ(G(V)) .

Proof. Let n be an integer and let φ ∈ Z^{n}(C); that is, φ : Id_{C} → Σ^{n} is a natural transformation satisfying φΣ = (−1)^{n}Σφ. We need to show that tr_{F}(φ)Δ = (−1)^{n}Δtr_{F}(φ). To see this, we consider the following diagram of functors and natural transformations:

[figure: Δ; Δf; ΔFG; ΔFId_{C}G; ΔFφG; aId_{C}G; ΔFΣ^{n}G; ≅; aΣ^{n}G; Δ^{n+1}FG; Δ^{n+1}g′; Δ^{n}aG; Δ^{n+1}; (−1)^{n}; FΣId_{C}G; FΣφG; FΣΣ^{n}G; ≅; (−1)^{n}; Δ^{n}FΣG; (−1)^{n}; FId_{C}ΣG; FφΣGΔ; FΣ^{n}ΣG; ≅; Δ^{n}FΣG; Δ; fΔ; FGΔ; FId_{C}GΔ; FId_{C}b; FφGΔ; FΣ^{n}GΔ; ≅; FΣ^{n}b; Δ^{n}FGΔ; Δ^{n}Fb; Δ^{n}g′Δ; Δ^{n+1}]

The first row is Δtr_{F}(φ) and the last row is tr_{F}(φ)Δ. The left and right rectangle of the diagram commute by the compatibility assumption. The commutativity of the remaining squares is easily checked by using either the naturality, the compatibility condition or the assumption φΣ = (−1)^{n}Σφ. The unmarked isomorphisms are obtained from iterating a. We need to check the commutativity of the diagram in the statement of the theorem. The left side of that diagram is tr_{F}(φ) evaluated at V, hence equal to the composition

V ⟶^{f(V)} FG(V) ⟶^{FφG(V)} FΣ^{n}G(V) ≅ Δ^{n}FG(V) ⟶^{Δ^{n}g′(V)} Δ^{n}(V)

This is clearly equal to the map tr_{F}(V, V) evaluated at φ(G(V)). □

Corollary 4.4. With the notation of 4.3, setting π_{F} = tr_{F}(Id_{Id_{C}}), we have π_{F} ∈ Z^{0}(D).

Proof. The identity transformation Id_{Id_{C}} commutes obviously with Σ, hence belongs to Z^{0}(C) and thus its image π_{F} = tr_{F}(Id_{Id_{C}}) belongs to Z^{0}(D) by 4.3. □

Definition 4.5. With the notation above, we call π_{F} = tr_{F}(Id_{Id_{C}}) the relatively F-projective element in Z^{0}(D).

The transfer map tr_{F} depends additively on F:

Proposition 4.6. Let C, D be R-linear categories endowed with R-linear self-equivalences Σ, Δ, respectively. Let F, F′ : C → D and G, G′ : D → C be R-linear functors. Suppose that F, G and F′, G′ are two pairs of adjoint functors which are both compatible with Σ, Δ with respect to natural isomorphisms of functors a : Δ ∘ F ≅ F ∘ Σ, b : G ∘ Δ ≅ Σ ∘ G and a′ : Δ ∘ F′ ≅ F′ ∘ Σ b′ : G′ ∘ Δ ≅ Σ ∘ G′.

(i) We have tr_{F⊕F′} = tr_{F} + tr_{F′} . (ii) For any two objects V, W in D, any α ∈ Hom_{C}(G(V), G(W)) and any α′ ∈ Hom_{C}(G′(V), G′(W)) we have tr_{F⊕F′}(V, W)(α ⊕ α′) = tr_{F}(V, W)(α) + tr_{F′}(V, W)(α′) .

Proof. Straightforward verification using 3.6. □

Transfer maps compose as expected:

Proposition 4.7. Let C, D, E be R-linear categories endowed with R-linear self-equivalences Σ, Δ, Γ, respectively. Let F : C → D, G : D → C be adjoint R-linear functors, compatible with Σ, Δ with respect to isomorphisms a : FΣ ≅ ΔF, b : GΔ ≅ ΣG. Let F′ : D → E, G′ : E → D be adjoint R-linear functors, compatible with Δ, Γ with respect to isomorphisms a′ : F′Δ ≅ ΓF′, b′ : G′Γ ≅ ΔG′.

