TENSOR PRODUCTS AND SUPPORT VARIETIES FOR SOME NONCOCOMMUTATIVE HOPF ALGEBRAS

JULIA YAEL PLAVNIK AND SARAH WITHERSPOON

Abstract. We explore questions of projectivity and tensor products of modules for finite dimensional Hopf algebras. We construct many classes of examples in which tensor powers of nonprojective modules are projective and tensor products of modules in one order are projective but in the other order are not. Our examples are smash coproducts with duals of group algebras, some having algebra and coalgebra structures twisted by a cocycle. We develop a theory of support varieties for these Hopf algebras to use as a tool in our investigations.

1. Introduction

Tensor products of modules for finite dimensional cocommutative Hopf algebras (equivalently finite group schemes) are well behaved: The tensor product is commutative up to natural isomorphism. Tensor powers of nonprojective modules are nonprojective. There is a well developed theory of support varieties—a very fruitful tool originating from finite groups—and the variety of a tensor product of modules is the intersection of their varieties, as shown by Friedlander and Pevtsova [10]. All of these phenomena occur in positive characteristic. In characteristic 0, some noncocommutative Hopf algebras share this good behavior, such as the quantum elementary abelian groups in [20] by Pevtsova and the second author.

By contrast, there are examples of finite dimensional noncocommutative Hopf algebras for which the tensor product of modules is not so well behaved: Benson and the second author [6] showed that some Hopf algebras constructed from finite groups in positive characteristic have nonprojective modules with projective tensor powers and modules whose tensor products in one order are projective but in the other order are not. A support variety theory for finite dimensional selfinjective algebras [7, 22] applies to these examples. The varieties are reasonably well behaved, yet the variety of a tensor product of modules is not the intersection of their varieties.

Date: November 28, 2016.

1991 Mathematics Subject Classification. 16T05, 18D10.

Key words and phrases. projective modules; nonsemisimple Hopf algebra; support varieties; smash coproduct.

The first author was partially supported by NSF grant DMS-1410144. The second author was partially supported by NSF grant DMS-1401016.

These results lead us to ask: What hypotheses on a noncocommutative finite dimensional Hopf algebra are necessary or sufficient to ensure that (1) the tensor product of modules in one order is projective if and only if it is projective in the other order, (2) tensor powers of nonprojective modules are nonprojective, or (3) the support variety of a tensor product of modules is the intersection of their varieties? For a given Hopf algebra, a positive answer to (3) implies a positive answer to (1) and (2); this follows, e.g., from Theorem 3.1(i) below, a restatement of a theorem of Erdmann, Holloway, Snashall, Solberg, and Taillefer [7].

In this paper we answer some of these questions for some types of Hopf algebras, and develop their support variety theory. We generalize the examples of [6] to present many more Hopf algebras with reasonable support variety theory and yet negative answers to all three questions. In particular we give examples in characteristic 0. These are in Section 4, and are smash coproducts of quantum elementary abelian groups with duals of group algebras. We use the variety theory developed in [19, 20] for quantum elementary abelian groups (that is, tensor products of Taft algebras) to handle these examples. Our general Theorem 4.3 shows that these types of examples are ubiquitous in any characteristic, and are not just isolated anomalies: It implies that any finite dimensional nonsemisimple Hopf algebra having a positive answer to question (3) above can be embedded in a Hopf algebra having a negative answer to (3).

In Section 5 we give many new examples in positive characteristic by generalizing some of the results in [6] to crossed coproducts of group algebras and duals of group algebras—these are smash coproducts in which the algebra and coalgebra structures are twisted by cocycles. Again we find a good support variety theory and yet negative answers to all three questions above. Group cohomology features prominently in our methods, as it did in [6], exploiting our Theorem 5.2 that Hochschild cohomology of a twisted group algebra is isomorphic to group cohomology with coefficients in the adjoint module given by the twisted group algebra. Even so, the behavior of the representations is far removed from the usual behavior of those of finite groups or of finite group schemes.

In contrast to the results of Sections 4 and 5, we show in Theorem 6.2 that for quasitriangular Hopf algebras, the answer to question (2) above is yes. The answer to (1) is automatically yes since the tensor product of modules is commutative up to natural isomorphism. For (2), we give an elementary argument that applies more generally to any module of a Hopf algebra for which the tensor product of the module with its dual commutes up to isomorphism. Question (3) is open for quasitriangular Hopf algebras.

The known support variety theories needed to make sense of question (3) in general require some homological assumptions. We recall these assumptions and the definitions in Section 3. In Section 2 we define smash coproducts and crossed coproducts. In Theorem 2.11 we give a general description of tensor products and duals of modules for a crossed coproduct of a group algebra with the dual of a group algebra. In Section 4, we give general results on support variety theory for smash coproducts with duals of group algebras, and present our examples involving quantum elementary abelian groups. Support varieties for crossed coproducts of group algebras and duals of group algebras are in Section 5. In Section 6, we look at consequences of commutativity, up to isomorphism, of the tensor product of some modules.

We will work over an algebraically closed field k of arbitrary characteristic, restricting either to characteristic 0 or to positive characteristic for some classes of examples. All modules will be finite dimensional left modules unless otherwise indicated.

2. Smash coproduct and crossed coproduct

Smash coproduct. We recall the definition and principal representation theoretic properties of the smash coproduct Hopf algebras first defined by Molnar [16]. He defined them for any commutative Hopf algebra, and here we restrict to the Hopf algebras dual to group algebras.

