TWISTED SUPPORT VARIETIES

PETTER ANDREAS BERGH

Abstract. We define and study twisted support varieties for modules over an Artin algebra, where the twist is induced by an automorphism of the algebra. Under a certain finite generation hypothesis we show that the twisted variety of a module satisfies Dade’s Lemma and is one dimensional precisely when the module is periodic with respect to the twisting automorphism. As a special case we obtain results on D Tr-periodic modules over Frobenius algebras.

1. Introduction

In this paper we define and study a new type of cohomological support varieties, called twisted support varieties, for modules over Artin algebras, varieties sharing many of the properties of those defined for Artin algebras in [EHSST] and [SnS]. In those papers the underlying geometric object used to define support varieties was the Hochschild cohomology ring of an algebra, whereas our geometric object is a “twisted” Hochschild cohomology ring, the twist being induced by an automorphism of the algebra.

By introducing a finite generation hypothesis similar to the one used in [Be1] we are able to relate the dimension of the twisted variety of a module to the polynomial growth of the lengths of the modules in its minimal projective resolution. In particular Dade’s Lemma holds, that is, the twisted variety of a module is trivial if and only if the module has finite projective dimension. This property is highly desirable in any cohomological variety theory. Moreover, the modules whose twisted varieties are one dimensional are precisely those which are periodic with respect to the twisting automorphism, a concept defined in Section 4. As a special case we obtain results on D Tr-periodic modules over Frobenius algebras.

Both when studying the twisted Hochschild cohomology ring and twisted support varieties we illustrate the theory with examples taken from the well known representation theory of the four dimensional Frobenius algebra

k⟨x, y⟩/(x^{2}, xy + qyx, y^{2}),

where k is a field and q ∈ k is not a root of unity. Indeed, detecting the D Tr-periodic modules over this algebra by means of support varieties was the motivation for this paper in the first place.

2. Preliminaries

Throughout we let k be a commutative Artin ring and Λ an indecomposable Artin k-projective k-algebra with Jacobson radical 𝔯. Unless otherwise specified all modules considered are finitely generated left modules. We denote by mod Λ the category of all finitely generated left Λ-modules, and we fix a nonzero module M ∈ mod Λ with minimal projective resolution

ℙ: ⋯ → P_{2} ⟶^{d_{2}} P_{1} ⟶^{d_{1}} P_{0} ⟶^{d_{0}} M → 0.

2000 Mathematics Subject Classification. 16E05, 16E30, 16E40, 16P90.

Key words and phrases. Support varieties, twisting automorphism.

Let F: mod Λ → mod Λ be an exact k-functor. It follows from a theorem of Watts (see [Rot, Corollary 3.34]) that there exists a bimodule Q having the property that F is naturally equivalent to the functor Q ⊗_{Λ} −, and that Q may be chosen to be F (Λ). Since Q is in mod Λ it has finite k-length and is therefore finitely generated also as a right module, and this implies that Q is projective as a right module (it is flat). Suppose Q is projective also as a left module (but not necessarily as a bimodule). Then if P is a projective Λ-module the functor Hom_{Λ}(F (P), −), being naturally isomorphic to Hom_{Λ}(P, Hom_{Λ}(Q, −)), is exact, hence F (P) is projective.

For such a functor F and a positive integer t ∈ ℕ, define a homogeneous product in the graded k-module

Ext_{Λ}^{t∗}(F^{∗}(M), M) = ⨁_{n=0}^{∞} Ext_{Λ}^{tn}(F^{n}(M), M)

as follows; if η and θ are two homogeneous elements in Ext_{Λ}^{t∗}(F^{∗}(M), M), then

ηθ =^{def} η ∘ F^{|η|/t}(θ),

where “∘” denotes the Yoneda product. In other words, if η and θ are given as exact sequences

η: 0 → M → A_{tn−1} → ⋯ → A_{0} → F^{n}(M) → 0
θ: 0 → M → B_{tm−1} → ⋯ → B_{0} → F^{m}(M) → 0,

then ηθ is given as the exact sequence obtained from the Yoneda product of the sequence η with the sequence

F^{n}(θ): 0 → F^{n}(M) → F^{n}(B_{tm−1}) → ⋯ → F^{n}(B_{0}) → F^{m+n}(M) → 0.

Furthermore, for a Λ-module N, define a homogeneous right scalar action on Ext_{Λ}^{t∗}(F^{∗}(M), N) from Ext_{Λ}^{t∗}(F^{∗}(M), M) as follows; if μ ∈ Ext_{Λ}^{t∗}(F^{∗}(M), N) and η ∈ Ext_{Λ}^{t∗}(F^{∗}(M), M) are homogeneous elements, then μη =^{def} μ ∘ F^{|μ|/t}(η).

Lemma 2.1. Extending the homogeneous product and scalar product defined above to all graded elements makes Ext_{Λ}^{t∗}(F^{∗}(M), M) into a graded k-algebra and Ext_{Λ}^{t∗}(F^{∗}(M), N) into a graded right Ext_{Λ}^{t∗}(F^{∗}(M), M)-module.

Proof. The product in Ext_{Λ}^{t∗}(F^{∗}(M), M) is associative, and the right distributive law holds; if θ_{1}, θ_{2} and η are homogeneous elements such that θ_{1} and θ_{2} are of the same degree tn, then

(θ_{1} + θ_{2})η = (θ_{1} + θ_{2}) ∘ F^{n}(η) = θ_{1} ∘ F^{n}(η) + θ_{2} ∘ F^{n}(η) = θ_{1}η + θ_{2}η.

Now suppose the degree of η is tm. We can represent θ_{i} as a map

Ω_{Λ}^{tn}(F^{n}(M)) ⟶^{f_{θ_{i}}} M,

and since the functor F^{m} preserves direct limits (in particular pushouts) and projective resolutions (though it may not preserve minimal resolutions), we see that the map

F^{m}(Ω_{Λ}^{tn}(F^{n}(M))) ⟶^{F^{m}(f_{θ_{i}})} F^{m}(M)

represents F^{m}(θ_{i}). As F^{m} is additive we have F^{m}(f_{θ_{1}}+f_{θ_{2}}) = F^{m}(f_{θ_{1}})+F^{m}(f_{θ_{2}}) as elements of Hom_{Λ}(F^{m}(Ω_{Λ}^{tn}(F^{n}(M))), F^{m}(M)), and this implies that F^{m}(θ_{1}+θ_{2}) = F^{m}(θ_{1})+F^{m}(θ_{2}). Therefore the left distributive law also holds for the homogeneous product;

η(θ_{1} + θ_{2}) = η ∘ (F^{m}(θ_{1} + θ_{2})) = η ∘ (F^{m}(θ_{1}) + F^{m}(θ_{2}))
= η ∘ F^{m}(θ_{1}) + η ∘ F^{m}(θ_{2}) = ηθ_{1} + ηθ_{2}.

Extending this homogeneous product in the natural way we see that Ext_{Λ}^{t∗}(F^{∗}(M), M) becomes a graded k-algebra, and similar arguments show that Ext_{Λ}^{t∗}(F^{∗}(M), N) is a graded right Ext_{Λ}^{t∗}(F^{∗}(M), M)-module. □

Example. Suppose k is a field and Λ is selfinjective, and consider a bimodule B in the stable category \underline{mod} Λ^{e} of finitely generated Λ^{e}-modules modulo projectives. The Auslander-Reiten translation τ_{Λ^{e}} = D Tr is a self-equivalence on \underline{mod} Λ^{e}, and we can endow the graded k-vector space

𝔸(B, τ_{Λ^{e}}) = Hom_{Λ^{e}}(B, B) ⊕ ⨁_{i=1}^{∞} \underline{Hom}_{Λ^{e}}(τ_{Λ^{e}}^{i}(B), B)

with a product as follows; for two homogeneous elements f ∈ 𝔸(B, τ_{Λ^{e}})_{m} and g ∈ 𝔸(B, τ_{Λ^{e}})_{n} we define fg to be the composition f ∘ τ_{Λ^{e}}^{m}(g) ∈ 𝔸(B, τ_{Λ^{e}})_{m+n}. In this way 𝔸(B, τ_{Λ^{e}}) becomes a graded k-algebra. Now for any selfinjective Artin algebra Γ the functors τ_{Γ}^{i} and Ω_{Γ}^{2i}N^{i} are isomorphic by [ARS, Proposition IV.3.7], where N is the Nakayama functor D Hom_{Γ}(−, Γ). Therefore the orbit algebra 𝔸(B, τ_{Λ^{e}}) of the bimodule B has the form

Hom_{Λ^{e}}(B, B) ⊕ ⨁_{i=1}^{∞} \underline{Hom}_{Λ^{e}}(Ω_{Λ^{e}}^{2i}(N^{i}B), B),

giving an isomorphism

𝔸(B, τ_{Λ^{e}}) ≃ ⨁_{i=0}^{∞} Ext_{Λ^{e}}^{2i}(N^{i}B, B)

of graded k-algebras.

The algebra 𝔸(Λ, τ_{Λ^{e}}) is called the Auslander-Reiten orbit algebra of Λ, and these algebras have been extensively studied by Z. Pogorzały. In [Po1] it was shown that they are invariant under stable equivalences of Morita type between symmetric algebras; if Γ and Δ are finite dimensional symmetric K-algebras (where K is a field) stably equivalent of Morita type, then 𝔸(Γ, τ_{Γ^{e}}) and 𝔸(Δ, τ_{Δ^{e}}) are isomorphic K-algebras. This was generalized in [Po2] to arbitrary finite dimensional selfinjective algebras. In [Po4] the Auslander-Reiten orbit algebras of a class of finite dimensional basic connected selfinjective Nakayama K-algebras were computed. Namely, it was shown that if Γ is such an algebra of τ_{Γ^{e}}-period 1, then 𝔸(Γ, τ_{Γ^{e}}) ≃ K[x] if Γ is a radical square zero algebra, and if not then there exists a natural number t such that 𝔸(Γ, τ_{Γ^{e}}) ≃ K[x, y]/(y^{t}). In [Po3] τ-periodicity was investigated using similar techniques as was used in [Sch] to study syzygy-periodicity.

3. Twisted Hochschild cohomology

The underlying geometric object used in [EHSST] and [SnS] to define support varieties was the Hochschild cohomology ring HH^{∗}(Λ, Λ) = Ext_{Λ^{e}}^{∗}(Λ, Λ) = ⨁_{n=0}^{∞} Ext_{Λ^{e}}^{n}(Λ, Λ) of Λ (where Λ^{e} is the enveloping algebra Λ ⊗_{k} Λ^{op} of Λ). For an Artin k-algebra Γ and any pair of Λ-Γ-bimodules X and Y the tensor map

− ⊗_{Λ} X: Ext_{Λ^{e}}^{∗}(Λ, Λ) → Ext_{Λ ⊗_{k}Γ^{op}}^{∗}(X, X)

is a homomorphism of graded k-algebras, making Ext_{Λ ⊗_{k}Γ^{op}}^{∗}(X, Y) a left and right Ext_{Λ^{e}}^{∗}(Λ, Λ)-module via the maps − ⊗_{Λ} Y and − ⊗_{Λ} X, respectively (followed by Yoneda product). Now since Λ is projective as a k-module, it follows from [Yon, Proposition 3] that for any two Λ-Λ bimodules B and B′ which are both projective as right Λ-modules, and for any homogeneous elements η ∈ Ext_{Λ^{e}}^{∗}(B, B′) and θ ∈ Ext_{Λ ⊗_{k}Γ^{op}}^{∗}(X, Y), the Yoneda relation

(†)  (η ⊗_{Λ} Y) ∘ (B ⊗_{Λ} θ) = (−1)^{|η||θ|}(B′ ⊗_{Λ} θ) ∘ (η ⊗_{Λ} X)

holds (see also [SnS, Theorem 1.1]). By specializing to the case Γ = Λ and B = B′ = X = Y = Λ we see that the Hochshild cohomology ring HH^{∗}(Λ, Λ) of Λ is graded commutative, whereas the case Γ = k and B = B′ = Λ shows that for any pair of Λ-modules X and Y the left and right scalar actions from HH^{∗}(Λ, Λ) on Ext_{Λ}^{∗}(X, Y) are related graded commutatively.