(i) We have tr_{F′∘F} = tr_{F′} ∘ tr_{F} . (ii) For any two objects V, W in E we have tr_{F′∘F}(V, W) = tr_{F′}(V, W) ∘ tr_{F}(G(V), G(W)) .

Proof. Straightforward verification, using 3.7. □

The class of morphisms which are in the image of a transfer map form an “ideal”:

Proposition 4.8. Let C, D be R-linear categories endowed with R-linear self-equivalences Σ, Δ, respectively. Let F : C → D and G : D → C be R-linear functors which are adjoint to each other, compatible with Σ, Δ with respect to natural isomorphisms of functors a : Δ ∘ F ≅ F ∘ Σ and b : G ∘ Δ ≅ Σ ∘ G. Let U, V, W be objects in D, and let α ∈ Ext_{C}^{∗}(G(V), G(W)).

(i) For any β ∈ Ext_{D}^{∗}(U, V) we have tr_{F}(V, W)(α)β = tr_{F}(U, W)(αG(β)). (ii) For any γ ∈ Ext_{D}^{∗}(W, U) we have γtr_{F}(V, W)(α) = tr_{F}(V, U)(G(γ)α).

Proof. We prove (i); the proof of (ii) is analogous. We may assume that α and β are homogeneous of degree m and n, respectively. That is,

α : G(V) ⟶ Σ^{m}G(W)

is a morphism in C and

β : U ⟶ Δ^{n}(V)

is a morphism in D. The product αG(β) is the composition of morphisms

G(U) ⟶^{G(β)} GΔ^{n}(V) ≅ Σ^{n}G(V) ⟶^{Σ^{n}(α)} Σ^{n+m}G(W)

With the notation of 4.3, applying tr_{F}(U, W) to αG(β) yields the composition of morphisms

U ⟶^{f(U)} FG(U) ⟶^{FG(β)} FGΔ^{n}(V) ≅ FΣ^{n}G(V) ⟶^{FΣ^{n}(α)}
FΣ^{n+m}G(W) ≅ Δ^{n+m}FG(W) ⟶^{Δ^{n+m}g′(W)} Δ^{n+m}(W)

We need to compare this to the product tr_{F}(V, W)(α)β, which is equal to the composition

U ⟶^{β} Δ^{n}(V) ⟶^{Δ^{n}f(V)} Δ^{n}FG(V) ⟶^{Δ^{n}F(α)}
Δ^{n}FΣ^{m}G(W) ≅ Δ^{n+m}FG(W) ⟶^{Δ^{n+m}g′(W)} Δ^{n+m}(W)

The right ends of these two morphisms coincide because FΣ^{n}(α) and Δ^{n}F(α) are equal modulo the identifications induced by a and b. We need to show that the two left morphisms compose to the same morphism modulo identifications; that is, we need to show that the diagram

U ⟶^{f(U)} FG(U) ⟶^{FG(β)} FGΔ^{n}(V)
∥  ↓≅
U ⟶^{β} Δ^{n}(V) ⟶^{Δ^{n}f(V)} Δ^{n}FG(V)

commutes. Now we have a commutative diagram

U  ⟶^{f(U)} FG(U)
β↓  ↓^{FG(β)}
Δ^{n}(V) ⟶_{f(Δ^{n}(V))} FGΔ^{n}(V)

because f is a natural transformation, and we have a commutative diagram

Δ^{n}(V) ⟶_{f(Δ^{n}(V))} FGΔ^{n}(V)
∥  ↓≅
Δ^{n}(V) ⟶_{Δ^{n}f(V)} Δ^{n}FG(V)

because of the compatibility conditions in the definition 4.1. Together they yield the required statement. □

5 Stable elements

Definition 5.1. Let C, D be R-linear categories endowed with R-linear self-equivalences Σ, Δ, respectively. Let F : C → D be an R-linear functor such that there is a natural isomorphism a : FΣ ≅ ΔF. Given an integer n, a natural transformation φ : Id_{C} → Σ^{n} is called F-stable, if there is a natural transformation ψ : Id_{D} → Δ^{n} such that the diagram

F ⟶^{Fφ} FΣ^{n}
∥  ↓≅
F ⟶_{ψF} Δ^{n}F

commutes, where the isomorphism FΣ^{n} ≅ Δ^{n}F is induced by a. An element φ ∈ Z^{∗}(C) is called F-stable, if there is ψ ∈ Z^{∗}(D) such that the previous diagram is commutative for the components of φ and ψ in any degree n; in that case we write abusively Fφ = ψF. We denote by Z_{F}^{∗}(C) the set of F-stable elements in Z^{∗}(C).