Let A be a Hopf algebra over k. Let G be a finite group acting on A by Hopf algebra automorphisms. Let k^{G} be the Hopf algebra dual to the group algebra kG, so that k^{G} = Hom_{k}(kG, k) as a vector space. Letting {p_{x} : x ∈ G} denote the basis of k^{G} dual to G, multiplication is given by p_{x}p_{y} = δ_{x,y}p_{x} for all x, y ∈ G. In particular, the multiplicative identity of k^{G} is 1 = ∑_{x∈G} p_{x}. Comultiplication is defined by

Δ(p_{x}) = ∑_{y∈G} p_{y} ⊗ p_{y^{−1}x},

and counit and antipode by ε(p_{x}) = δ_{1,x} and S(p_{x}) = p_{x^{−1}} for all x ∈ G. The smash coproduct Hopf algebra [16] is denoted

K = A♮k^{G},

and given as follows. The algebra structure is the usual tensor product A ⊗ k^{G} of algebras. Let a♮p_{x} denote the element a ⊗ p_{x} in K, for each a ∈ A and x ∈ G. Comultiplication is defined by

(2.1)  Δ(a♮p_{x}) = ∑_{y∈G}(a_{1}♮p_{y}) ⊗ ((y^{−1} · a_{2})♮p_{y^{−1}x}),

for all x ∈ G, a ∈ A. The counit and antipode are defined by

ε(a♮p_{x}) = δ_{1,x}ε(a) and S(a♮p_{x}) = (x^{−1} · S(a))♮p_{x^{−1}}.

Since A and k^{G} are subalgebras of K, for simplicity we will sometimes write a and p_{x} in place of the elements a♮1 and 1♮p_{x} in K.

Examples of smash coproducts in [6] are of the form A♮k^{G} with A = kL, the group algebra of a finite group L. In Section 4, we will give some general results for representations of smash coproducts A♮k^{G}, and look at examples in which A is a tensor product of copies of a Taft algebra. We will need some representation theoretic results from [18], and we recall them here.

Note that the elements p_{x} are orthogonal central idempotents of K. Given a K-module M and x ∈ G, we will denote by M_{x} the K-submodule p_{x} · M of M, which we also view as an A-module by restriction of action to A. Then

(2.2)  M = ⨁_{x∈G} M_{x}.

In this way, K-modules are graded by the group G. We will refer to the component of a K-module associated to x ∈ G under this grading as the x-component of the module. For y ∈ G, let ^{y}M_{x} denote the conjugate K-module: This is M_{x} as a vector space in the x-component, with action of A given by a ·_{y} m = (y^{−1} · a) · m for all a ∈ A, m ∈ M. Denote by M^{∗} the dual of the K-module M, that is, M^{∗} = Hom_{k}(M, k) with module structure given by (b · f)(m) = f(S(b) · m) for all b ∈ K, f ∈ M^{∗}, m ∈ M.

The following theorem describes what happens to the G-grading when taking tensor products and duals of K-modules.

Theorem 2.3. [18, Theorem 7.1] Let M, N be A♮k^{G}-modules. For each x ∈ G, there are isomorphisms of A♮k^{G}-modules:

(i) (M ⊗ N)_{x} ≃ ⨁_{y,z∈G,yz=x} M_{y} ⊗ ^{y}N_{z}, and (ii) (M^{∗})_{x} ≃ ^{x}(M_{x^{−1}})^{∗}.

Crossed coproduct Hopf Algebra. We next consider a crossed coproduct that is analogous to a smash coproduct but with algebra and coalgebra structures twisted by cocycles. For our examples we are interested in the case A = kL, the group algebra of a finite group L, and we restrict to that case here for simplicity of presentation.

Let G be a finite group acting on L by automorphisms, and consider L to act on G trivially. There is a version of the construction for nontrivial actions of L on G; see, e.g., [1]. The representation theory of the resulting crossed coproduct is more complicated, and we do not consider it here. See, e.g., [14, 23] for the representation theory in a special case when the action of L on G is nontrivial, while the action of G on L is trivial.

Let (k^{G})^{×} denote the group of units of k^{G}. Let σ : L × L → (k^{G})^{×} be a normalized 2-cocycle, that is, a function for which

(2.4)  σ(l, m)σ(lm, n) = σ(m, n)σ(l, mn)

and σ(1, l) = σ(l, 1) = 1 for all l, m, n ∈ L. We write

σ(l, m) = ∑_{x∈G} σ_{x}(l, m)p_{x}

for functions σ_{x} : L × L → k^{×}. As the elements p_{x} are orthogonal idempotents, the functions σ_{x} are normalized 2-cocycles with values in k^{×}. It follows from Equation (2.4) with m = l^{−1} and n = l that

σ_{x}(l, l^{−1}) = σ_{x}(l^{−1}, l)

for all x ∈ G and l ∈ L.

Let τ : L → k^{G} ⊗ k^{G} be a normalized 2-cocycle, that is, a function determined by

τ(l) = ∑_{x,y∈G} τ_{x,y}(l)p_{x} ⊗ p_{y}

for τ_{x,y} : L → k^{×} satisfying

(2.5)  τ_{xy,z}(l)τ_{x,y}(l) = τ_{x,yz}(l)τ_{y,z}(x^{−1} · l)

and τ_{1,x}(l) = τ_{x,1}(l) = 1 for all x, y, z ∈ G, l ∈ L.