Denote the commutative even subalgebra ⨁_{n=0}^{∞} Ext_{Λ^{e}}^{2n}(Λ, Λ) of HH^{∗}(Λ, Λ) by H^{ev}. The support variety of the Λ-module M is the subset

V(M) =^{def} {𝔪 ∈ MaxSpec H^{ev} | Ann_{H^{ev}} Ext_{Λ}^{∗}(M, M) ⊆ 𝔪}

of the maximal ideal spectrum of H^{ev}. As shown in [EHSST] and [SnS], the theory of support varieties is rich and in many ways similar to the theory of cohomological varieties for groups, especially under the hypothesis that H^{ev} is Noetherian and that Ext_{Λ}^{∗}(X, Y) is a finitely generated H^{ev}-module for all Λ-modules X and Y.

In order to obtain a partly similar theory of twisted support varieties, the underlying geometric object we use is a “twisted” version of the Hochschild cohomology ring. Let Γ be any ring and let ρ, ϕ: Γ → Γ be two ring automorphisms. If X is a left Γ-module and B is a Γ-Γ-bimodule, denote by _{ϕ}X and _{ϕ}B_{ρ} the left module and bimodule whose scalar actions are defined by γ ·x = ϕ(γ)x and γ_{1} ·b·γ_{2} = ϕ(γ_{1})bρ(γ_{2}) for γ, γ_{1}, γ_{2} ∈ Γ, x ∈ X, b ∈ B. The functor assigning to each Γ-module X the twisted module_{ρ}X is exact and isomorphic to the functor_{ρ}Γ_{1}⊗_{Γ}−, and it preserves projective modules and minimal resolutions when the latter makes sense.

Fix a k-algebra automorphism

ψ: Λ → Λ

and a positive integer t ∈ ℕ, and consider the graded k-module

HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) = Ext_{Λ^{e}}^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) = ⨁_{n=0}^{∞} Ext_{Λ^{e}}^{tn}(_{ψ^{n}}Λ_{1}, Λ).

By the above and Lemma 2.1 the pairing

Ext_{Λ^{e}}^{tm}(_{ψ^{m}}Λ_{1}, Λ) × Ext_{Λ^{e}}^{tn}(_{ψ^{n}}Λ_{1}, Λ) → Ext_{Λ^{e}}^{t(m+n)}(_{ψ^{m+n}}Λ_{1}, Λ)
(η, θ) ↦ η ∘_{ψ^{m}} θ

defines a multiplication under which HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) becomes a graded k-algebra, i.e. if η and θ are homogeneous elements given as exact sequences

η: 0 → Λ → E_{tm−1} → ⋯ → E_{0} → _{ψ^{m}} Λ_{1} → 0
θ: 0 → Λ → T_{tn−1} → ⋯ → T_{0} → _{ψ^{n}} Λ_{1} → 0,

then their product ηθ is given as the Yoneda product of the sequence η with the sequence

ψ^{m}θ: 0 → _{ψ^{m}} Λ_{1} → _{ψ^{m}}(T_{tn−1})_{1} → ⋯ → _{ψ^{m}}(T_{0})_{1} → _{ψ^{m+n}} Λ_{1} → 0.

Before giving an example, recall that a finite dimensional algebra Γ over a field K is Frobenius if _{Γ}Γ and D(Γ_{Γ}) are isomorphic as left Γ-modules, where D denotes the usual K-dual Hom_{K}(−, K). For such an algebra Γ, let φ_{l}: _{Γ}Γ → D(Γ_{Γ}) be an isomorphism. Let y ∈ Γ be any element, and consider the linear functional φ_{l}(1) · y ∈ D(Γ), i.e. the K-linear map Γ → K defined by γ ↦ φ_{l}(1)(yγ). Since φ_{l} is surjective there is an element x ∈ Γ having the property that φ_{l}(x) = φ_{l}(1) · y, giving x · φ_{l}(1) = φ_{l}(1) · y since φ_{l} is a map of left Γ-modules. It is not difficult to show that the map y ↦ x defines a K-algebra automorphism on Γ, and its inverse ν_{Γ} is called the Nakayama automorphism of Γ (with respect to φ_{l}). Thus ν_{Γ} is defined by φ_{l}(1)(γx) = φ_{l}(1)(ν_{Γ}(x)γ) for all γ ∈ Γ. This automorphism is unique up to an inner automorphism; if φ′_{l}: _{Γ}Γ → D(Γ_{Γ}) is another isomorphism of left modules yielding a Nakayama automorphism ν′_{Γ}, then there exists an invertible element z ∈ Γ such that ν_{Γ} = zν′_{Γ}z^{−1}. Note that φ_{l} is an isomorphism between the bimodules _{1}Γ_{ν_{Γ}^{−1}} and D(Γ), and since ν_{Γ}^{−1} : _{ν_{Γ}}Γ_{1} →_{1} Γ_{ν_{Γ}^{−1}} is an isomorphism of bimodules we see that _{ν_{Γ}}Γ_{1} ≃ D(Γ).

As D(Γ_{Γ}) is an injective left Γ-module, we see that a Frobenuis algebra is always left selfinjective, but in fact the definition is left-right symmetric. For if φ_{l}: _{Γ}Γ → D(Γ_{Γ}) is an isomorphism of left Γ-modules, we can dualize and get an isomorphism D(φ_{l}): D^{2}(Γ_{Γ}) → D(_{Γ}Γ) of right modules, and composing with the natural isomorphism Γ_{Γ} ≃ D^{2}(Γ_{Γ}) we get an isomorphism φ_{r}: Γ_{Γ} → D(_{Γ}Γ) of right Γ-modules. Moreover, we can view this last map as an isomorphism φ_{r}: _{Γ^{op}}Γ^{op} → D(Γ_{Γ^{op}}^{op}) of left Γ^{op}-modules, thereby giving an isomorphism φ_{l} ⊗ φ_{r}: Γ ⊗_{K} Γ^{op} → D(Γ_{Γ}) ⊗_{K} D(Γ_{Γ^{op}}^{op}) of Γ-Γ-bimodules. As D(Γ_{Γ}) ⊗_{K} D(Γ_{Γ^{op}}^{op}) is isomorphic to D(Γ⊗_{K} Γ^{op}) as a Γ-Γ-bimodule, we see that the enveloping algebra Γ^{e} = Γ ⊗_{K} Γ^{op} of Γ is also Frobenius.

Example. Suppose k is a field and Λ is Frobenius. Recall from the example in Section 2 that for a bimodule B in the stable category \underline{mod} Λ^{e} the orbit algebra

𝔸(B, τ_{Λ^{e}}) = Hom_{Λ^{e}}(B, B) ⊕ ⨁_{i=1}^{∞} \underline{Hom}_{Λ^{e}}(τ_{Λ^{e}}^{i}(B), B)

is isomorphic to the algebra

⨁_{i=0}^{∞} Ext_{Λ^{e}}^{2i}(N^{i}B, B),

where N is the Nakayama functor D Hom_{Λ^{e}}(−, Λ^{e}). This functor maps the projective cover of a simple Λ^{e}-module to its injective envelope, and is therefore isomorphic as a functor to D(Λ^{e}) ⊗_{Λ^{e}} −. As Λ is a Frobenius algebra, so is Λ^{e}, and we know that D(Λ^{e}) is isomorphic to _{ν_{Λ^{e}}} Λ^{e}_{1} as a Λ^{e}-Λ^{e}-bimodule, giving an isomorphism

𝔸(B, τ_{Λ^{e}}) ≃ ⨁_{i=0}^{∞} Ext_{Λ^{e}}^{2i}(_{ν_{Λ^{e}}^{i}} B, B)

of graded k-algebras.

Next we address the question of commutativity in HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ). For two homogeneous elements

η: 0 → Λ → E_{tm−1} → ⋯ → E_{0} → _{ψ^{m}} Λ_{1} → 0
θ: 0 → Λ → T_{tn−1} → ⋯ → T_{0} → _{ψ^{n}} Λ_{1} → 0

the Yoneda relation (†) gives the equality

η ∘ (_{ψ^{m}}Λ_{1} ⊗_{Λ}θ) = (−1)^{|η||θ|}θ ∘ (η ⊗_{Λ} _{ψ^{n}} Λ_{1}),

in which the left hand side is the product ηθ (we can identify _{ψ^{m}} Λ_{1} ⊗_{Λ}θ with ψ^{m}θ). However, the extension on the right hand side is in general not the product θη; it is the Yoneda product of θ with η_{ψ^{−n}}, and although we have bimodule isomorphisms_{1} Λ_{ψ^{−n}} ≃ _{ψ^{n}} Λ_{1} and_{ψ^{m}} Λ_{ψ^{−n}} ≃ _{ψ^{m+n}} Λ_{1} we cannot in general identify η_{ψ^{−n}} with _{ψ^{n}}η unless n = 0. In fact, the following example shows that the algebra HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) in general is not graded commutative.

Example. Suppose k is an algebraically closed field not of characteristic 2, and for any natural numbers m and n, with n ≥ 2, let Q_{m,n} be the quiver

[figure: ·; α_{2}^{1}; ⋯; α_{n−1}^{1}; ·; α_{n}^{1}; ·; α_{2}^{2}; ⋯; α_{n−1}^{2}; ·; α_{n}^{2}; ·; α_{2}^{m}; ⋯; α_{n−1}^{m}; ·; α_{n}^{m}; 1·; α_{1}^{1}; β_{1}^{1}; ·; α_{1}^{2}; β_{1}^{2}; ·; ⋯; ·; α_{1}^{m}; β_{1}^{m}; 1·; ·; β_{2}^{1}; ⋯; β_{n−1}^{1}; ·; β_{n}^{1}; ·; β_{2}^{2}; ⋯; β_{n−1}^{2}; ·; β_{n}^{2}; ·; β_{2}^{m}; ⋯; β_{n−1}^{m}; ·; β_{n}^{m}]

in which the leftmost vertex is the same as the rightmost vertex. Denote by 𝔞_{m,n} the ideal in the path algebra kQ_{m,n} generated by the paths of length n+1, the sums of the parallel paths of length n, and the paths {β_{1}^{1}α_{n}^{m}, α_{1}^{1}β_{n}^{m}, β_{1}^{i+1}α_{n}^{i}, α_{1}^{i+1}β_{n}^{i}}_{i=1}^{m−1}. Finally, denote the finite dimensional algebra kQ_{m,n}/ 𝔞_{m,n} by Λ_{m,n}. It was shown in [Rie] and [ScW] that the algebras {Λ_{m,n}}_{1≤m,2≤n} represent the distinct stable equivalence classes of Möbius algebras. Moreover, in [Asa] and [M-H] it was shown that these algebras also represent the distinct derived equivalent classes of Möbius algebras.

Let Λ be an algebra of the above type. By [EHS, Theorem 3.5] there is an automorphism Λ ⟶^{ψ} Λ such that Ω_{Λ^{e}}^{p}(Λ) ≃ _{ψ^{−1}} Λ_{1}, where p = 2n − 1. Thus the initial part of the minimal bimodule resolution of Λ has the form

0 → _{ψ^{−1}} Λ_{1} → Q_{p−1} → ⋯ → Q_{0} → Λ → 0,

and twisting this exact sequence with ψ we obtain the initial part

0 → Λ → _{ψ}(Q_{p−1})_{1} → ⋯ → _{ψ}(Q_{0})_{1} → _{ψ} Λ_{1} → 0

of the minimal bimodule resolution of _{ψ} Λ_{1}. The latter exact sequence corresponds to an element μ in HH^{p∗}(_{ψ^{∗}}Λ_{1}, Λ), and if this algebra is graded commutative then the relation μ^{2} = (−1)^{p^{2}}μ^{2} implies μ^{2} = 0 (since p is odd). However, since Ω_{Λ^{e}}^{2p}(_{ψ^{2}} Λ_{1}) = Λ, we may represent the element μ^{2} ∈ Ext_{Λ^{e}}^{2p}(_{ψ^{2}} Λ_{1}, Λ) by the identity

map Ω_{Λ^{e}}^{2p}(_{ψ^{2}} Λ_{1}) ⟶^{id} Λ, and so if μ^{2} = 0 then the identity on Λ factors through a projective bimodule. Then Λ itself is projective as a bimodule, and therefore semisimple as an algebra. This is obviously impossible, hence the algebra HH^{p∗}(_{ψ^{∗}}Λ_{1}, Λ) cannot be graded commutative.