The notion of F-stability depends on the choice of the natural isomorphism a : FΣ ≅ ΔF.

Proposition 5.2. Let C, D be R−linear categories endowed with R-linear self-equivalences Σ, Δ, respectively. Let F : C → D be an R−linear functor such that there is a natural isomorphism a : FΣ ≅ ΔF. Then the F-stable elements Z_{F}^{∗}(C) form a graded subalgebra of Z^{∗}(C).

Proof. This follows from adding and composing commutative diagrams as in definition 5.1. □

The image of a transfer map is a bimodule for the subalgebra of stable elements. This is a “general abstract nonsense version” of well-known Frobenius reciprocity type statements in cohomology such as [10, 3.4] in Hochschild cohomology, for instance.

Theorem 5.3. Let C, D be R-linear categories endowed with R-linear self-equivalences Σ, Δ, respectively. Let F : C → D and G : D → C be R-linear adjoint functors compatible with Σ, Δ. Let φ ∈ Z_{F}^{∗}(C) and ψ ∈ Z^{∗}(D) such that Fφ = ψF.

(i) We have φG = Gψ; in particular, ψ ∈ Z_{G}^{∗}(D). (ii) For any τ ∈ Z^{∗}(C) we have tr_{F}(φτ) = ψtr_{F}(τ) and tr_{F}(τφ) = tr_{F}(τ)ψ. (iii) We have tr_{F}(φ) = π_{F}ψ. In particular, Im(tr_{F}) is a sub-Z_{G}^{∗}(D)-Z_{G}^{∗}(D)-bimodule of Z^{∗}(D) and tr_{F}(Z_{F}^{∗}(C)) = π_{F}Z_{G}^{∗}(D).

Proof. We may assume that φ, ψ are homogeneous of degree n and that τ is homogeneous of degree m. In order to prove (i) we will show that there is a commutative diagram of the following form:

G ⟶^{φG} Σ^{n}G
↓  ↓
GFG ⟶^{GFφG} GFΣ^{n}G
∥  ↓≅
GFG ⟶_{GψFG} GΔ^{n}FG
↓  ↓
G ⟶_{Gψ} GΔ^{n}

Indeed, the upper square is obtained from the square of adjunction units

Id_{C} ⟶^{φ} Σ^{n}
↓  ↓
GF ⟶^{GFφ} GFΣ^{n}

composed with G on the right. The lower square is obtained from the square of adjunction counits

FG ⟶_{ψFG} Δ^{n}FG
↓  ↓
Id_{D} ⟶_{ψ} Δ^{n}

composed with G on the left. For the square in the middle, we use the hypothesis on φ, ψ according to which we have a commutative diagram

F ⟶^{Fφ} FΣ^{n}
∥  ↓≅
F ⟶_{ψF} Δ^{n}F

which we then compose with G on both sides. Since the composition of adjunction units and counits G → GFG → G is the identity transformation, this proves (i). For the first equality in (ii) we will show that there is a commutative diagram of the form

Id_{D} ⟶ FG ⟶^{FφτG} FΣ^{n+m}G ⟶^{≅} Δ^{n+m}FG ⟶ Δ^{n+m}
∥  ∥  ↓≅  ∥  ∥
Id_{D} ⟶ FG ⟶^{ψFτG} Δ^{n}FΣ^{m}G ⟶^{≅} Δ^{n+m}FG ⟶ Δ^{n+m}
ψ↓  ↓^{ψFG}  ∥  ∥  ∥
Δ^{n} ⟶ Δ^{n}FG ⟶^{Δ^{n}FτG} Δ^{n}FΣ^{m}G ⟶^{≅} Δ^{n+m}FG ⟶ Δ^{n+m}

obtained as follows. The first row is tr_{F}(φτ), where one notes that φτ = Σ^{m}(φ) ∘ τ : Id_{C} → Σ^{n+m}. The second row is obtained from the first by making use of the hypothesis ψF = Fφ. The last row uses that ψ is a natural transformation. The last row is easily seen to be equal to Δ^{n}(tr_{F}(τ)) and hence upon composition with ψ (the left lower vertical arrow in the diagram) is equal to ψtr_{F}(τ) as claimed. A similar argument proves the second equality in (ii). One can show this also using 4.3 and 4.8. Statement (iii) follows from (ii) applied to τ = Id_{Id_{C}}. The last two statements follow from (ii) and (iii), respectively. □

Corollary 5.4. Let C, D be R-linear categories endowed with R-linear self-equivalences Σ, Δ, respectively. Let F : C → D and G : D → C be R-linear adjoint functors compatible with Σ, Δ. We have Z_{F}^{∗}(C) ⊆ Z_{GF}^{∗}(C).