Assume additionally that

(2.6)  σ_{xy}(l, m)τ_{x,y}(lm) = σ_{x}(l, m)σ_{y}(x^{−1} · l, x^{−1} · m)τ_{x,y}(l)τ_{x,y}(m)

for all l, m ∈ L and x, y ∈ G. In particular, setting x = 1, this implies that σ_{1}(l, m) = 1, and setting l = 1, this implies that τ_{x,y}(1) = 1 for all l, m ∈ L and x, y ∈ G. It also follows from (2.6), setting y = x^{−1}, that

(2.7)  σ_{x}^{−1}(l, m) = σ_{x^{−1}}(x^{−1} · l, x^{−1} · m)τ_{x,x^{−1}}(l)τ_{x,x^{−1}}(m)τ_{x,x^{−1}}^{−1}(lm)

and setting m = l^{−1}, that

(2.8)  τ_{x,y}^{−1}(l) = σ_{xy}^{−1}(l, l^{−1})σ_{x}(l, l^{−1})σ_{y}(x^{−1} · l, x^{−1} · l^{−1})τ_{x,y}(l^{−1}).

We define the crossed coproduct Hopf algebra

K = kL♮_{σ}^{τ}k^{G}

as follows. As a vector space, it is kL ⊗ k^{G}. The product is twisted by σ:

(2.9)  (l♮p_{x})(m♮p_{y}) = lm♮σ(l, m)p_{x}p_{y} = δ_{x,y}σ_{x}(l, m)lm♮p_{x},

for l, m ∈ L and x, y ∈ G. This is the standard crossed product algebra structure [17, Chapter 7] (with trivial action of L on k^{G}). The coproduct is twisted by τ:

Δ(l♮p_{x}) = ∑_{y∈G} τ_{y,y^{−1}x}(l)(l♮p_{y}) ⊗ ((y^{−1} · l)♮p_{y^{−1}x})

for l ∈ L, x ∈ G. The counit is given by

ε(l♮p_{x}) = δ_{1,x},

and the antipode twisted by σ and τ is given by

S(l♮p_{x}) = τ_{x,x^{−1}}(l^{−1})σ_{x}(l^{−1}, l)(x^{−1} · l^{−1})♮p_{x^{−1}}

for l ∈ L, x ∈ G.

It may be checked directly that H is a Hopf algebra. Verification calls on assumption (2.6) as well as the cocycle defining properties (2.4) and (2.5) of σ and τ. This crossed coproduct Hopf algebra K = kL♮_{σ}^{τ}k^{G} is related to that given in [1, Proposition 3.1.12], but the coalgebra structure is slightly different. We choose this version since it generalizes Molnar’s smash coproduct as used in [6]. See also [15, Section 4] and [2] for earlier versions of [1, Proposition 3.1.12]. For comparison, in the language and notation of [1], A = k^{G}, B = kL, the weak coaction (dual to the action by automorphisms of G on L that we started with) is given by ρ(l) = ∑_{x∈G} x^{−1} · l ⊗ p_{x}, and the weak action ⇀ of L on G is the trivial action. Condition (3.1.9) in [1] holds in our case because the action of the group G on L is by automorphisms, and condition (3.1.11) is satisfied in our case due to condition (2.6).

We will see that twisted group algebras arise naturally within these Hopf algebras. In general, for α : L × L → k^{×} a 2-cocycle, the twisted group algebra k^{α}L is defined to be kL as a vector space. Multiplication is determined by \overline{l}·\overline{m} = α(l, m)\overline{lm} for all l, m ∈ L, where \overline{l} is the basis element of k^{α}L corresponding to l. Since α is a 2-cocycle, we find that

\overline{l^{−1}} = α(l^{−1}, l)(\overline{l})^{−1} = α(l, l^{−1})(\overline{l})^{−1}

for all l ∈ L.

In the Hopf algebra K, again the elements p_{x} (that is, 1♮p_{x}) are orthogonal central idempotents, and

K = kL♮_{σ}^{τ}k^{G} ≃ ⨁_{x∈G} kL · p_{x} ≃ ⨁_{x∈G} k^{σ_{x}}L,

a direct sum of ideals, where k^{σ_{x}}L is the twisted group algebra defined above. The subalgebra kL · p_{x} or kL ⊗ kp_{x} of K is indeed isomorphic to k^{σ_{x}}L since (l♮p_{x})(m♮p_{x}) = σ_{x}(l, m)lm♮p_{x} for all l, m ∈ L. Thus for a K-module M:

M ≃ ⨁_{x∈G} M_{x},

where each M_{x} = p_{x} · M is naturally a k^{σ_{x}}L-module.

In general, given a 2-cocycle α for L with coefficients in k^{×}, and an element y of G, we can define a new 2-cocycle ^{y}α for L by

yα(l, m) = α(y^{−1} · l, y^{−1} · m)

for all l, m ∈ L. We say that α is G-invariant if ^{y}α = α for all y ∈ G. For each x ∈ G, the group G acts on the vector space k^{σ_{x}}L: Define y · \overline{l} = \overline{y · l} for all y ∈ G and l ∈ L. A calculation shows that when σ_{x} is G-invariant, this action of G on k^{σ_{x}}L is by algebra automorphisms.