The above example shows that the twisted Hochschild cohomology ring in general is not graded commutative. In light of this example and the Yoneda relation (†), all we can say about commutativity in HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) is that a homogeneous degree zero element commutes with everything, which is not unexpected since HH^{0}(_{ψ^{0}}Λ_{1}, Λ) = Ext_{Λ^{e}}^{0}(Λ, Λ) = Z(Λ) (the center of Λ). The next result gives a criterion under which commutativity relations hold, but first recall the following from [CaE, page 174-175]: for each n ≥ 0 let Q_{n} denote the n-fold tensor product of Λ over k (with Q_{0} = k), and define

B^{n} = Q_{n+2} = \underbrace{Λ ⊗_{k} ⋯ ⊗_{k} Λ}_{n+2 copies of Λ} .

We give B^{n} a Λ^{e}-module structure by defining

λ(λ_{0} ⊗ ⋯ ⊗ λ_{n+1})λ′ = λλ_{0} ⊗ ⋯ ⊗ λ_{n+1}λ′,

and as A^{e}-modules we then have an isomorphism B^{n} ≃ Λ^{e} ⊗_{k}Q_{n}. Note that B^{n} is Λ^{e}-projective since the functor Hom_{Λ^{e}}(Λ^{e} ⊗_{k}Q_{n}, −) is naturally isomorphic to Hom_{k}(Q_{n}, −) by adjointness. Now for each n ≥ 1, define d: B^{n} → B^{n−1} by

λ_{0} ⊗ ⋯ ⊗ λ_{n+1} ↦ ∑_{i=0}^{n}(−1)^{i}λ_{0} ⊗ ⋯ ⊗ λ_{i}λ_{i+1} ⊗ ⋯ ⊗ λ_{n+1}.

The sequence

𝔹 : ⋯ → B^{3} ⟶^{d} B^{2} ⟶^{d} B^{1} ⟶^{d} B^{0} ⟶^{μ} Λ → 0,

where μ is the “multiplication map”, is an exact Λ^{e}-projective resolution called the standard resolution (or Bar-resolution) of Λ.

Proposition 3.1. Let η ∈ HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) be a homogeneous element of degree tm represented by the map f_{η}: _{ψ^{m}}B_{1}^{tm} → Λ. If for some n ≥ 1

f_{η}(ψ^{−n}(λ_{0}) ⊗ ⋯ ⊗ ψ^{−n}(λ_{tm}) ⊗ 1) = ψ^{−n}f_{η}(λ_{0} ⊗ ⋯ ⊗ λ_{tm} ⊗ 1)

for all λ_{0} ⊗ ⋯ ⊗ λ_{tm} ⊗ 1 in _{ψ^{m}}B_{1}^{tm}, then η_{ψ^{−n}} = _{ψ^{n}}η, and consequently ηθ = (−1)^{|η||θ|}θη for every homogeneous element θ ∈ HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) such that tn divides |θ|. In particular, if

f_{η}(ψ^{−1}(λ_{0}) ⊗ ⋯ ⊗ ψ^{−1}(λ_{tm}) ⊗ 1) = ψ^{−1}f_{η}(λ_{0} ⊗ ⋯ ⊗ λ_{tm} ⊗ 1)

for all λ_{0} ⊗ ⋯ ⊗ λ_{tm} ⊗ 1 in _{ψ^{m}}B_{1}^{tm}, then η belongs to the graded center of HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ).

Proof. Suppose the given condition on f_{η} holds for n. When viewing η_{ψ^{−n}} as a tm-fold extension of _{ψ^{n}} Λ_{1} by _{ψ^{m+n}} Λ_{1}, we use the bimodule isomorphisms ψ^{−n}: _{ψ^{m+n}} Λ_{1} →_{ψ^{m}} Λ_{ψ^{−n}} and ψ^{n}: _{1} Λ_{ψ^{−n}} →_{ψ^{n}} Λ_{1}. A lifting of the former along ψ^{m+n}𝔹_{1} is given by the maps

(ψ^{−n})^{⊗i+2} : _{ψ^{m+n}}B_{1}^{i} → _{ψ^{m}}B_{ψ^{−n}}^{i}
λ_{0} ⊗ ⋯ ⊗ λ_{i+1} ↦ ψ^{−n}(λ_{0}) ⊗ ⋯ ⊗ ψ^{−n}(λ_{i+1})

for i ≥ 0, giving the commutative diagram

\xymatrix{⋯ \ar[r] & _{ψ^{m+n}}B_{1}^{tm} \ar[d]^{(ψ^{−n})^{⊗tm+2}} \ar[r]^{d} & _{ψ^{m+n}}B_{1}^{tm−1} \ar[d]^{(ψ^{−n})^{⊗tm+1}} \ar[r]^{d} & ⋯ \ar[r] & _{ψ^{m+n}} Λ_{1} \ar[d]^{ψ^{−n}} \ar[r] & 0 \\ ⋯ \ar[r] & _{ψ^{m}}B_{ψ^{−n}}^{tm} \ar[r]^{d} \ar[d]^{f_{η}} & _{ψ^{m}}B_{ψ^{−n}}^{tm−1} \ar[r]^{d} & ⋯ \ar[r] & _{_{ψ^{m}} Λ_{ψ^{−n}}} \ar[r] & 0 \\  & _{_{1} Λ_{ψ^{−n}}} \ar[d]^{ψ^{n}} \\  & _{_{ψ^{n}} Λ_{1}}}

with exact rows. Therefore the extension η_{ψ^{−n}} is represented by the map composite map

ψ^{n} ∘ f_{η} ∘ (ψ^{−n})^{⊗tm+2} : _{ψ^{m+n}}B_{1}^{tm} →_{ψ^{n}} Λ_{1},

under which the image of an element λ_{0} ⊗ ⋯ ⊗ λ_{tm+1} ∈ _{ψ^{m+n}}B_{1}^{tm} is easily seen to be f_{η}(λ_{0} ⊗ ⋯ ⊗ λ_{tm+1}) because of the assumption on f_{η}.

This shows that we may identify the extensions η_{ψ^{−n}} and _{ψ^{n}}η, and by induction we see that η_{ψ^{−jn}} = _{ψ^{jn}}η for all j ≥ 1. Consequently, if θ ∈ HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) is a homogeneous element such that tn divides |θ|, say |θ| = tnj, then the Yoneda relation gives

ηθ = (−1)^{|η||θ|}θ ∘ η_{ψ^{−jn}} = (−1)^{|η||θ|}θ ∘_{ψ^{jn}}η = (−1)^{|η||θ|}θη.
□

For reasons to be explained in the next section, we make the following definition motivated by the above proposition.

Definition 3.2. A commutative graded subalgebra H ⊆ HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) is strongly commutative if η_{ψ^{−1}} = _{ψ}η for all homogeneous elements η ∈ H.

We end this section with an example of a Frobenius algebra for which the twisted Hochschild cohomology ring with respect to the Nakayama automorphism and t = 2 is strongly commutative.

Example. Let k be a field not of characteristic 2 and q ∈ k a nonzero element which is not a root of unity, and denote by Λ the k-algebra

Λ = k⟨x, y⟩/(x^{2}, xy + qyx, y^{2}).

Denote by D the usual k-dual Hom_{k}(−, k), and consider the map φ: _{Λ} Λ → D(Λ_{Λ}) of left Λ-modules defined by

φ(1)(α + βx + γy + δyx) =^{def} δ.

It is easy to show that this is an injective map and hence also an isomorphism since dim_{k} Λ = dim_{k} D(Λ), and therefore Λ is a Frobenius algebra by definition. Straightforward calculations show that x · φ(1) = φ(1) · (−q^{−1}x) and y · φ(1) = φ(1) · (−qy), hence since x and y generate Λ over k we see that the Nakayama automorphism ν (with respect to φ) is the degree preserving map defined by

x ↦ −q^{−1}x, y ↦ −qy.

In [Be2] the Hochschild cohomology of this 4-dimensional graded Koszul algebra was studied, and for every degree preserving k-algebra automorphism ψ: Λ → Λ the cohomology groups HH^{∗}(Λ,_{1}Λ_{ψ}) = Ext_{Λ^{e}}^{∗}(Λ,_{1}Λ_{ψ}) were computed. In particular, from [Be2, 3.4(iv)] we get

dim_{k} Ext_{Λ^{e}}^{2n}(Λ,_{1}Λ_{ν^{n}}) = \{ \begin{matrix}1 & for n even \\ 0 & for n odd \end{matrix}

when n > 0, whereas Ext_{Λ^{e}}^{0}(Λ,_{1}Λ_{ν^{0}}) = Z(Λ) is two dimensional (the elements 1 and yx form a basis for the center of Λ). Moreover, it is not difficult to see that Ext_{Λ^{e}}^{2n}(Λ,_{1}Λ_{ν^{n}}) is isomorphic to Ext_{Λ^{e}}^{2n}(_{ν^{n}}Λ_{1}, Λ). Therefore the algebra HH^{4∗}(_{ν^{2∗}}Λ_{1}, Λ) = ⨁_{n=0}^{∞} Ext_{Λ^{e}}^{4n}(_{ν^{2n}}Λ_{1}, Λ) is two dimensional in degree zero and one dimensional in the positive degrees.

We now recall the construction of the minimal bimodule projective resolution of Λ from [BGMS]. Define the elements

f_{0}^{0} = 1, f_{0}^{1} = x, f_{1}^{1} = y,
f_{−1}^{n} = 0 = f_{n+1}^{n} for each n ≥ 0,

and for each n ≥ 2 define elements \{f_{i}^{n}\}_{i=0}^{n} ⊆ \underbrace{Λ ⊗_{k} ⋯ ⊗_{k} Λ}_{n copies} inductively by

f_{i}^{n} = f_{i−1}^{n−1} ⊗ y + q^{i}f_{i}^{n−1} ⊗ x.

Denote by F^{n} the Λ^{e}-projective module ⨁_{i=0}^{n} Λ ⊗_{k}f_{i}^{n} ⊗_{k} Λ, and by f̃_{i}^{n} the element 1 ⊗ f_{i}^{n} ⊗ 1 ∈ F^{n} (and f̃_{0}^{0} = 1 ⊗ 1). The set {f̃_{i}^{n}}_{i=0}^{n} generates F^{n} as a Λ^{e}-module. Now define a map δ: F^{n} → F^{n−1} by

f̃_{i}^{n} ↦ [xf̃_{i}^{n−1} + (−1)^{n}q^{i}f̃_{i}^{n−1}x] + [q^{n−i}yf̃_{i−1}^{n−1} + (−1)^{n}f̃_{i−1}^{n−1}y] .

It is shown in [BGMS] that

(𝔽, δ): ⋯ → F^{n+1} ⟶^{δ} F^{n} ⟶^{δ} F^{n−1} → ⋯

is a minimal Λ^{e}-projective resolution of Λ.

For each m ≥ 1 consider the Λ^{e}-linear map

g_{4m}: _{ν^{2m}}F_{1}^{4m} → Λ
f̃_{i}^{4m} ↦ \{ \begin{matrix}1 & for i = 2m \\ 0 & for i ≠ 2m. \end{matrix}

The only generators in _{ν^{2m}}F_{1}^{4m+1} which can possibly map to f̃_{2m}^{4m} under the map ν^{2m}F_{1}^{4m+1} ⟶^{δ}_{ν^{2m}} F_{1}^{4m} are f̃_{2m}^{4m+1} and f̃_{2m+1}^{4m+1}, and so it follows from the equalities

g_{4m} ∘ δ(f̃_{2m}^{4m+1}) = g_{4m}(xf̃_{2m}^{4m} − q^{2m}f̃_{2m}^{4m}x)
= g_{4m} ([q^{2m}x] · f̃_{2m}^{4m} − f̃_{2m}^{4m} · [q^{2m}x])

and

g_{4m} ∘ δ(f̃_{2m+1}^{4m+1}) = g_{4m}(q^{2m}yf̃_{2m}^{4m} − f̃_{2m}^{4m}y)
= g_{4m} (y · f̃_{2m}^{4m} − f̃_{2m}^{4m} · y)
= 0

that g_{4m} belongs to the kernel of the map δ^{∗}: Hom_{Λ^{e}}(_{ν^{2m}}F_{1}^{4m}, Λ) → Hom_{Λ^{e}}(_{ν^{2m}}F_{1}^{4m+1}, Λ). Moreover, there cannot exist a Λ^{e}-linear map _{ν^{2m}}F_{1}^{4m−1} → Λ making the diagram

\xymatrix{_{ν^{2m}}F_{1}^{4m} \ar[r]^{δ} \ar[d]^{g_{4m}} & _{ν^{2m}}F_{1}^{4m−1} \ar[dl] \\ Λ}

commute, since δ is a map of graded degree 1, whereas the degree of g_{4m} is 0. This shows that g_{4m} represents a generator for the one dimensional space Ext_{Λ^{e}}^{4m}(_{ν^{2m}}Λ_{1}, Λ). Note that if we multiply this map with yx, then there does exist a map making the above diagram commute, hence whenever we multiply any homogeneous element of positive degree in HH^{4∗}(_{ν^{2∗}}Λ_{1}, Λ) with the element yx ∈ HH^{0}(_{ν^{0}}Λ_{1}, Λ) we get zero.