Proof. Let φ ∈ Z^{n}(C) and ψ ∈ Z^{n}(D) such that Fφ = ψF, where n is an integer. By 5.3 we get Gψ = φG, and hence (GF)φ = GψF = φ(GF), which shows that φ is GF-stable. □

§6 Normalised Transfer

We extend some of the machinery in [10, §3] to the present context. The following is [10, 3.1.(ii)] in the case of Hochschild cohomology.

Definition 6.1 Let C, D be R-linear categories endowed with R-linear equivalences Σ : C → C and Δ : D → D. Let F : C → D and G : D → C be R-linear functors which are adjoint to each other, compatible with Σ and Δ. Suppose that the relatively projective element π_{F} = tr_{F}(Id_{Id_{C}}) is invertible in Z^{0}(D). The normalised transfer map Tr_{F} : Z^{∗}(C) ⟶ Z^{∗}(D) is the graded R-linear map defined by Tr_{F}(φ) = (π_{F})^{−1}tr_{F}(φ) for all φ ∈ Z^{∗}(C).

This makes sense as Z^{∗}(D) is a module over Z^{0}(D). Unlike tr_{F}, the normalised transfer Tr_{F} restricted to appropriate subalgebras of stable elements does no longer depend on the choices of a compatible adjunction (but still depends on the choice of the transformations a, b), provided that the relatively projective elements are invertible; this generalises [10, 3.6]:

Theorem 6.2. Let C, D be R-linear categories endowed with R-linear equivalences Σ : C → C and Δ : D → D. Let F : C → D and G : D → C be R-linear functors which are adjoint to each other, compatible with Σ and Δ with respect to isomorphisms a : FΣ ≅ ΔF and b : GΣ ≅ ΔF. Suppose that the relatively projective element π_{F} = tr_{F}(Id_{Id_{C}}) is invertible in Z^{0}(D). The normalised transfer map Tr_{F} induces a surjective homomorphism of graded R-algebras

R_{F} : Z_{F}^{∗}(C) ⟶ Z_{G}^{∗}(D)

which is independent of the choice of a compatible adjunction with respect to a, b for which the relatively projective element π_{F} is invertible. In addition, if both π_{F} and π_{G} are invertible in Z^{0}(D) and Z^{0}(C), respectively, then R_{F} and R_{G} are graded R-algebra isomorphisms

Z_{F}^{∗}(C) ≅ Z_{G}^{∗}(D)

which are inverse to each other.

Proof. Let φ ∈ Z^{n}(C) and ψ ∈ Z^{n}(D) such that Fφ = ψF, with the notation as in 5.1. By 5.3.(iii) we have

Tr_{F}(φ) = ψ .

This proves that R_{F} is a surjective algebra homomorphisms and that R_{G} is inverse to R_{F} if π_{G} is invertible as well. Since the equality Fφ = ψF does not involve the adjunction isomorphism (but the transformation a) the independence statement in 6.2 follows as well. □

The next Theorem generalises the cancellation property from [10, 3.8] to this context.

Theorem 6.3. Let C, D, E be R-linear categories endowed with R-linear self-equivalences Σ, Δ, Γ, respectively. Let F : C → D, G : D → C be adjoint functors compatible with Σ, Δ, and let F′ : D → E, G′ : E → D be adjoint functors compatible with Δ, Γ. Suppose that the relatively projective element π_{G′} in Z^{0}(D) is invertible. Let n be a nonnegative integer, let ζ ∈ Z^{n}(C) and let τ ∈ Z^{n}(E) such that F′Fζ = τF′F. Then

Fζ = Tr_{G′}(τ)F

In particular, we have Z_{F′F}^{∗}(C) ⊆ Z_{F}^{∗}(C) and Tr_{G′}(Z_{GG′}^{∗}(E)) ⊆ Z_{G}^{∗}(D).