We will need to use the following operations on cocycles. Two 2-cocycles α, β : L × L → k^{×} may be multiplied:

(αβ)(l, m) = α(l, m)β(l, m)

for all l, m ∈ L. The product αβ is again a 2-cocycle. The 2-cocycle α has a multiplicative inverse given by

α^{−1}(l, m) = (α(l, m))^{−1}

for all l, m ∈ L.

Given a K-module M and y ∈ G, we define the conjugate module ^{y}M_{x} to be M_{x} as a vector space, with k^{^{y}σ_{x}}L-module structure given by \overline{l} ·_{y} v = (y^{−1} · \overline{l}) · v, for all l ∈ L, v ∈ M. If ^{y}σ_{x} ≠ σ_{x}, then ^{y}M_{x} is not a k^{σ_{x}}L-module.

Notice that (2.6) may be rewritten as

(2.10)  σ_{xy}(l, m) = σ_{x}(l, m)σ_{y}(x^{−1} · l, x^{−1} · m)τ_{x,y}(l)τ_{x,y}(m)τ_{x,y}^{−1}(lm),

or σ_{xy} = σ_{x}(^{x}σ_{y})β_{xy} where β_{xy} is the coboundary given by

β_{x,y}(l, m) = τ_{x,y}(l)τ_{x,y}(m)τ_{x,y}^{−1}(lm)

for all x, y ∈ G, l, m ∈ L. That is, σ_{xy} and σ_{x}(^{x}σ_{y}) are cohomologous. Then, by [13, Lemma 1.1], there is an isomorphism of twisted group algebras k^{σ_{xy}}L ⟶^{∼} k^{σ_{x}·(^{x}σ_{y})}L given by \overline{l} → τ_{x,y}(l)\overline{l}.

We next give a version of Theorem 2.3 in the context of crossed coproduct Hopf algebras. It is stated slightly differently as in this setting, the subalgebras k^{σ_{x}}L of K are not Hopf subalgebras, and the action of G on k^{σ_{x}}L is not necessarily an action by algebra automorphisms. In general for 2-cocycles α, β on L with values in k^{×}, the tensor product of a k^{α}L-module with a k^{β}L-module has the structure of a k^{αβ}L-module where \overline{l} in k^{αβ}L acts as \overline{l} ⊗ \overline{l} (the first tensor factor in k^{α}L, the second in k^{β}L) by [13, Lemma 5.1]. In general for a 2-cocycle α on L with values in k^{×}, the dual of a k^{α}L-module M has the structure of a k^{α^{−1}}L-module where \overline{l} in k^{α^{−1}}L acts as (\overline{l} · f)(m) = f((\overline{l})^{−1} · m) for all m ∈ M, f ∈ M^{∗} by [13, Theorem 4.6].

Theorem 2.11. Let M, N be kL♮_{σ}^{τ}k^{G}-modules. For each x ∈ G, there are isomorphisms of k^{σ_{x}}L-modules:

(i) (M ⊗ N)_{x} ≃ ⨁_{y,z∈G,yz=x} M_{y} ⊗ ^{y}N_{z}, and (ii) (M^{∗})_{x} ≃ ^{x}(M_{x^{−1}})^{∗}.

Proof. The proof is very similar to that of [18, Theorem 7.1]. We include details to show the effects of σ and τ.

We will prove statement (i) for modules of the form M = M_{y}, N = N_{z} for y, z ∈ G. In principle, M_{y} ⊗ ^{y}N_{z} is a k^{σ_{y}(^{y}σ_{z})}L-module. By equation (2.10), σ_{y}(^{y}σ_{z}) and σ_{yz} are cohomologous. Therefore k^{σ_{y}(^{y}σ_{z})}L and k^{σ_{yz}}L are isomorphic as explained above. This means that M_{y} ⊗^{y}N_{z} can also be regarded as a k^{σ_{x}}Lmodule for x = yz. We must prove that the k^{σ_{x}}L-module (M ⊗ N)_{x} is isomorphic to the tensor product of the k^{σ_{y}}L-module M_{y} and the k^{^{y}σ_{z}}L-module ^{y}N_{z} under this isomorphism of twisted group algebras. The target module M_{y} ⊗ ^{y}N_{z} is a K-module on which p_{yz} acts as the identity and p_{w} acts as 0 for w ≠ yz. We consider the action of l♮p_{x}, identified with \overline{l} in k^{σ_{x}}L, for l in L. By applying Δ to l♮p_{x}, we obtain

(l♮p_{x}) · (m ⊗ n) = δ_{x,yz}τ_{y,z}(l)(\overline{l} · m) ⊗ ((y^{−1} · \overline{l}) · n)

for all m ∈ M_{y} and n ∈ N_{z}. On the other hand, the action of \overline{l} in k^{σ_{y}(^{y}σ_{z})}L on the tensor product of the k^{σ_{y}}L-module M_{y} and k^{^{y}σ_{z}}L-module ^{y}N_{z} is given by

\overline{l} · (m ⊗ n) = (\overline{l} · m) ⊗ (\overline{l} ·_{y} n) = (\overline{l} · m) ⊗ ((y^{−1} · \overline{l}) · n).

Considering the isomorphism k^{σ_{x}}L ⟶^{∼} k^{σ_{y}(^{y}σ_{z})}L given by \overline{l} → τ_{y,z}(l)\overline{l}, the k^{σ_{x}}Lmodule structure on the vector space M_{y} ⊗^{y}N_{z} is precisely that arising from this isomorphism and the action of k^{σ_{y}(^{y}σ_{z})}L on this space.