For each 0 ≤ i ≤ 4 define Λ^{e}-linear maps ḡ_{4m+i}: _{ν^{2(m+1)}}F_{1}^{4m+i} →_{ν^{2}} F_{1}^{i} by

ḡ_{4m}: f̃_{i}^{4m} ↦ \{ \begin{matrix}f̃_{0}^{0} & for i = 2m \\ 0 & otherwise \end{matrix}

ḡ_{4m+1} : f̃_{i}^{4m+1} ↦ \{ \begin{matrix}q^{2m}f̃_{0}^{1} & for i = 2m \\ f̃_{1}^{1} & for i = 2m + 1 \\ 0 & otherwise \end{matrix}

ḡ_{4m+2} : f̃_{i}^{4m+2} ↦ \{ \begin{matrix}q^{4m}f̃_{0}^{2} & for i = 2m \\ q^{2m}f̃_{1}^{2} & for i = 2m + 1 \\ f̃_{2}^{2} & for i = 2m + 2 \\ 0 & otherwise \end{matrix}

ḡ_{4m+3} : f̃_{i}^{4m+3} ↦ \{ \begin{matrix}q^{4m}f̃_{1}^{3} & for i = 2m + 1 \\ q^{2m}f̃_{2}^{3} & for i = 2m + 2 \\ 0 & otherwise \end{matrix}

ḡ_{4m+4} : f̃_{i}^{4m+4} ↦ \{ \begin{matrix}q^{4m}f̃_{2}^{4} & for i = 2m \\ 0 & otherwise. \end{matrix}

Then for each 1 ≤ i ≤ 4 the diagram

\xymatrix{_{ν^{2(m+1)}}F_{1}^{4m+i} \ar[d]^{ḡ_{4m+i}} \ar[r]^{δ} & _{ν^{2(m+1)}}F_{1}^{4m+i−1} \ar[d]^{ḡ_{4m+i−1}} \\ _{ν^{2}}F_{1}^{i} \ar[r]^{δ} & _{ν^{2}}F_{1}^{i−1}}

commutes, hence the maps ḡ_{4m+i} for 0 ≤ i ≤ 4 provide a lifting of ν^{2(m+1)}F_{1}^{4m} ⟶^{_{ν^{2}} g_{4m}} _{ν^{2}} Λ_{1} along the projective bimodule resolution _{ν^{2(m+1)}}𝔽_{1} of ν^{2(m+1)} Λ_{1}. Moreover, the composition g_{4} ∘ḡ_{4m+4} equals the map q^{4m}g_{4m+4}. Therefore, if we denote by θ the extension in Ext_{Λ^{e}}^{4}(_{ν^{2}}Λ_{1}, Λ) corresponding to the map g_{4}, we see that for each m ≥ 2 the element θ^{m} is the (nonzero) extension in Ext_{Λ^{e}}^{4m}(_{ν^{2m}}Λ_{1}, Λ) corresponding to the map q^{4(1+2+···+m−1)}g_{4m}.

As a result, we see that we have an isomorphism

HH^{4∗}(_{ν^{2∗}}Λ_{1}, Λ) ≃ k[x] ×_{k} k

of graded k-algebras, with x of degree 4, and in particular HH^{4∗}(_{ν^{2∗}}Λ_{1}, Λ) is commutative. Moreover, it is easily shown that we may identify the extensions θ_{ν^{−2}} and _{ν^{2}}θ, and so HH^{4∗}(_{ν^{2∗}}Λ_{1}, Λ) is actually strongly commutative.

4. Twisted support varieties

In order to obtain a theory of twisted support varieties, the underlying geometric object we shall use is the twisted Hochschild cohomology ring HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) =

Ext_{Λ^{e}}^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) studied in the previous section (where Λ ⟶^{ψ} Λ is an automorphism and t ∈ ℕ). Similarly as for HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ), the graded k-module Ext_{Λ}^{t∗}(_{ψ^{∗}}M, M) = ⨁_{n=0}^{∞} Ext_{Λ}^{tn}(_{ψ^{n}}M, M) is a k-algebra, and for each Λ-module N the graded k-module Ext_{Λ}^{t∗}(_{ψ^{∗}}M, N) = ⨁_{n=0}^{∞} Ext_{Λ}^{tn}(_{ψ^{n}}M, N) becomes a graded right Ext_{Λ}^{t∗}(_{ψ^{∗}}M, M)- module by defining

ζμ =^{def} ζ ∘_{ψ^{m}}μ

for ζ ∈ Ext_{Λ}^{tm}(_{ψ^{m}}M, N) and μ ∈ Ext_{Λ}^{tn}(_{ψ^{n}}M, M). Moreover, the tensor map

− ⊗_{Λ} M: HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) → Ext_{Λ}^{t∗}(_{ψ^{∗}}M, M)
η ↦ η ⊗_{Λ} M

is a homomorphism of graded k-algebras, thus making Ext_{Λ}^{t∗}(_{ψ^{∗}}M, N) a graded right HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ)-module. Similarly Ext_{Λ}^{t∗}(_{ψ^{∗}}M, N) becomes a graded left HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) via the homomorphism

− ⊗_{Λ} N: HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) → Ext_{Λ}^{t∗}(_{ψ^{∗}}N, N)
η ↦ η ⊗_{Λ} N

of algebras.

We now make the following assumption:

Assumption. Given the automorphism Λ ⟶^{ψ} Λ and the integer t ∈ ℕ, the graded subalgebra H = ⨁_{n=0}^{∞} H^{tn} of HH^{t∗}(_{ψ^{∗}} Λ_{1}, Λ) is strongly commutative with H^{0} = HH^{0}(Λ, Λ) = Z(Λ).

Why do we restrict ourselves to strongly commutative algebras instead of “ordinary” commutative algebras? The answer is that we want the bifunction Ext_{Λ}^{t∗}(_{ψ^{∗}}−, −), which maps the pair (M, N) of Λ-modules to the H-module Ext_{Λ}^{t∗}(_{ψ^{∗}}M, N), to preserve maps. Namely, consider two homogeneous elements ζ ∈ Ext_{Λ}^{tm}(_{ψ^{m}}M, N) and η ∈ Ext_{Λ^{e}}^{tn}(_{ψ^{n}} Λ_{1}, Λ). By definition ζ · η is the extension ζ ∘ _{ψ^{m}}(η ⊗_{Λ} M) = ζ ∘ (_{ψ^{m}}η ⊗_{Λ} M), and if _{ψ^{m}}η = η_{ψ^{−m}} we get _{ψ^{m}}η ⊗_{Λ} M = η_{ψ^{−m}} ⊗_{Λ} M = η ⊗_{Λ} _{ψ^{m}}M. Hence in this case we may identify the right scalar product ζ · η with (Λ ⊗_{Λ}ζ) ∘ (η ⊗_{Λ} _{ψ^{m}}M), and then by the Yoneda relation (†) from the previous section we get

ζ · η = (Λ ⊗_{Λ}ζ) ∘ (η ⊗_{Λ} _{ψ^{m}}M) = (−1)^{|ζ||η|}(η ⊗_{Λ} N) ∘ (_{ψ^{n}} Λ_{1} ⊗_{Λ} ζ).

However, the extension (η ⊗_{Λ} N) ∘ (_{ψ^{n}} Λ_{1} ⊗_{Λ} ζ) is precisely the left scalar product η · ζ, and so if H is strongly commutative we see that

ζ · η = (−1)^{|ζ||η|}η · ζ

for all homogeneous elements η ∈ H and ζ ∈ Ext_{Λ}^{t∗}(_{ψ^{∗}}M, N).

The fact that the left and right scalar multiplications basically coincide is what makes Ext_{Λ}^{t∗}(_{ψ^{∗}}−, −) preserve maps. To see this, let f: M → M′ be a Λ-homomorphism. This map induces a homomorphism f̂: Ext_{Λ}^{t∗}(_{ψ^{∗}}M′, N) → Ext_{Λ}^{t∗}(_{ψ^{∗}}M, N) of graded groups, under which the image of a homogeneous element

θ: 0 → N → X_{tn} → ⋯ → X_{1} → _{ψ^{n}}M′ → 0

is the extension θf given by the commutative diagram

[figure: θf: 0; N; X_{tn}; ⋯; X_{2}; Y; _{ψ^{n}}M; f; 0; θ: 0; N; X_{tn}; ⋯; X_{2}; X_{1}; _{ψ^{n}}M′; 0]

in which the module Y is a pullback. For a homogeneous element η ∈ H^{tm} we then get

f̂(θ · η) = (−1)^{|η||θ|}f̂(η · θ)
= (−1)^{|η||θ|}f̂((η ⊗_{Λ} N) ∘_{ψ^{m}}θ)
= (−1)^{|η||θ|}(η ⊗_{Λ} N) ∘_{ψ^{m}}(θf)
= (−1)^{|η||θ|}η · f̂(θ)
= f̂(θ) · η,

showing f̂ is a homomorphism of H-modules. Similarly Ext_{Λ}^{t∗}(_{ψ^{∗}}−, −) preserves maps in the second argument.

Remark. It should also be noted that in many cases considering only strongly commutative algebras is not a severe restriction. For example, let H be a commutative Noetherian graded subalgebra of HH^{t∗}(_{ψ^{∗}} Λ_{1}, Λ), as will be the case in most of the results in this section. Then by [Be1, Proposition 2.1] there exists an integer w and a homogeneous element η ∈ H of positive degree, say |η| = tn, such that the multiplication map

H^{ti} ⟶^{η·} H^{t(i+n)}

is injective for i ≥ w. Now for any homogeneous element θ ∈ H^{tm} the product θ · η is by definition the extension θ∘_{ψ^{m}}η = (θ⊗_{Λ}Λ)∘(_{ψ^{m}} Λ_{1}⊗_{Λ}η), which by the Yoneda relation (†) from Section 3 equals the extension (−1)^{|η||θ|}(Λ ⊗_{Λ}η) ∘ (θ ⊗_{Λ} _{ψ^{n}} Λ_{1}) = (−1)^{|η||θ|}η ∘ θ_{ψ^{−n}}. However, the extension η ∘ θ_{ψ^{−n}} corresponds to the product η ·_{ψ^{−n}}θ_{ψ^{−n}} in H, and so since H is commutative we get

η · θ = θ · η = (−1)^{|η||θ|}η ·_{ψ^{−n}}θ_{ψ^{−n}}.

Now if |θ| ≥ tw then θ = (−1)^{|η||θ|}_{ψ^{−n}}θ_{ψ^{−n}} due to the injectivity of the multiplication map induced by η, and twisting both sides in this equality by ψ^{n} from the left we get _{ψ^{n}}θ = (−1)^{|η||θ|}θ_{ψ^{−n}}.

The algebra H, being Noetherian, is generated over H^{0} by a finite set of homogeneous elements, say {x_{0}, …, x_{r}}. Suppose now that k contains an infinite field. Then from the proof of [Be1, Theorem 2.5] we see that we may choose the element η above such that |η| = lcm{|x_{0}|, …, |x_{r}|} In particular, if all the generators x_{0}, …, x_{r} are of the same degree, i.e. |x_{i}| = t, then the subalgebra H^{0} ⊕ ⨁_{i=w}^{∞} H^{ti} of H is strongly commutative.