Proof. We will show that there is a commutative diagram of the following form:

[figure: F; Fζ; FΣ^{n}; ≅; Δ^{n}F; π_{G′} Id_{Δ^{n}F}; G′F′F; G′F′Fζ; G′F′FΣ^{n}; ≅; G′F′F; G′τF′F; G′Γ^{n}F′F; ≅; Δ^{n}G′F′F; F; tr_{G′} (τ)F; Δ^{n}F]

The unmarked arrows are induced by the adjunction unit Id_{D} → G′F′ and adjunction counit G′F′ → Id_{D}. The composition of these two is the relatively projective element π_{G′}, which is invertible by the assumptions. This accounts for the commutativity of the right triangle in the diagram. The smaller square in the middle of the diagram is commutative, because this is the equality F′Fζ = τF′F composed with G′ on the left. The lower part of the diagram is commutative, because this is the definition of tr_{G′}(τ) composed with F on the right. The pentagon in the upper right area of the diagram is commutative by the compatibility assumptions. Consider now the outer square of this diagram and invert the right vertical arrow (which is possible as π_{G′} is invertible). The resulting square expresses the equality Fζ = Tr_{G′}(τ)F as stated. Thus in particular, ζ ∈ Z_{F}^{n}(C), which yields the inlcusion Z_{F′F}^{∗}(C) ⊆ Z_{F}^{∗}(C), and by 5.3.(i) we also have Tr_{G′}(τ) ∈ Z_{G}^{n}(D), which proves the rest. □

Corollary 6.4. Let C, D be R-linear categories endowed with R-linear self-equivalences Σ, Δ, respectively. Let F : C → D, G : D → C be adjoint functors compatible with Σ, Δ. Suppose that the relatively projective element π_{F} in H^{0}(D) is invertible. Then Z_{F}^{∗}(C) = Z_{GF}^{∗}(C).

Proof. By 6.3 we have Z_{GF}^{∗}(C) ⊆ Z_{F}^{∗}(C). The other inclusion follows from 5.4. □

Remark 6.5. When applied to the degree zero component and the case where Σ, Δ are the identity functors on R-linear categories C, D, the notion of stability (with the notation of 5.1) reads as follows: an elements φ ∈ Z(C) is called F-stable, if there is ψ ∈ Z(D) such that Fφ = ψF; that is, such that F(φ(U)) = ψ(F(U)) for any object U in C. The set Z_{F}(C) of F-stable elements in Z(C) is a subalgebra of Z(C), and Theorem 6.2 specialises to the following statements:

Corollary 6.6. Let C, D be R-linear categories, let F : C → D and G : D → C be R-linear functors which are adjoint to each other. Suppose there is an adjunction isomorphism such that that the relative projective elements π_{F} and π_{G} are invertible in Z(D) and Z(C), respectively. Then, for any φ ∈ Z(C) and any ψ ∈ Z(D), we have Fφ = ψF if and only if Gψ = φG, and the correspondence mapping φ to ψ satisfying these equalities is an algebra isomorphism Z_{F}(C) ≅ Z_{G}(D) .

§7 Symmetric algebras

7.1. Let k be a field. A k-algebra A is called symmetric if A is isomorphic to its k-dual A^{∗} = Hom_{k}(A, k) as A-A-bimodule. In particular, A is self-injective; that is, A is both projective and injective as left and as right A-module. Any choice of a bimodule isomorphism Φ : A ≅ A^{∗} determines a symmetrising form s : A → k by s = Φ(1). Since A is generated as left and right A-module by 1, A^{∗} is generated as left and right A-module by s; that is, for any a ∈ A we have Φ(a) = a.s = s.a, where (a.s)(b) = s(ba) and (s.a)(b) = s(ab) for any b ∈ A. Thus, in particular, s(ab) = s(ba) for all a, b ∈ A, and Φ is determined by s. For any A-module U we have a natural isomorphism Hom_{A}(U, A) ≅ Hom(U, k) = U^{∗} sending φ ∈ Hom_{A}(U, A) to s ∘ φ. Indeed, this map is functorial in U and an isomorphism for U = A; as both sides are exact in U it follows that this is an isomorphism for all U. Given two symmetric k-algebras A, B and a B-A-bimodule M which is finitely generated projective as left B-module and as right A-module, the two functors M ⊗_{A}− : Mod(A) ⟶ Mod(B) and M^{∗} ⊗_{B}− : Mod(B) ⟶ Mod(A) are left and right adjoint to each

other (all this is well-known; see e.g. [10, Appendix] for a very brief review with some proofs). The algebra A ⊗_{k} A^{0} is symmetric as well, and if we take (C, Σ) = (\overline{mod}(A ⊗_{k} A^{0}), Σ_{A⊗A^{0}}), we get the

Tate analogue of Hochschild cohomology Ext_{C}^{∗}(A, A) = ĤH^{∗}(A). Applying the transfer formalism to D^{b}(A), \overline{mod}(A), and related Ext-rings yields the well-known transfer maps in group cohomology, Tate cohomology and the standard reciprocity statements.