To prove (ii), we assume M = M_{y} for some y ∈ G. We first show that the dual K-module M^{∗} satisfies M^{∗} = (M^{∗})_{y^{−1}}. Let x ∈ G, f ∈ M^{∗}, and m ∈ M. Then

((1♮p_{x}) · (f))(m) = f(S(1♮p_{x})m) = f((1♮p_{x^{−1}})m) = δ_{x^{−1},y}f(m).

So M^{∗} = (M_{y})^{∗} = (M^{∗})_{y^{−1}}.

Setting x = y^{−1}, we now want to determine the k^{σ_{x}}L-module structure on (M^{∗})_{x}. There is an isomorphism k^{σ_{x}}L ⟶^{∼} k^{^{x}σ_{y}^{−1}}L, given by \overline{l} → τ_{x,y}^{−1}(l)\overline{l}, by (2.7). Now M_{y} is a k^{σ_{y}}L-module, so (M_{y})^{∗} has the structure of a k^{σ_{y}^{−1}}L-module and x(M_{y})^{∗} has the structure of a k^{^{x}σ_{y}^{−1}}L-module. Recall that if \overline{l} ∈ k^{σ_{z}}L, then \overline{l^{−1}} = σ_{z}(l^{−1}, l)\overline{l}^{−1}. Then, for m ∈ M_{y} and f ∈ M_{y}^{∗},

((l♮p_{y^{−1}}) · f)(m) = f(S(l♮p_{y^{−1}})m)
= f(τ_{y^{−1},y}(l^{−1})σ_{y^{−1}}(l^{−1}, l)((y · l^{−1})♮p_{y})m)
= τ_{y^{−1},y}(l^{−1})σ_{y^{−1}}(l^{−1}, l)f (\overline{(y · l^{−1})}m)
= τ_{y^{−1},y}(l^{−1})σ_{y^{−1}}(l^{−1}, l)σ_{y}(y · l^{−1}, y · l)f (\overline{(y · l)}^{−1}m)
= τ_{y^{−1},y}^{−1}(l)f (\overline{(y · l)}^{−1}m)
= τ_{y^{−1},y}^{−1}(l)(\overline{(y · l)} · f)(m)
= τ_{y^{−1},y}^{−1}(l)((y · \overline{l}) · f)(m),

by (2.8) with x = y^{−1}, where the overline notation is used first for elements in k^{σ_{y}}L, and in the last two lines for elements in k^{σ_{y}^{−1}}L. The action of y on \overline{l} in the last line indicates that M_{y}^{∗} is a k^{^{x}σ_{y}^{−1}}L-module. Considering the isomorphism k^{σ_{x}}L ⟶^{∼} k^{^{x}σ_{y}^{−1}}L given by \overline{l} ↦ τ_{x,y}^{−1}(l)\overline{l}, with x = y^{−1}, the k^{σ_{x}}L-module structure on (M^{∗})_{x} is as claimed. □

3. Support varieties

We will recall here some of the definitions and results given in [7, 22] on support varieties based on Hochschild cohomology, particularly for self-injective algebras. We will apply this theory to finite dimensional Hopf algebras, which are Frobenius algebras, and therefore self-injective. There is a parallel support variety theory using Hopf algebra cohomology (that is, cohomology of an augmented algebra) instead of Hochschild cohomology, a direct generalization of support varieties for finite groups based on group cohomology. Consequences for the representation theory of Hopf algebras are the same in either theory, although the tensor product property (3.2) below may not be. We choose the Hochschild cohomology approach for our examples here. These support varieties contain some useful extra information, namely the G-components for modules graded by the group G as in (2.2).

Let A be a finite dimensional self-injective algebra over k, and let A^{e} = A ⊗ A^{op} (the enveloping algebra of A), that is the tensor product algebra of A with its opposite A^{op}. The Hochschild cohomology ring HH^{}(A) is isomorphic to Ext_{A^{e}}(A, A) since k is a field. Moreover, for an A-bimodule M, the Hochschild cohomology space HH^{}(A, M) is isomorphic to Ext_{A^{e}}(A, M).

For any left A-module M, there is a graded ring homomorphism from HH^{}(A) to Ext_{A}(M, M) given by −⊗_{A} M. That is, for each n, view an element of HH^{n}(A) as an equivalence class of n-extensions of A by A, and apply −⊗_{A} M. Followed by Yoneda composition with generalized extensions of N by M, this homomorphism induces an action of HH^{}(A) on Ext^{}(M, N) for any left A-module N.

In order to define support varieties, we make assumptions (fg1) and (fg2) below. Let Jac(A) denote the Jacobson radical of A.

Assumption (fg1): There is a graded subalgebra H_{A} of HH^{}(A) such that H_{A} is a finitely generated commutative algebra and H_{A}^{0} = HH^{0}(A).

For any two left A-modules M, N, we view Ext_{A}(M, N) as an H_{A}-module via restriction of the action of HH^{}(A) described above.

Assumption (fg2): Ext_{A}(A/ Jac(A), A/ Jac(A)) is a finitely generated H_{A}-module.

The assumption (fg2) is equivalent to either of the following two statements by [7, Proposition 2.4]:

(i) For all finite dimensional A-modules M and N, Ext_{A}(M, N) is finitely generated over H_{A}.