Having made all the necessary assumptions, we are now ready to define twisted support varieties. For two Λ-modules X and Y, denote by A_{H}^{ψ}(X, Y) the ideal Ann_{H} Ext_{Λ}^{t∗}(_{ψ^{∗}}X, Y), i.e. the annihilator in H of Ext_{Λ}^{t∗}(_{ψ^{∗}}X, Y). This ideal is graded since Ext_{Λ}^{t∗}(_{ψ^{∗}}X, Y) is a graded H-module. As for ordinary support varieties, the defining ideal for the twisted support variety of M is A_{H}^{ψ}(M, Λ / 𝔯).

Definition 4.1. The twisted support variety V_{H}^{ψ}(M) of M (with respect to the automorphism ψ, the integer t and the ring H) is the subset

V_{H}^{ψ}(M) =^{def} {𝔪 ∈ MaxSpec H | A_{H}^{ψ}(M, Λ / 𝔯) ⊆ 𝔪}

of the maximal ideal spectrum of H.

Note that since M and therefore also Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯) is nonzero, the degree zero part of A_{H}^{ψ}(M, Λ / 𝔯) must be contained in the radical rad H^{0} of the local ring H^{0} = Z(Λ) (if not then A_{H}^{ψ}(M, Λ / 𝔯) contains the identity in H). Hence the twisted variety V_{H}^{ψ}(M) always contains the unique graded maximal ideal

𝔪_{H} = rad H^{0} ⊕ H^{t} ⊕ H^{2t} ⊕ ⋯

of H, and we shall say the variety is trivial if it does not contain any other element.

The following proposition lists some elementary and more or less expected properties of twisted varieties. Proposition 4.2. (i) The ideal A_{H}^{ψ}(M, M) is also a defining ideal for the

twisted variety of M, i.e.

V_{H}^{ψ}(M) = {𝔪 ∈ MaxSpec H | A_{H}^{ψ}(M, M) ⊆ 𝔪}.

(ii) If Ext_{Λ}^{tn}(_{ψ^{n}}M, M) = 0 for n ≫ 0, then V_{H}^{ψ}(M) is trivial. In particular, the twisted variety of M is trivial whenever the projective or injective dimension of M is finite.

(iii) Given nonzero Λ-modules M′ and M′′ and an exact sequence

0 → M′ → M → M′′ → 0,

the relation

V_{H}^{ψ}(M) ⊆ V_{H}^{ψ}(M′) ∪ V_{H}^{ψ}(M′′)

holds. Moreover, if Ω_{Λ}^{t−1}(M′) ≠ 0, then the relation

V_{H}^{ψ}(M′′) ⊆ V_{H}^{ψ}(M) ∪ V_{H}^{ψ}(Ω_{Λ}^{t−1}(_{ψ}M′))

holds, and if M′′ = Ω_{Λ}^{t−1}(M′′′) for some module M′′′, then the relation

V_{H}^{ψ}(M′) ⊆ V_{H}^{ψ}(M) ∪ V_{H}^{ψ}(_{ψ^{−1}}M′′′)

holds. (iv) If M′ and M′′ are nonzero Λ-modules such that M = M′ ⊕ M′′, then

V_{H}^{ψ}(M) = V_{H}^{ψ}(M′) ∪ V_{H}^{ψ}(M′′).

Proof. For a Λ-module N, denote by V_{H}^{ψ}(M, N) the variety whose defining ideal is A_{H}^{ψ}(M, N). To prove (i), we must show that V_{H}^{ψ}(M, Λ / 𝔯) and V_{H}^{ψ}(M, M) are equal, and we start by proving by induction on the length of N that V_{H}^{ψ}(M, N) is contained in V_{H}^{ψ}(M, Λ / 𝔯). For any simple Λ-module S the ideal A_{H}^{ψ}(M, Λ / 𝔯) is contained in A_{H}^{ψ}(M, S) since S is a summand of Λ / 𝔯, hence V_{H}^{ψ}(M, S) ⊆ V_{H}^{ψ}(M, Λ / 𝔯). If the length of N is greater than 2, take any simple submodule S ⊂ N and consider the exact sequence

0 → S → N → N/S → 0.

This sequence induces the exact sequence

Ext_{Λ}^{t∗}(_{ψ^{∗}}M, S) → Ext_{Λ}^{t∗}(_{ψ^{∗}}M, N) → Ext_{Λ}^{t∗}(_{ψ^{∗}}M, N/S)

of H-modules, from which the inclusion A_{H}^{ψ}(M, S) · A_{H}^{ψ}(M, N/S) ⊆ A_{H}^{ψ}(M, N)

follows. This implies that V_{H}^{ψ}(M, N) ⊆ V_{H}^{ψ}(M, S) ∪ V_{H}^{ψ}(M, N/S), and since both V_{H}^{ψ}(M, S) and V_{H}^{ψ}(M, N/S) are contained in V_{H}^{ψ}(M, Λ / 𝔯) by induction, we get V_{H}^{ψ}(M, N) ⊆ V_{H}^{ψ}(M, Λ / 𝔯). In particular V_{H}^{ψ}(M, M) is contained in V_{H}^{ψ}(M, Λ / 𝔯). The reverse inclusion V_{H}^{ψ}(M, Λ / 𝔯) ⊆ V_{H}^{ψ}(M, M) follows from the inclusion A_{H}^{ψ}(M, M) ⊆ A_{H}^{ψ}(M, Λ / 𝔯) of ideals in H, thus proving (i).

Now suppose Ext_{Λ}^{tn}(_{ψ^{n}}M, M) = 0 for n ≫ 0, let 𝔪 ⊂ H be a maximal ideal containing A_{H}^{ψ}(M, Λ / 𝔯), and let x ∈ H^{+} = ⨁_{n=1}^{∞} H^{tn} be a homogeneous element.

As the scalar action from H on Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯) factors through Ext_{Λ}^{t∗}(_{ψ^{∗}}M, M), some power of x must lie in A_{H}^{ψ}(M, Λ / 𝔯). Therefore x must be an element of 𝔪, implying 𝔪 is a graded ideal. But 𝔪_{H} is the only graded maximal ideal of H. This proves (ii).

Next suppose we are given an exact sequence as in (iii). The sequence induces the exact sequence

Ext_{Λ}^{t∗}(_{ψ^{∗}}M′′, Λ / 𝔯) → Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯) → Ext_{Λ}^{t∗}(_{ψ^{∗}}M′, Λ / 𝔯)

of H-modules, giving the inclusion

A_{H}^{ψ}(M′, Λ / 𝔯) · A_{H}^{ψ}(M′′, Λ / 𝔯) ⊆ A_{H}^{ψ}(M, Λ / 𝔯)

of ideals of H. The relation V_{H}^{ψ}(M) ⊆ V_{H}^{ψ}(M′) ∪ V_{H}^{ψ}(M′′) now follows. The original exact sequence also induces the two exact sequences

0 → Hom_{Λ}(M′′, Λ / 𝔯) → Hom_{Λ}(M, Λ / 𝔯)
Ext_{Λ}^{tn−1}(_{ψ^{n}}M′, Λ / 𝔯) → Ext_{Λ}^{tn}(_{ψ^{n}}M′′, Λ / 𝔯) → Ext_{Λ}^{tn}(_{ψ^{n}}M, Λ / 𝔯)

for n ≥ 1, in which we may identify Ext_{Λ}^{tn−1}(_{ψ^{n}}M′, Λ / 𝔯) with Ext_{Λ}^{t(n−1)}(_{ψ^{n−1}}Ω_{Λ}^{t−1}(_{ψ}M′), Λ / 𝔯). Consequently the inclusion

A_{H}^{ψ}(M, Λ / 𝔯) · A_{H}^{ψ}(Ω_{Λ}^{t−1}(_{ψ}M′), Λ / 𝔯) ⊆ A_{H}^{ψ}(M′′, Λ / 𝔯)

holds whenever Ω_{Λ}^{t−1}(M′) ≠ 0, giving V_{H}^{ψ}(M′′) ⊆ V_{H}^{ψ}(M) ∪ V_{H}^{ψ}(Ω_{Λ}^{t−1}(_{ψ}M′)). Finally, the short exact sequence induces the exact sequence

Ext_{Λ}^{tn}(_{ψ^{n}}M, Λ / 𝔯) → Ext_{Λ}^{tn}(_{ψ^{n}}M′, Λ / 𝔯) → Ext_{Λ}^{tn+1}(_{ψ^{n}}M′′, Λ / 𝔯)

for n ≥ 0. If M′′ = Ω_{Λ}^{t−1}(M′′′) for some module M′′′, then we may identify Ext_{Λ}^{tn+1}(_{ψ^{n}}M′′, Λ / 𝔯) with Ext_{Λ}^{t(n+1)}(_{ψ^{n+1}}(_{ψ^{−1}}M′′′), Λ / 𝔯), and the inclusion

A_{H}^{ψ}(M, Λ / 𝔯) · A_{H}^{ψ}(_{ψ^{−1}}M′′′, Λ / 𝔯) ⊆ A_{H}^{ψ}(M′, Λ / 𝔯)

holds. This gives V_{H}^{ψ}(M′) ⊆ V_{H}^{ψ}(M) ∪ V_{H}^{ψ}(_{ψ^{−1}}M′′′), and the proof of (iii) is complete.

To prove (iv), note that by (iii) the inclusion V_{H}^{ψ}(M) ⊆ V_{H}^{ψ}(M′) ∪ V_{H}^{ψ}(M′′) holds, whereas the inclusions A_{H}^{ψ}(M, Λ / 𝔯) ⊆ A_{H}^{ψ}(M′, Λ / 𝔯) and A_{H}^{ψ}(M, Λ / 𝔯) ⊆ A_{H}^{ψ}(M′′, Λ / 𝔯) give V_{H}^{ψ}(M′) ⊆ V_{H}^{ψ}(M) and V_{H}^{ψ}(M′′) ⊆ V_{H}^{ψ}(M). □

As a corollary, we obtain a result whose analogue in the theory of support varieties says that varieties are invariant under the syzygy operator.

Corollary 4.3. If Ω_{Λ}^{t}(M) is nonzero, then V_{H}^{ψ}(M) = V_{H}^{ψ}(Ω_{Λ}^{t}(_{ψ}M)).

Proof. The exact sequence

0 → Ω_{Λ}^{1}(M) → P_{0} → M → 0

gives V_{H}^{ψ}(M) ⊆ V_{H}^{ψ}(P_{0}) ∪ V_{H}^{ψ}(Ω_{Λ}^{t−1}(_{ψ}Ω_{Λ}^{1}(M))), and since V_{H}^{ψ}(P_{0}) is trivial we get V_{H}^{ψ}(M) ⊆ V_{H}^{ψ}(Ω_{Λ}^{t}(_{ψ}M)). On the other hand, the exact sequence

0 → Ω_{Λ}^{t}(_{ψ}M) → _{ψ}P_{t−1} → Ω_{Λ}^{t−1}(_{ψ}M) → 0

gives V_{H}^{ψ}(Ω_{Λ}^{t}(_{ψ}M)) ⊆ V_{H}^{ψ}(_{ψ}P_{t−1}) ∪ V_{H}^{ψ}(_{ψ^{−1}}(_{ψ}M)), and since V_{H}^{ψ}(_{ψ}P_{t−1}) is trivial we get V_{H}^{ψ}(Ω_{Λ}^{t}(_{ψ}M)) ⊆ V_{H}^{ψ}(M). □

We illustrate this last result with an example.

Example. Suppose k is a field and Λ is a Frobenius algebra, let ψ = ν_{Λ} be the Nakayama automorphism of Λ, and take t = 2. We saw in the example following Lemma 2.1 that the Auslander-Reiten translation τ = D Tr is isomorphic to Ω_{Λ}^{2}N, where N is the Nakayama functor D Hom_{Λ}(−, Λ). Moreover, in the example prior to Proposition 3.1 we saw that the latter is isomorphic to _{ν} Λ_{1} ⊗_{Λ}−. Therefore, from the corollary above we get

V_{H}^{ν}(M) = V_{H}^{ν}(Ω_{Λ}^{2}(_{ν}M)) = V_{H}^{ν}(Ω_{Λ}^{2}(_{ν}Λ_{1} ⊗_{Λ}M)) = V_{H}^{ν}(τ(M))

whenever Ω_{Λ}^{2}(M) is nonzero.