7.2. For (C, Σ) = (D^{b}(A ⊗_{k} A^{0}), [1]) and U = V = A the notation in 2.6 yields Hochschild co-

homology; that is, Ext_{C}^{∗}(A, A) ≅ HH^{∗}(A). Specialising the definition of transfer maps in 7.1 to this situation yields the transfer maps in Hochschild cohomology introduced in [10]. In order to describe this more precisely, let A, B two symmetric k-algebras. Let X be a bounded complex of B-A-bimodules which are finitely generated projective as left B-modules and as right A-modules. The k-dual X^{∗} = Hom_{k}(X, k) is then a bounded complex of A-B-bimodules which are finitely generated projective as left A-modules and as right B-modules; this uses the symmetry of A and B. The functors F = X ⊗_{A} − : D^{b}(A) ⟶ D^{b}(B) and G = X^{∗} ⊗_{B} − : D^{b}(B) ⟶ D^{b}(A)

are left and right adjoint to each other; more precisely, any choice of bimodule isomorphisms A ≅ A^{∗} and B ≅ B^{∗} determines adjunction isomorphisms which commute with the shift functors. The transfer map tr_{X} : HH^{∗}(A) → HH^{∗}(B) in [10, 2.9] coincides with the transfer map tr_{F}(A, A) : Ext_{D^{b}(A⊗_{k}A^{0})}^{∗}(A, A) → Ext_{D^{b}(B⊗_{k}B^{0})}(B, B) with the notation from 7.1. It is an imme-

diate consequence of the formal properties of the definition of transfer maps and stable elements that the canonical map HH^{∗}(A) → Z^{∗}(D^{b}(A)) restricts to an algebra homomorphism on stable elements HH_{X^{∗}}^{∗}(A) → Z_{F}^{∗}(D^{b}(A)) and we have a commutative diagram

7.3.

HH^{∗}(A) ⟶ Z^{∗}(D^{b}(A))
tr_{X^{∗}} ↓  ↓^{tr}_{F}
HH^{∗}(B) ⟶ Z^{∗}(D^{b}(B))

The image of the relative projective element π_{X^{∗}} ∈ Z(A) = HH^{0}(A) in Z^{0}(D^{b}(A)) is the relatively G-projective element π_{G}; in particular, if π_{X^{∗}} is invertible, so is π_{G}. Similarly, π_{F} is the image of π_{X}. Therfore, if both π_{X} and π_{X^{∗}} are invertible, we get a commutative diagram of graded algebras

7.4.

HH_{X^{∗}}^{∗}(A) ⟶ Z_{F}^{∗}(D^{b}(A))
R_{X} ↓  ↓^{R_{F}}
HH_{X}^{∗}(B) ⟶ Z_{G}^{∗}(D^{b}(B))

where the vertical arrows are isomorphisms by [10, 3.6] and 6.2, respectively.

7.5. With the notation in 7.2, let X′ be a direct summand of the complex of B-A-bimodules X. The functors F′ : X′ ⊗_{A} − : D^{b}(A) ⟶ D^{b}(B) and G′ : (X′)^{∗} ⊗_{B} − : D^{b}(B) ⟶ D^{b}(A) are again left

and right adjoint to each other.

Lemma 7.6. With the notation above, let n be an integer, let ζ ∈ Z^{n}(D^{b}(A)) be F-stable and let τ ∈ Z^{n}(D^{b}(B)) such that Fζ = τF. Then F′ζ = τF′. In particular, we have Z_{F}^{∗}(D^{b}(A)) ⊆ Z_{F′}^{∗}(D^{b}(A)).