(ii) For all finite dimensional A-bimodules M, HH^{}(A, M) is finitely generated

over H_{A}.

Further, the above statement (i) is equivalent to the corresponding statement in which M = N, since we may apply the latter to the direct sum of two finite dimensional modules to obtain (i).

Many finite dimensional Hopf algebras A are known to satisfy (fg1) and (fg2) when H_{A} is the subalgebra of HH^{}(A) generated by Ext_{A}(k, k) and HH^{0}(A): For this choice, we view Ext_{A}(k, k) as a subalgebra of HH^{}(A) via an embedding as in [12, 19]. For example, group algebras and more generally cocommutative Hopf algebras [11] and quantum elementary abelian groups [19] satisfy (fg1) and (fg2).

It is an open question, related to a conjecture of Etingof and Ostrik [8] about finite generation of Ext_{A}(k, k), as to whether all finite dimensional Hopf algebras satisfy these assumptions.

Assume (fg1) and (fg2) hold. Let

V_{H_{A}} = Max(H_{A}),

the maximal ideal spectrum of the finitely generated commutative algebra H_{A}. We will also use the notation V_{A} when it is clear from context which algebra H_{A} is involved.

Let M be a finite dimensional A-module. Let I_{H_{A}}(M) be the annihilator of the action of H_{A} on Ext_{A}(M, M). The action may be taken to be either the left or right action; their annihilators coincide according to [22, Lemma 2.1]. Then I_{H_{A}}(M) is a homogeneous ideal of H_{A}. The support variety of M is

V_{A}(M) = Max(H_{A}/I_{H_{A}}(M)),

the maximal ideal spectrum of H_{A}/I_{H_{A}}(M), as in [22, Definition 2.3]. We identify V_{A}(M) with the subset of V_{A} consisting of ideals containing I_{H_{A}}(M).

An action of a group G on A by automorphisms induces an action on Hochschild cohomology HH^{}(A) via the tensor product action on the components of the bar resolution of A. It also induces an action on the category A-mod of finite dimensional A-modules. If the action of G on HH^{}(A) preserves H_{A}, then G also acts on subvarieties of V_{A} in such a way that V_{A}(^{g}M) = ^{g}V_{A}(M) for all finite dimensional A-modules M and group elements g ∈ G.

We will use the following result from [7, 22]. Part (i) follows from [7, Theorem 2.5(c)], which states more generally that the dimension of the support variety is the complexity of the module (that is, the rate of growth of a minimal projective resolution). Part (ii) is [22, Proposition 3.4(f)].

Theorem 3.1. [7, 22] Let A be a finite dimensional self-injective algebra satisfying (fg1) and (fg2), and let M and N be finite dimensional A-modules. Then

(i) dim(V_{A}(M)) = 0 if and only if M is projective.

(ii) V_{A}(M ⊕ N) = V_{A}(M) ∪ V_{A}(N).

Now let A be a finite dimensional Hopf algebra that satisfies (fg1) and (fg2). We say that A has the tensor product property with respect to the choice of subalgebra

H_{A} of HH^{}(A) if

(3.2)  V_{A}(M ⊗ N) = V_{A}(M) ∩ V_{A}(N)

for all finite dimensional A-modules M and N. For example, the tensor product property holds if A is the group algebra of a finite p-group G over a field k of characteristic p and H_{A} is the subalgebra of HH^{}(A) generated by group cohomology Ext_{kG}(k, k) and HH^{0}(kG). (For this, we use a standard embedding of group cohomology into Hochschild cohomology of the group algebra; see, e.g., Corollary 5.3 below, taking the cocycle α to be trivial.) For finite groups in general, the tensor product property holds for the classical choice of support variety theory based on group cohomology instead. (If G is not a p-group, HH^{0}(kG) is no longer a local ring, and its primitive central idempotents contribute to the support varieties in the Hochschild cohomology version.) Cocommutative Hopf algebras generally [10, Proposition 3.2] have the tensor product property for support varieties defined in terms of their cohomology as augmented algebras. Quantum elementary abelian groups [20, Theorem 1.1] have the tensor product property for either version of support varieties (these algebras have trivial centers).

4. Properties of modules for smash coproducts

Let K = A♮k^{G} be a smash coproduct arising from a finite dimensional Hopf algebra A with an action of a finite group G by Hopf algebra automorphisms, as described in Section 2. In this section, we study homological and representation theoretic properties of K.

Smash coproducts and finite generation assumptions. We will assume that A satisfies the conditions (fg1) and (fg2) of Section 3. The following theorem shows that the smash coproduct K then satisfies these two conditions as well, with an appropriate choice of H_{K}.

Theorem 4.1. Let A be a finite dimensional Hopf algebra with an action of a finite group G by Hopf algebra automorphisms. Assume that A satisfies conditions (fg1) and (fg2) from Section 3. Then the smash coproduct K = A♮k^{G} also satisfies conditions (fg1) and (fg2).

Proof. Given that A satisfies condition (fg1), there exists a graded subalgebra H_{A} of HH^{}(A) such that H_{A} is a finitely generated commutative algebra and H_{A}^{0} = HH^{0}(A). As an algebra, K is the tensor product algebra A ⊗ k^{G}, and so

HH^{}(K) = HH^{}(A ⊗ k^{G}) ≃ HH^{}(A) ⊗ HH^{}(k^{G}) ≃ HH^{}(A) ⊗ k^{G}.