A fundamental feature within the theory of support varieties for modules over both group algebras of finite groups and complete intersections is the finite generation of Ext^{∗}(X, Y) (where X and Y are modules over the ring in question) as a module over the commutative Noetherian graded ring of cohomological operators (see [Ben] and [Car] for the group ring case and [Avr] and [AvB] for the complete intersection case). In order to obtain a similar theory for selfinjective algebras in [EHSST], it was necessary to assume that the same finite generation hypothesis held, since there exist selfinjective algebras for which it does not hold.

We now introduce a finite generation hypothesis similar to that used in [Be1], where instead of assuming finite generation of Ext^{∗}(X, Y) for all modules X and Y, a local variant focusing on a single module was used.

Assumption (Fg(M, H, ψ, t)). Given the automorphism Λ ⟶^{ψ} Λ and the integer t ∈ ℕ, there exists a strongly commutative Noetherian graded subalgebra H = ⨁_{n=0}^{∞} H^{tn} of HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) such that H^{0} = HH^{0}(Λ, Λ) = Z(Λ), and with the property that Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯) is a finitely generated H-module.

As mentioned in the remark prior to Definition 4.1, considering only strongly commutative algebras instead of “ordinary” commutative algebras is not a severe restriction. In fact, the following result shows that if Fg(M, H, ψ, t) holds for a commutative Noetherian graded algebra H which is not necessarily strongly commutative, then there exist a positive integer s and a strongly commutative Noetherian subalgebra H′ ⊆ H for which Fg(M, H′, ψ^{s}, ts) holds.

Proposition 4.4. Let Λ ⟶^{ψ} Λ be an automorphism and t ∈ ℕ an integer, and suppose there exists a commutative Noetherian graded subalgebra H = ⨁_{n=0}^{∞} H^{tn} of HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) such that H^{0} = HH^{0}(Λ, Λ) = Z(Λ), and with the property that Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯) is a finitely generated H-module. Then there exist a positive integer s and a strongly commutative Noetherian graded subalgebra H′ of H for which Fg(M, H′, ψ^{s}, ts) holds

Proof. From the remark prior to Definition 4.1 we see that there exist two positive integers n and w such that _{ψ^{n}}θ = θ_{ψ^{−n}} for every homogeneous element θ ∈ H with |θ| ≥ tw. Let η_{1}, …, η_{r} ∈ H be homogeneous elements of positive degrees generating H as an algebra over H^{0}, and denote the integer wn by s. Then the subalgebra H′ = H^{0}[η_{1}^{s}, …, η_{r}^{s}] of H is strongly commutative with respect to the automorphism ψ^{s}, and Ext_{Λ}^{(ts)∗}(_{(ψ^{s})^{∗}}M, Λ / 𝔯) is a finitely generated H′-module. □

We now show that introducing the above finite generation hypothesis enables us to compute the dimension of the twisted variety of a module. Recall first that if X = ⨁_{n=0}^{∞} X_{n} is a graded k-module of finite type (that is, the k-length of X_{n} is finite for all n), then the rate of growth of X, denoted γ(X), is defined as

γ(X) =^{def} inf{c ∈ ℕ ∪ {0} | ∃a ∈ ℝ such that ℓ_{k}(X_{n}) ≤ an^{c−1} for n ≫ 0}.

For a Λ-module Y with minimal projective resolution ⋯ → Q_{1} → Q_{0} → Y → 0, we define the m-complexity of Y (where m ≥ 1 is a number) as the rate of growth of ⨁_{n=0}^{∞} Q_{mn}, and denote it by cx^{m} Y. Note that the 1-complexity of a module coincides with the usual notion of complexity, and that the identity

cx^{m} Y = γ (Ext_{Λ}^{m∗}(Y, Λ / 𝔯))

always holds (the latter can be seen by adopting the arguments given in [Be1, Section 3]).

The following result allows us to compute the dimension of V_{H}^{ψ}(M) in terms of the t-complexity of M provided Fg(M, H, ψ, t) is satisfied. In particular Dade’s Lemma holds.

Proposition 4.5. If Fg(M, H, ψ, t) holds, then dim V_{H}^{ψ}(M) = cx^{t} M. In particular V_{H}^{ψ}(M) is trivial if and only if M has finite projective dimension.

Proof. Adopting the arguments used to prove [Ben, Proposition 5.7.2] and [EHSST, Proposition 2.1] gives dim V_{H}^{ψ}(M) = γ (Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯)). For any Λ-module Y

and any k-automorphism Λ ⟶^{ϕ} Λ there is an isomorphism Y ≃ _{ϕ}Y of k-modules, and so if ⋯ → Q_{1} → Q_{0} → Y → 0 is a minimal projective resolution we see that the m-complexity of Y equals the rate of growth of ⨁_{n=0}^{∞} _{ϕ^{n}}(Q_{mn}). In particular the equalities

cx^{t} M = γ (⨁_{n=0}^{∞}_{ψ^{n}}(P_{tn})) = γ (Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯))

hold, where

ℙ: ⋯ → P_{2} ⟶^{d_{2}} P_{1} ⟶^{d_{1}} P_{0} ⟶^{d_{0}} M → 0

is the minimal projective resolution of M. □

Note that whenever Fg(M, H, ψ, t) is satisfied the dimension of V_{H}^{ψ}(M), and therefore also the t-complexity of M, must be finite; since H is commutative graded Noetherian it is generated as an algebra over H^{0} by a finite set {x_{0}, …, x_{r}} of homogeneous elements of positive degrees, giving γ (Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯)) ≤ r (see the discussion prior to [Be1, Proposition 3.1]). It then follows from the proof of the above proposition that cx^{t} M ≤ r.

The next result gives a sufficient and necessary condition for the variety to be one dimensional. Recall that a Λ-module Y is periodic if there exists a positive integer p such that Y ≃ Ω_{Λ}^{p}(Y), whereas it is eventually periodic if Ω_{Λ}^{i}(Y) is periodic for some

i ≥ 0. For a k-automorphism Λ ⟶^{ϕ} Λ we define Y to be ϕ-periodic if there exists a positive integer p such that Y ≃ Ω_{Λ}^{p}(_{ϕ}Y), and eventually ϕ-periodic if Ω_{Λ}^{i}(Y) is ϕ-periodic for some i ≥ 0.

Proposition 4.6. If Fg(M, H, ψ, t) holds, then dim V_{H}^{ψ}(M) = 1 if and only if M is eventually ψ^{i}-periodic for some i ≥ 1. Moreover, when this occurs there is a positive integer w such that Ω_{Λ}^{tj}(M) ≃ Ω_{Λ}^{t(j+w)}(_{ψ^{w}}M) for some j ≥ 0.

Proof. If M is eventually ψ^{i}-periodic, then the sequence

ℓ_{k}(P_{0}), ℓ_{k}(P_{1}), ℓ_{k}(P_{2}), …

must be bounded, that is, the 1-complexity of M is 1. But then the sequence

ℓ_{k}(P_{0}), ℓ_{k}(P_{t}), ℓ_{k}(P_{2t}), …

is also bounded, that is, the t-complexity of M is also 1, and consequently dim V_{H}^{ψ}(M) = 1.

Conversely, suppose the latter of the above sequences is bounded. By [Be1, Proposition 2.1] there exists a homogeneous element η ∈ H of positive degree, say |η| = tw, such that the multiplication map

Ext_{Λ}^{ti}(_{ψ^{i}}M, Λ / 𝔯) ⟶^{·η} Ext_{Λ}^{t(i+w)}(_{ψ^{i+w}}M, Λ / 𝔯)

is injective for i ≫ 0. Represent the element η ⊗_{Λ} M ∈ Ext_{Λ}^{t∗}(_{ψ^{∗}}M, M) by a map f_{η}: Ω_{Λ}^{tw}(_{ψ^{w}}M) → M. Then for each i ≥ 0 there is a map f_{i}: _{ψ^{w}}(P_{tw+i}) → P_{i} making the diagram

\xymatrix{⋯ \ar[r] & _{ψ^{w}}(P_{tw+2}) \ar[d]^{f_{2}} \ar[r]^{d_{tw+2}} & _{ψ^{w}}(P_{tw+1}) \ar[d]^{f_{1}} \ar[r]^{d_{tw+1}} & _{ψ^{w}}(P_{tw}) \ar[d]^{f_{0}} \ar[r] & Ω_{Λ}^{tw}(_{ψ^{w}}M) \ar[d]^{f_{η}} \ar[r] & 0 \\ ⋯ \ar[r] & P_{2} \ar[r]^{d_{2}} & P_{1} \ar[r]^{d_{1}} & P_{0} \ar[r]^{d_{0}} & M \ar[r] & 0}

with exact rows commute. If θ ∈ Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯) is a homogeneous element, say |θ| = tn, and represented by a map f_{θ}: _{ψ^{n}}(P_{tn}) → Λ / 𝔯, then θη ∈ Ext_{Λ}^{t(w+n)}(_{ψ^{w+n}}M, Λ / 𝔯) is represented by the composite map

ψ^{w+n}(P_{t(w+n)}) ⟶^{_{ψ^{n}} f_{tn}} _{ψ^{n}}(P_{tn}) ⟶^{f_{θ}} Λ / 𝔯 .

For any i ≥ 0 the complex _{ψ^{i}}ℙ is a minimal projective resolution of _{ψ^{i}}M, and therefore Ext_{Λ}^{ti}(_{ψ^{i}}M, Λ / 𝔯) = Hom_{Λ}(_{ψ^{i}}(P_{ti}), Λ / 𝔯). Moreover, the multiplication map

Ext_{Λ}^{ti}(_{ψ^{i}}M, Λ / 𝔯) ⟶^{·η} Ext_{Λ}^{t(i+w)}(_{ψ^{i+w}}M, Λ / 𝔯)

is just the map

Hom_{Λ}(_{ψ^{i}}(P_{ti}), Λ / 𝔯) ⟶^{(_{ψ^{i}} f_{ti})^{∗}} Hom_{Λ}(_{ψ^{i+w}}(P_{t(i+w)}), Λ / 𝔯)
g  ↦  g ∘_{ψ^{i}} f_{ti},

and since the exact sequence

ψ^{i+w}(P_{t(i+w)}) ⟶^{_{ψ^{i}} f_{ti}} _{ψ^{i}}(P_{ti}) → _{ψ^{i}}(Coker f_{ti}) → 0

shows that the kernel of the multiplication map is isomorphic to Hom_{Λ}(_{ψ^{i}}(Coker f_{ti}), Λ / 𝔯), we see that Coker f_{ti} = 0 for i ≫ 0. Consequently, for each i ≫ 0 there exists a surjective map Ω_{Λ}^{t(w+i)}(_{ψ^{w}}M) ↠ Ω_{Λ}^{ti}(M) and therefore also a sequence

⋯ ↠ Ω_{Λ}^{t(3w+i)}(_{ψ^{3w}}M) ↠ Ω_{Λ}^{t(2w+i)}(_{ψ^{2w}}M) ↠ Ω_{Λ}^{t(w+i)}(_{ψ^{w}}M) ↠ Ω_{Λ}^{ti}(M)

of surjections. However, by assumption the sequence

ℓ_{k}(P_{0}), ℓ_{k}(P_{t}), ℓ_{k}(P_{2t}), …

is bounded, and therefore Ω_{Λ}^{t((q+1)w+i)}(_{ψ^{(q+1)w}}M) must be isomorphic to Ω_{Λ}^{t(qw+i)}(_{ψ^{qw}}M) for large q and i. By setting j = qw + i and twisting with the automorphism ψ^{−qw}, we see that Ω_{Λ}^{tj}(M) ≃ Ω_{Λ}^{t(j+w)}(_{ψ^{w}}M) for j ≫ 0. □

As a particular case of the proposition we obtain the following result on D Trperiodicity over a Frobenius algebra.

Proposition 4.7. Suppose k is a field and Λ is a Frobenius algebra, and let Λ ⟶^{ν} Λ be a Nakayama automorphism. If M does not have a nonzero projective summand and Fg(M, H, ν^{n}, 2n) holds for some n ≥ 1, then dim V_{H}^{ν}(M) = 1 if and only if M ≃ τ^{p}(M) for some p ≥ 1.