Proof. The assumption Fζ = τF means that for any bounded complex of A-modules U the diagram

X ⊗_{A} U ⟶^{Id_{X}⊗ζ(U)} X ⊗_{A} U[n]
∥  ∥
X ⊗_{A} U ⟶_{τ(X⊗_{A}U)} X ⊗_{A} U[n]

is commutative. Choose chain maps of B-A-modules ι : X′ → X and π : X → X′ such that π ∘ ι = Id_{X′}. Using that τ is a natural transformation we get a commutative diagram

X′ ⊗_{A} U ⟶^{Id_{X′} ⊗ζ(U)} X′ ⊗_{A} U[n]
ι⊗Id_{U} ↓  ↓^{ι⊗Id_{U} [n]}
X ⊗_{A} U ⟶^{Id_{X}⊗ζ(U)} X ⊗_{A} U[n]
∥  ∥
X ⊗_{A} U ⟶_{τ(X⊗_{A}U)} X ⊗_{A} U[n]
π⊗Id_{U} ↓  ↓^{π⊗Id_{U} [n]}
X′ ⊗_{A} U ⟶_{τ(X′⊗_{A}U)} X′ ⊗_{A} U[n]

This shows that F′ζ = τF′ and hence that ζ is also F′-stable. □

§8 Proof of Theorem 1.1

Let k be an algebraically closed field of prime characteristic p, let G be a finite group and let B be a source algebra of a block b of kG with P as defect group; that is, B = ikGi for some primitive idempotent i in (kGb)^{P} satisfying Br_{P} (i) ≠ 0, where Br_{P} : (kG)^{P} → kC_{G}(P) is the Brauer homomorphism. The canonical symmetrising form on kG (sending 1_{G} to 1_{k} and x ∈ G − {1_{G}} to 0) restricts to a symmetrising form s : B → k. Set X = B, viewed as B-kP -bimodule. Then X^{∗} ≅ B, as kP -B-bimodule because B is symmetric. In other words, if we consider as in 7.2 the functors F = X ⊗_{kP} − : D^{b}(kP) → D^{b}(B) and G = X^{∗} ⊗_{B} − : D^{b}(B) → D^{b}(kP) then G is the

restriction functor. By [10, 5.6.(iii)], the canonical map from block cohomology H^{∗}(B) to Hochschild cohomology HH^{∗}(B) sends H^{∗}(B) to HH_{X}^{∗}(B). By 7.2, the canonical map from HH^{∗}(B) to the graded center H^{∗}(D^{b}(B)) sends HH_{X}^{∗}(B) to H_{G}^{∗}(D^{b}(B)).

Lemma 8.1. With the notation above, the relative projective elements π_{X} ∈ Z(B), π_{X^{∗}} ∈ Z(kP), π_{F} ∈ Z^{0}(D^{b}(B)), π_{G} ∈ Z^{0}(D^{b}(kP)) are all invertible.

Proof. For π_{X} and π_{X^{∗}} this is proved in [10, 5.6.(i)]. Since π_{F} and π_{G} are their images in Z^{0}(D^{b}(B)) and Z^{0}(D^{b}(kP)), respectively (cf. 7.2), the result follows. □

Denote by γ : H^{∗}(B) → HH_{X^{∗}}^{∗}(B) the canonical algebra homomorphism from [10, 5.6.(iii)]; this is obtained by composing the inclusion H^{∗}(B) ⊆ H^{∗}(P, k) with the canonical “diganonal induction” map δ_{P} : H^{∗}(P, k) → HH^{∗}(kP) followed by the normalised transfer map T_{X} : HH^{∗}(kP) → HH^{∗}(B). Denote further by ϵ : Z^{∗}(D^{b}(kP)) → H^{∗}(P, k) the canonical graded algebra homomorphism obtained from evaluating natural transformations at the trivial kP -module k, as in 2.8. By 1.2 the kernel of ϵ is a nilpotent ideal in Z^{∗}(D^{b}(kP)). Define η : Z_{F}^{∗}(D^{b}(B)) → H^{∗}(P, k) as the unique algebra homomorphism obtained from restricting ϵ to Z_{F}^{∗}(D^{b}(kP)) and precomposing it with the normalised transfer T_{G} : Z_{G}^{∗}(D^{b}(B)) ≅ Z_{F}^{∗}(D^{b}(kP)) (this is an isomorphism by 8.1 and 6.2.). Using the last commutative diagram in 7.2 we get altogether a commutative diagram of graded algebra homomorphisms 8.2.