The last isomorphism follows from the semisimplicity and commutativity of k^{G}; the tensor factor k^{G} is concentrated in homological degree 0. Consider the subalgebra

H_{K} = H_{A} ⊗ k^{G}

of HH^{}(A)⊗k^{G} ≃ HH^{}(K). By its definition, H_{K} is a graded subalgebra of HH^{}(K) and H_{K}^{0} = H_{A}^{0} ⊗ k^{G} coincides with HH^{0}(A) ⊗ k^{G} = HH^{0}(K). In addition, H_{K} is a finitely generated commutative algebra since both H_{A} and k^{G} are. In this way, K satisfies (fg1) via H_{K}.

To check that K also satisfies (fg2), let M and N be finite dimensional Kmodules. Recall that M = ⊕_{x∈G}M_{x} and N = ⊕_{x∈G}N_{x} where M_{x} = p_{x} · M and N_{x} = p_{x} · N for each x ∈ G. Since {p_{x} : x ∈ G} is a set of orthogonal central idempotents in K and Kp_{x} ≃ A for each x, it also follows that

Ext_{K}(M, N) ≃ ⨁_{x∈G} Ext_{K}(M_{x}, M_{x}) ≃ ⨁_{x∈G} Ext_{A}(M_{x}, N_{x}),

and the action of HH^{}(K) on Ext_{K}(M_{x}, N_{x}) corresponds to the action of HH^{}(A). By hypothesis, for each x ∈ G, Ext_{A}(M_{x}, N_{x}) is finitely generated as an H_{A}- module, and since H_{A}⊗kp_{x} ⊂ H_{K}, Ext_{K}(M, N) is finitely generated over H_{K}. □

Since the smash coproduct K = A♮k^{G} satisfies (fg1) and (fg2) once A does, we may define support varieties for K-modules, as in Section 3, for some classes of Hopf algebras A.

Smash coproducts and the tensor product property. The following theorem is a consequence of Theorem 3.1(ii). As in the proof of Theorem 4.1, we choose H_{K} = H_{A} ⊗ k^{G}. Then V_{K}, the maximal ideal spectrum of H_{K}, may be identified with V_{A} × G, whose elements we denote by I × x for I a maximal ideal of H_{A} and x ∈ G. For simplicity of language we do not mention the choices H_{A}, H_{K} explicitly in theorem statements.

Theorem 4.2. Let A be a finite dimensional Hopf algebra with an action of a finite group G by Hopf algebra automorphisms. Assume that A satisfies (fg1) and (fg2) and has the tensor product property (3.2). Let M and N be finite dimensional A♮k^{G}-modules. Then

(i) V_{K}(M) = ⋃_{x∈G} V_{A}(M_{x}) × x.

(ii) V_{A}((M ⊗ N)_{x}) = ⋃_{y,z∈G, yz=x} V_{A}(M_{y}) ⋂ ^{y}(V_{A}(N_{z})).

Proof. To prove statement (i), note that the variety of a direct sum is the union of the varieties, by Theorem 3.1(ii). Applying this result to M = ⊕_{x∈G}M_{x}, we have V_{K}(M) = ⋃_{x∈G} V_{K}(M_{x}) = ⋃_{x∈G} V_{A}(M_{x}) × x.

Since A has the tensor product property, statement (ii) is a consequence of Theorem 2.3(i) and Theorem 3.1(ii):

V_{A}((M ⊗ N)_{x}) = ⋃_{y,z∈G, yz=x} V_{A}(M_{y} ⊗ ^{y}N_{z})
= ⨁_{y,z∈G, yz=x} V_{A}(M_{y}) ∩ V_{A}(^{y}N_{z})
= ⨁_{y,z∈G, yz=x} V_{A}(M_{y}) ∩ (^{y}V_{A}(N_{z})).
□

We next show that for any finite dimensional nonsemisimple Hopf algebra A having the tensor product property, there is a Hopf algebra containing A as a subalgebra that does not. There are many such Hopf algebras, but we will take a smash coproduct K = (A ⊗ A)♮k^{ℤ_{2}} where ℤ_{2} is the group of order two in which the nonidentity element interchanges the tensor factors.

Theorem 4.3. Let A be a finite dimensional nonsemisimple Hopf algebra satisfying (fg1) and (fg2) and the tensor product property (3.2). Let K = (A ⊗ A)♮k^{ℤ_{2}}. Then K does not have the tensor product property. Moreover, there are nonprojective K-modules M and N such that M ⊗ M and M ⊗ N are projective, and N ⊗ M is not projective.

The statement and proof of the theorem is based on choices H_{A} and H_{K} discussed above. However the last statement in the theorem implies that there is no choice of H_{K} with respect to which K has the tensor product property. Even further, it implies that replacing these support varieties with those defined via Hopf algebra cohomology Ext_{K}(k, k), the tensor product property still will not hold for K, due to Theorem 3.1(i).

Proof. Note that HH^{}(A ⊗ A) ≃ HH^{}(A) ⊗ HH^{}(A). We choose H_{A⊗A} = H_{A} ⊗ H_{A}, which satisfies (fg1) and (fg2) for the Hopf algebra A ⊗ A: Condition (fg1) holds since H_{A⊗A} is the tensor product of two copies of H_{A}. For (fg2), note that

Ext_{A⊗A}((A ⊗ A)/ Jac(A ⊗ A), (A ⊗ A)/ Jac(A ⊗ A))
≃ Ext_{A}(A/ Jac(A), A/ Jac(A)) ⊗ Ext_{A}(A/ Jac(A), A/ Jac(A)).