Proof. By the previous proposition the variety V_{H}^{ν}(M) is one dimensional if and only if there is a positive integer w such that Ω_{Λ}^{2nj}(M) ≃ Ω_{Λ}^{2n(j+w)}(_{ν^{nw}}M) for some j ≥ 0. Taking cosyzygies we see that the latter happens precisely when M ≃ Ω_{Λ}^{2nw}(_{ν^{nw}}M), that is, when M ≃ τ^{nw}(M). □

We illustrate this last result with an example.

Example. Let k be an algebraically closed field of odd characteristic and q ∈ k a nonzero element which is not a root of unity, and denote by Λ the k-algebra

Λ = k⟨x, y⟩/(x^{2}, xy + qyx, y^{2}).

We saw in the example following Proposition 3.1 that the Nakayama automorphism ν of Λ is defined by

x ↦ −q^{−1}x, y ↦ −qy,

and that HH^{4∗}(_{ν^{2∗}}Λ_{1}, Λ) is isomorphic to the fibre product k[θ]×_{k}k with θ of degree 4. In particular HH^{4∗}(_{ν^{2∗}}Λ_{1}, Λ) is strongly commutative, and we denote this ring by H.

For elements α, β ∈ k, denote by M_{(α,β)} the Λ-module Λ(αx + βy) (see [Sma] for a counterexample, using the module M_{(1,1)}, to a question raised by Auslander, a question for which a counterexample was first given in [JoS]). Consider the module M = M_{(1,β)} for β ≠ 0, and for each i ≥ 0 let P_{i} = Λ. The sequence

ℙ: ⋯ → P_{3} ⟶^{·(x+q^{3}βy)} P_{2} ⟶^{·(x+q^{2}βy)} P_{1} ⟶^{·(x+qβy)} P_{0} ⟶^{·(x+βy)} M → 0

is a minimal projective resolution of M, hence since _{ν^{n}}ℙ is a minimal projective resolution of _{ν^{n}}M for any n ≥ 1 we see that Ext_{Λ}^{2n}(_{ν^{n}}M, k) = Hom_{Λ}(_{ν^{n}}Λ, k) is one dimensional. We shall prove that Fg(M, H, ν, 2) holds.

Recall from the example following Proposition 3.1 the minimal bimodule projective resolution

(𝔽, δ): ⋯ → F^{n+1} ⟶^{δ} F^{n} ⟶^{δ} F^{n−1} → ⋯

of Λ, where the set {f̃_{i}^{n}}_{i=0}^{n} generates F^{n} as a Λ^{e}-module and the differential δ: F^{n} → F^{n−1} is given by

f̃_{i}^{n} ↦ [xf̃_{i}^{n−1} + (−1)^{n}q^{i}f̃_{i}^{n−1}x] + [q^{n−i}yf̃_{i−1}^{n−1} + (−1)^{n}f̃_{i−1}^{n−1}y] .

The element θ ∈ H^{4} represented by the map

g_{4} : _{ν^{2}}F_{1}^{4} → Λ
f̃_{i}^{4} ↦ \{ \begin{matrix}1 & for i = 2 \\ 0 & for i ≠ 2 \end{matrix}

generates H as an algebra over H^{0}. The resolution 𝔽 ⊗_{Λ} M is also a projective resolution of M, and defining Λ-linear maps

h_{n}: P_{n} → F^{n} ⊗_{Λ} M
1 ↦ (∑_{i=0}^{n} q^{\frac{i(i+1)}{2}} β^{i}f̃_{i}^{n}) ⊗ (x + βy)

gives a commutative diagram

[figure: ⋯; P_{2}; h_{2}; ·(x+q^{2}βy); P_{1}; h_{1}; ·(x+qβy); P_{0}; h_{0}; ·(x+βy); M; 0; ⋯; F^{2} ⊗_{Λ} M; δ⊗1; F^{1} ⊗_{Λ} M; δ⊗1; F^{0} ⊗_{Λ} M; M; 0]

with exact rows. Consequently, the element θ⊗_{Λ} M ∈ Ext_{Λ}^{4}(_{ν^{2}}M, M) is represented by the composite map

f̄_{θ}: _{ν^{2}}(P_{4}) ⟶^{(g_{4}⊗1)∘h_{4}} Λ ⊗_{Λ}M ≃ M
1  ↦  q^{3}β^{2}(x + βy).

Now for each i ≥ 0 define f_{i}: _{ν^{2}}(P_{i+4}) →_{ν^{2}} (P_{i}) by 1 ↦ q^{2i+3}β^{2}. We then obtain a commutative diagram

\xymatrix{⋯ \ar[r] & _{ν^{2}}(P_{7}) \ar[d]^{f_{3}} \ar[r]^{·(x+q^{7}βy)} & _{ν^{2}}(P_{6}) \ar[d]^{f_{2}} \ar[r]^{·(x+q^{6}βy)} & _{ν^{2}}(P_{5}) \ar[d]^{f_{1}} \ar[r]^{·(x+q^{5}βy)} & _{ν^{2}}(P_{4}) \ar[d]^{f_{0}} \ar[r] \ar[dr]^{f̄_{θ}} & ⋯ \\ ⋯ \ar[r] & P_{3} \ar[r]^{·(x+q^{3}βy)} & P_{2} \ar[r]^{·(x+q^{2}βy)} & P_{1} \ar[r]^{·(x+qβy)} & P_{0} \ar[r]^{·(x+βy)} & M \ar[r] & 0}

with exact rows, hence if μ ∈ Ext_{Λ}^{2n}(_{ν^{n}}M, k) = Hom_{Λ}(_{ν^{n}}(P_{2n}), k) is an element represented by a map f̄_{μ}: _{ν^{n}}(P_{2n}) → k we see that the element μ · θ ∈ Ext_{Λ}^{2(n+2)}(_{ν^{(}n+2)}M, k) is represented by the composite map

ν^{(}n+2)(P_{2(n+2)}) ⟶^{_{ν^{n}} (f_{2n})} _{ν^{n}}(P_{2n}) ⟶^{f̄_{μ}} k.

Moreover, this composition is nonzero whenever μ is nonzero, since f_{2n} is just multiplication with q^{4n+3}β^{2}. Therefore, since Ext_{Λ}^{2n}(_{ν^{n}}M, k) is one dimensional for each n ≥ 1, the H-module Ext_{Λ}^{2∗}(_{ν^{∗}}M, k) is finitely generated; it is generated as an H-module by any k-basis in Ext_{Λ}^{0}(_{ν^{0}}M, k) = Hom_{Λ}(M, k) together with any nonzero elements μ_{1} ∈ Ext_{Λ}^{2}(_{ν}M, k) and μ_{2} ∈ Ext_{Λ}^{4}(_{ν^{2}}M, k).

The above shows that Fg(M_{(1,β)}, H, ν, 2) holds for β ≠ 0, and since the 2complexity of M_{(1,β)} is obviously 1 we see from Proposition 4.5 that the variety V_{H}^{ν}(M_{(1,β)}) is one dimensional. From Proposition 4.7 we conclude that M_{(1,β)} ≃ τ^{w}(M_{(1,β)}) for some w ≥ 1. Indeed, in [LiS] it is shown that M_{(1,β)} is isomorphic to τ(M_{(1,β)}).

Recall that each nonprojective indecomposable Λ-module is annihilated by yx, and is therefore a module over the algebra Λ /(yx) (see [Sch, Section 4]). The latter is stably equivalent to the Kronecker algebra, an equivalence under which the representation

\xymatrix{k \ar[r]^{β} \ar[r]_{α} & k}

corresponds to the module M_{(α,β)}. Denoting M_{(α,β)} by M_{(α,β)}^{1}, it follows from the well known representation theory of the Kronecker algebra (see, for example, [ARS] and [ASS]) that the indecomposable τ-periodic Λ-modules are divided into distinct countable classes {M_{(α,β)}^{i}}_{i=1}^{∞}, one for each pair (α, β) ∈ {(0, 1)}∪{(1, β)}_{β∈k}, such that for each i ≥ 1 there exists a short exact sequence

0 → M_{(α,β)}^{i} → M_{(α,β)}^{i−1} ⊕ M_{(α,β)}^{i+1} → M_{(α,β)}^{i} → 0,

where M_{(α,β)}^{0} = 0. Now each such exact sequence induces an exact sequence

Ext_{Λ}^{2∗}(_{ν^{∗}}M_{α}^{i}, k) → Ext_{Λ}^{2∗}(_{ν^{∗}}M_{α}^{i−1}, k) ⊕ Ext_{Λ}^{2∗}(_{ν^{∗}}M_{α}^{i+1}, k) → Ext_{Λ}^{2∗}(_{ν^{∗}}M_{α}^{i}, k)

of H-modules, and so since H is Noetherian and Ext_{Λ}^{2∗}(_{ν^{∗}}M_{(1,β)}^{1}, k) is a finitely generated H-module whenever β is nonzero, an induction argument shows that Ext_{Λ}^{2∗}(_{ν^{∗}}M_{(1,β)}^{i}, k) is a finitely generated H-module for any i ≥ 1 and β ≠ 0. We conclude that Fg(M_{(1,β)}^{i}, H, ν, 2) holds for all the modules {M_{(1,β)}^{i}}_{i=1}^{∞} when β is nonzero.

However, there are two more classes of indecomposable τ-periodic Λ-modules, namely {M_{(1,0)}^{i}}_{i=0}^{∞} and {M_{(0,1)}^{i}}_{i=0}^{∞}. Do these modules satisfy the finite generation hypothesis? The answer is no, and to see this, consider the module M_{(1,0)} = Λ x. Letting P_{i} = Λ for each i ≥ 0 we see that the sequence

⋯ → P_{3} ⟶^{·x} P_{2} ⟶^{·x} P_{1} ⟶^{·x} P_{0} ⟶^{·x} M_{(1,0)} → 0

is a minimal projective resolution of M_{(1,0)}, and defining Λ-linear maps

h_{n}: P_{n} → F^{n} ⊗_{Λ} M_{(1,0)}
1 ↦ f̃_{0}^{n} ⊗ x

gives a commutative diagram

[figure: ⋯; P_{1}; h_{1}; ·x; P_{0}; h_{0}; ·x; M_{(1,0)}; 0; ⋯; F^{1} ⊗_{Λ} M_{(1,0)}; δ⊗1; F^{0} ⊗_{Λ} M_{(1,0)}; M_{(1,0)}; 0]

with exact rows. Since the map h_{4} does not “hit” the generator f̃_{2}^{4} ⊗ x ∈ F^{4} ⊗_{Λ} M_{(1,0)}, we see that the element θ ⊗_{Λ} M ∈ Ext_{Λ}^{4}(_{ν^{2}}M, M) is represented by the zero map. This shows that Ext_{Λ}^{2∗}(_{ν^{∗}}M_{(1,0)}^{i}, k) cannot be a finitely generated H-module, and a similar argument shows that the same is true for the module M_{(0,1)}.

Note also that the finite generation condition cannot hold for any nonzero indecomposable nonprojective Λ-module which is not τ-periodic; if X is such a module and Ext_{Λ}^{2∗}(_{ν^{∗}}X, k) is finitely generated over H, then the rate of growth of Ext_{Λ}^{2∗}(_{ν^{∗}}X, k) is not more than that of H. However, the latter equals the Krull dimension of H, thus γ (Ext_{Λ}^{2∗}(_{ν^{∗}}X, k)) ≤ γ(H) = 1. Since X is nonprojective we conclude that the rate of growth of Ext_{Λ}^{2∗}(_{ν^{∗}}X, k) is 1, and so by Proposition 4.5 the variety V_{H}^{ν}(X) is one dimensional. But then Proposition 4.7 implies X is τ-periodic, a contradiction.