[figure: H^{∗}(B); γ; HH_{X}^{∗}(B); ≅; Z_{G}^{∗}(D^{b}(B)); η; ≅; H^{∗}(P, k); HH_{X^{∗}}^{∗}(kP); Z_{F}^{∗}(D^{b}(kP)); H^{∗}(P, k); HH^{∗}(kP); Z^{∗}(D^{b}(kP)); ϵ; H^{∗}(P, k)]

The unmarked vertical arrows in 8.2 are inclusions. Composing the three maps in the last row of this diagram yields the identity on H^{∗}(P, k). In order to show a similar statement for the first row of this diagram, we need to show the following statement:

Lemma 8.3. With the notation above, the homomorphism η maps Z_{G}^{∗}(D^{b}(B)) onto H^{∗}(B).

Proof. The relative projective elements determined by F, G are invertible by 8.1. Thus, by 6.2, we have Z_{G}^{∗}(D^{b}(B)) ≅ Z_{F}^{∗}(D^{b}(kP)). By 6.4, we have Z_{F}^{∗}(D^{b}(kP)) = Z_{GF}^{∗}(D^{b}(kP)). Thus it suffices to show that the canonical map ϵ in the above diagram sends Z_{GF}^{∗}(D^{b}(kP)) to H^{∗}(B). Let n be an integer and let ζ ∈ Z^{n}(D^{b}(kP)) such that there exists τ ∈ Z^{n}(D^{b}(kP)) satisfying GFζ = τGF. Since the functor GF is isomorphic to the functor B ⊗_{kP} − on D^{b}(kP) this is equivalent to the

commutativity of the diagrams

B ⊗_{kP} U ⟶^{Id⊗ζ(U)} B ⊗_{kP} U[n]
∥  ∥
B ⊗_{kP} U ⟶^{τ(B ⊗_{kP}U)} B ⊗_{kP} U[n]

for any bounded complex of kP -modules U, where B is regarded as kP -kP -bimodule. Any indecomposable direct summand of B is isomorphic to kP ⊗_{kQ} _{φ}kP for some subgroup Q of P and some

morphism φ : Q → P belonging to the fusion system of the source algebra B of b. By 7.6 the diagram

kP ⊗_{kQ} _{φ}kP ⊗_{kP} U  ⟶^{Id⊗ζ(U)}  kP ⊗_{kQ} _{φ}kP ⊗_{kP} U[n]
∥  ∥
kP ⊗_{kQ} _{φ}kP ⊗_{kP} U ⟶^{τ(kP ⊗_{kQ}_{φ}kP ⊗_{kP}U)} kP ⊗_{kQ} _{φ}kP ⊗_{kP} U[n]

is then still commutative. By the standard adjunction we get a commutative diagram in D^{b}(kQ) of the form

φU ⟶^{ζ(U)} _{φ}U[n]
∥  ∥
φU ⟶^{σ(_{φ}U)} _{φ}U[n]

where we abusively denote again by ζ(U) its restriction via φ and where σ(_{φ}U) is the map induced by τ. Now apply this to U = k, the trivial kP -module. Then Res_{Q}^{P}(U) = _{φ}U = k, the trivial kQ-

module. So in that case (kP ⊗_{kQ}_{φ}kP) ⊗_{kP} k = (kP ⊗_{kQ} kP) ⊗_{kP} k = kP ⊗_{kQ} k hence τ((kP ⊗_{kQ}_{φ}kP) ⊗_{kP} k) =

τ((kP ⊗_{kQ} kP) ⊗_{kP} k) = τ(kP ⊗_{kQ} k) and so σ(_{φ}k) does not depend on the choice of the homomorphism

φ from Q to P in the fusion system of iBi. But that means precisely that Res_{Q}^{P}(ζ(k)) = Res_{φ}(ζ(k)) for all morphisms φ in the fusion system of iBi, and thus ζ ∈ H^{n}(B). This shows that η maps Z_{G}^{∗}(D^{b}(B)) to H^{∗}(B). This map is onto because the last row in the diagram 8.2 is the identity. □

The proof of Theorem 1.1 concludes now as follows. Set N = ker(η). Then H^{∗}(B) ≅ Z_{G}^{∗}(D^{b}(B))/N by 8.3 and N is nilpotent as a consequence of the commutative diagram 8.2 in conjunction with the fact that ker(ϵ) is nilpotent by Proposition 1.3.

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Markus Linckelmann Department of Mathematical Sciences Meston Building Aberdeen, AB24 3UE

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