Returning to the general theory, we now impose the finite generation hypothesis on both M and Ω_{Λ}^{1}(M). The following result shows that, in this situation, if the variety of M is nontrivial (that is, when M does not have finite projective dimension) then there exists a homogeneous element in H “cutting down” the variety by one dimension. Recall first that if η ∈ HH^{t∗}(_{ψ^{∗}}Λ_{1}, Λ) is a homogeneous element, say η ∈ Ext_{Λ^{e}}^{tm}(_{ψ^{m}}Λ_{1}, Λ), then it can be represented by a Λ^{e}-linear map f_{η}: Ω_{Λ^{e}}^{tm}(_{ψ^{m}}Λ_{1}) → Λ. This map yields a commutative diagram

[figure: 0; Ω_{Λ^{e}}^{tm}(_{ψ^{m}}Λ_{1}); f_{η}; _{ψ^{m}}(P^{tm−1}); Ω_{Λ^{e}}^{tm−1}(_{ψ^{m}}Λ_{1}); 0; 0; Λ; K_{η}; Ω_{Λ^{e}}^{tm−1}(_{ψ^{m}}Λ_{1}); 0]

with exact rows, in which we have denoted by P^{i} the ith module in the minimal projective bimodule resolution of Λ. Note that up to isomorphism the module K_{η} is independent of the map f_{η} chosen to represent η.

Proposition 4.8. If both Fg(M, H, ψ, t) and Fg(Ω_{Λ}^{1}(M), H, ψ, t) hold and M does not have finite projective dimension, then there exists a homogeneous element η ∈ H of positive degree such that dim V_{H}^{ψ}(Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M) = dim V_{H}^{ψ}(M) − 1.

Proof. By assumption the H-modules Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯) and Ext_{Λ}^{t∗}(_{ψ^{∗}}Ω_{Λ}^{1}(M), Λ / 𝔯) are finitely generated, hence by slightly generalizing the proof of [Be1, Proposition 2.1] we see that there exists a homogeneous element η ∈ H of positive degree, say η ∈ H^{tm} ⊆ Ext_{Λ^{e}}^{tm}(_{ψ^{m}}Λ_{1}, Λ), such that the multiplication maps

Ext_{Λ}^{ti}(_{ψ^{i}}M, Λ / 𝔯) ⟶^{·η} Ext_{Λ}^{t(i+m)}(_{ψ^{i+m}}M, Λ / 𝔯)
Ext_{Λ}^{ti}(_{ψ^{i}}Ω_{Λ}^{1}(M), Λ / 𝔯) ⟶^{·η} Ext_{Λ}^{t(i+m)}(_{ψ^{i+m}}Ω_{Λ}^{1}(M), Λ / 𝔯)

are both k-monomorphisms for i ≫ 0. Consider the short exact sequence

0 → Λ → K_{η} → Ω_{Λ^{e}}^{tm−1}(_{ψ^{m}}Λ_{1}) → 0

obtained from η. As Ω_{Λ^{e}}^{tm−1}(_{ψ^{m}}Λ_{1}) is right Λ-projective, the sequence splits when considered as a sequence of right Λ-modules, and consequently the sequence

0 → M → K_{η} ⊗_{Λ} M → Ω_{Λ^{e}}^{tm−1}(_{ψ^{m}}Λ_{1}) ⊗_{Λ} M → 0

is exact. For each i ≥ 0 the latter sequence induces a long exact sequence

Ext_{Λ}^{ti}(_{ψ^{i}}(K_{η} ⊗_{Λ} M), Λ / 𝔯) → Ext_{Λ}^{ti}(_{ψ^{i}}M, Λ / 𝔯) ⟶^{∂_{ti}}

Ext_{Λ}^{t(i+m)}(_{ψ^{i+m}}M, Λ / 𝔯) → Ext_{Λ}^{ti+1}(_{ψ^{i}}(K_{η} ⊗_{Λ} M), Λ / 𝔯) →

Ext_{Λ}^{ti+1}(_{ψ^{i}}M, Λ / 𝔯) ⟶^{∂_{ti+1}} Ext_{Λ}^{t(i+m)+1}(_{ψ^{i+m}}M, Λ / 𝔯)

in which we have replaced Ext_{Λ}^{j}(_{ψ^{i}}(Ω_{Λ^{e}}^{tm−1}(_{ψ^{m}}Λ_{1}) ⊗_{Λ} M), Λ / 𝔯) with Ext_{Λ}^{j+tm−1}(_{ψ^{i+m}}M, Λ / 𝔯), due to the fact that _{ψ^{i}}(Ω_{Λ^{e}}^{tm−1}(_{ψ^{m}}Λ_{1}) ⊗_{Λ} M) is a (tm − 1)th syzygy of _{ψ^{i+m}}M. By [Mac, Theorem III.9.1] the connecting homomorphism ∂_{j} is then the Yoneda product with the extension (−1)^{j}_{ψ^{i}}(η ⊗_{Λ} M), in particular we see that ∂_{ti} is scalar multiplication with (−1)^{ti}η.

Now consider the connecting homomorphism ∂_{ti+1}. Applying the Yoneda relation (†) from Section 3 to η and the short exact sequence

θ: 0 → Ω_{Λ}^{1}(M) → P_{0} → M → 0

gives the relation

(η ⊗_{Λ} Ω_{Λ}^{1}(M)) ∘ (_{ψ^{m}} Λ_{1} ⊗_{Λ} θ) = (−1)^{tm}(Λ ⊗_{Λ}θ) ∘ (η ⊗_{Λ} M),

which we may twist by ψ^{i} to obtain the relation

ψ^{i}(η ⊗_{Λ} Ω_{Λ}^{1}(M)) ∘_{ψ^{i+m}}θ = (−1)^{tm}_{ψ^{i}}θ ∘_{ψ^{i}}(η ⊗_{Λ} M).

This gives a commutative diagram

\xymatrix{Ext_{Λ}^{ti}(_{ψ^{i}}Ω_{Λ}^{1}(M), Λ / 𝔯) \ar[d]^{·(−1)^{t(i+m)+1}η} \ar[r]^{_{ψ^{i}} θ} & Ext_{Λ}^{ti+1}(_{ψ^{i}}M, Λ / 𝔯) \ar[d]^{∂_{ti+1}} \\ Ext_{Λ}^{t(i+m)}(_{ψ^{i+m}}Ω_{Λ}^{1}(M), Λ / 𝔯) \ar[r]^{_{ψ^{i+m}} θ} & Ext_{Λ}^{t(i+m)+1}(_{ψ^{i+m}}M, Λ / 𝔯)}

in which the horizontal maps are isomorphisms, hence the connecting homomorphism ∂_{ti+1} is also basically just scalar multiplication with η, as was the case with ∂_{ti}. Consequently they are both injective for i ≫ 0, giving a short exact sequence

(††) 0 → _{Λ}^{ti}(_{ψ^{i}}M, Λ / 𝔯) ⟶^{·η} _{Λ}^{t(i+m)}(_{ψ^{i+m}}M, Λ / 𝔯) → _{Λ}^{ti+1}(_{ψ^{i}}(K_{η} ⊗_{Λ} M), Λ / 𝔯) → 0

for large i (in which we have used the short hand notion _{Λ}^{j}(−, −) for Ext_{Λ}^{j}(−, −)). Note that we may identify Ext_{Λ}^{ti+1}(_{ψ^{i}}(K_{η} ⊗_{Λ} M), Λ / 𝔯) with Ext_{Λ}^{ti}(_{ψ^{i}}(Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M), Λ / 𝔯); since K_{η} is right Λ-projective the projective bimodule cover

0 → Ω_{Λ^{e}}^{1}(K_{η}) → Q → K_{η} → 0

of K_{η} splits as a sequence of right Λ-modules, and therefore stays exact when tensored with M. In addition, the Λ-module Q ⊗_{Λ} M is projective, hence

Ext_{Λ}^{ti+1}(_{ψ^{i}}(K_{η} ⊗_{Λ} M), Λ / 𝔯) ≃ Ext_{Λ}^{ti}(_{ψ^{i}}(Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M), Λ / 𝔯).

Consider now the H-module Ext_{Λ}^{t∗}(_{ψ^{∗}}(Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M), Λ / 𝔯), and let w be an integer such that the sequence (††) is exact for i ≥ w. Then the submodule ⨁_{i=w}^{∞} Ext_{Λ}^{ti}(_{ψ^{i}}(Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M), Λ / 𝔯) is finitely generated over H, being a factor module of the submodule ⨁_{i=w}^{∞} Ext_{Λ}^{t(i+m)}(_{ψ^{i+m}}M, Λ / 𝔯) of the finitely generated H-module Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯). But then the H-module Ext_{Λ}^{t∗}(_{ψ^{∗}}(Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M), Λ / 𝔯) must be finitely generated itself, since each graded part Ext_{Λ}^{tj}(_{ψ^{j}}(Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M), Λ / 𝔯) is finitely generated over H^{0}. Also, from [Be1, Proposition 3.1] we get

γ (⨁_{i=w}^{∞} Ext_{Λ}^{ti}(_{ψ^{i}}(Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M), Λ / 𝔯)) = γ (⨁_{i=w}^{∞} Ext_{Λ}^{t(i+m)}(_{ψ^{i+m}}M, Λ / 𝔯)) − 1,

and since for any Λ-module X the rate of growth of Ext_{Λ}^{t∗}(_{ψ^{∗}}X, Λ / 𝔯) equals that of ⨁_{i=w}^{∞} Ext_{Λ}^{ti}(_{ψ^{i}}X, Λ / 𝔯) we get

γ (Ext_{Λ}^{t∗}(_{ψ^{∗}}(Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M), Λ / 𝔯)) = γ (Ext_{Λ}^{t∗}(_{ψ^{∗}}M, Λ / 𝔯)) − 1.

Therefore the equality cx^{t} (Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M) = cx^{t} M − 1 holds, and so from Proposition 4.5 we conclude that dim V_{H}^{ψ}(Ω_{Λ^{e}}^{1}(K_{η}) ⊗_{Λ} M) = dim V_{H}^{ψ}(M) − 1. □

Finally we turn to the setting in which Fg(X, H, ψ, t) holds for all Λ-modules X, and derive two corollaries from Proposition 4.8. Observe first that if Fg(Λ / 𝔯, H, ψ, t) holds, then Fg(S, H, ψ, t) holds for every simple Λ-module S, and so by induction on the length of a module we see that Fg(X, H, ψ, t) holds for every Λ-module X; namely, if ℓ(X) ≥ 2, choose a submodule Y ⊂ X such that ℓ(Y) = ℓ(X) − 1. The exact sequence

0 → Y → X → X/Y → 0

induces the exact sequence

Ext_{Λ}^{t∗}(_{ψ^{∗}}(X/Y), Λ / 𝔯) → Ext_{Λ}^{t∗}(_{ψ^{∗}}X, Λ / 𝔯) → Ext_{Λ}^{t∗}(_{ψ^{∗}}Y, Λ / 𝔯)

of H-modules, and since the end terms are finite over H, so is the middle term.

Corollary 4.9. If Fg(Λ / 𝔯, H, ψ, t) holds and dim V_{H}^{ψ}(M) = d > 0, then there exist homogeneous elements η_{1}, …, η_{d−1} ∈ H of positive degrees such that the module

Ω_{Λ^{e}}^{1}(K_{η_{d−1}}) ⊗_{Λ} ⋯ ⊗_{Λ} Ω_{Λ^{e}}^{1}(K_{η_{1}}) ⊗_{Λ} M

is eventually ψ^{i}-periodic for some i ≥ 1.

Proof. This is a direct consequence of Proposition 4.6 and Proposition 4.8. □

Corollary 4.10. Suppose k is a field and Λ is a Frobenius algebra, and let Λ ⟶^{ν} Λ be a Nakayama automorphism. If Fg(Λ / 𝔯, H, ν^{n}, 2n) holds for some n ≥ 1 and dim V_{H}^{ν}(M) = d > 0, then there exist homogeneous elements η_{1}, …, η_{d−1} ∈ H of positive degrees such that every nonzero nonprojective indecomposable summand of

Ω_{Λ^{e}}^{1}(K_{η_{d−1}}) ⊗_{Λ} ⋯ ⊗_{Λ} Ω_{Λ^{e}}^{1}(K_{η_{1}}) ⊗_{Λ} M

is τ-periodic.

Proof. This is a direct consequence of Proposition 4.7 and Proposition 4.8. □

Acknowledgement

I would like to thank my supervisor Øyvind Solberg for valuable suggestions and comments on this paper.

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Petter Andreas Bergh Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

   E-mail address: bergh@math.ntnu.no