Since A is not semisimple, there exists a nonprojective A-module U, and so by Theorem 3.1(i), dim(V_{A}(U)) ≠ 0. We will be interested in the two nonprojective A ⊗ A-modules given by U ⊗ A and A ⊗ U, viewing A as the left regular Amodule. Note that Ext_{A⊗A}(U ⊗ A, U ⊗ A) ≃ Ext_{A}(U, U), since A is projective as an A-module, where the subscript A is understood to be identified with A⊗k as a subalgebra of A⊗A. Similarly Ext_{A⊗A}(A⊗U, A⊗U) ≃ Ext_{A}(U, U), only this time the subscript A is identified with k⊗A. Then dim(V_{A⊗A}(U⊗A)∩V_{A⊗A}(A⊗U)) = 0.

Let h be the nonidentity element of ℤ_{2}, and let M = (U ⊗ A) ⊗ kp_{h} as a Kmodule. Then M is nonprojective since U ⊗ A is a nonprojective A-module. We have dim(V_{K}(M)) = dim(V_{A⊗A}(U ⊗ A) × h) ≠ 0. Note that ^{h}(U ⊗ A) ≃ A ⊗ U since h interchanges the two factors of A in the tensor product A ⊗ A. Thus by Theorem 2.3(i),

M ⊗ M ≃ ((U ⊗ A) ⊗ (A ⊗ U)) ⊗ kp_{1}.

It follows from Theorem 4.2(i) and (ii) that dim(V_{K}(M ⊗ M)) = 0. By Theorem 3.1(i), M ⊗ M is projective. On the other hand, dim(V_{K}(M) ∩ V_{K}(M)) ≠ 0 since M is not projective. Therefore K does not have the tensor product property. Consider the K-module N = (U ⊗A)⊗kp_{1}, also nonprojective. By Theorem 2.3(i),

M ⊗ N ≃ ((U ⊗ A) ⊗ (A ⊗ U)) ⊗ kp_{h},

and

N ⊗ M ≃ ((U ⊗ A) ⊗ (U ⊗ A)) ⊗ kp_{h}.

Then, dim(V_{K}(M ⊗ N)) = 0 and dim(V_{K}(N ⊗ M)) = dim(V_{A}(U ⊗ A)) ≠ 0 by Theorem 4.2(i) and (ii) and since U ⊗ A is a nonprojective A-module. This finishes the proof, since projectivity is determined by the dimension of the variety by Theorem 3.1(i). □

Next we show that A embeds into K as a subalgebra. Examination of the coproduct (2.1) shows that A is not a Hopf subalgebra.

Corollary 4.4. Let A be a finite dimensional nonsemisimple Hopf algebra satisfying (fg1) and (fg2) and the tensor product property. Then A can be embedded as a subalgebra of a Hopf algebra that does not have the tensor product property.

Proof. Let K = (A ⊗ A)♮k^{ℤ_{2}} as in the theorem. Then (A ⊗ k)♮k and (k ⊗ A)♮k are subalgebras of K that are isomorphic to A. □

Next we will give some specific examples of varieties and of further constructions involving projective and nonprojective modules.

Smash coproducts with quantum elementary abelian groups. Let k have characteristic 0. Let n ≥ 2 be a positive integer, and let q be a primitive nth root of 1 in k. The Taft algebra T_{n} is the algebra generated by symbols g and x with relations x^{n} = 0, g^{n} = 1 and gx = qxg. It is a Hopf algebra with comultiplication given by Δ(g) = g ⊗ g and Δ(x) = 1 ⊗ x + x ⊗ g, counit ε(g) = 1 and ε(x) = 0, and antipode S(g) = g^{−1} and S(x) = −xg^{−1}.

A quantum elementary abelian group is a tensor product A of m copies of the Taft algebra T_{n}, for some positive integer m. It is isomorphic to a skew group algebra Λ ⋊ G, where the group G ≃ (ℤ_{n})^{m} is elementary abelian with generators g_{1}, …, g_{m}, and Λ = k[x_{1}, x_{2}, …, x_{m}]/(x_{1}^{n}, x_{2}^{n}, …, x_{m}^{n}), see [19]. In this way, we view A as a Radford biproduct (or bosonization of a Nichols algebra), see [9, Exercise 8.25.12].

The cohomology of the quantum elementary abelian group A is

H^{}(A, k) = Ext_{A}(k, k) ≃ Ext_{Λ}(k, k)^{G} ≃ k[y_{1}, …, y_{m}]

as a graded algebra, where deg(y_{i}) = 2. This was shown explicitly for the case m = 1 for example in [19, Section 4] and the general case follows by applying the Künneth formula. Let H_{A} = H^{}(A, k). Condition (fg1) is satisfied; note that the center of a Taft algebra is just all scalar multiples of 1. Condition (fg2) holds as well: This is a direct consequence of the isomorphism

Ext_{A}(M, N) ≃ Ext_{A}(k, N ⊗ M^{∗})

and [19, Lemma 7]. It was shown in [20, Theorem 1.1] that A has the tensor product property.

In [19], the authors study support varieties and rank varieties for A-modules, and show that the support variety of a finite dimensional A-module is homeomorphic to its rank variety. The rank variety of a finite dimensional A-module U is defined as follows. For each m-tuple of scalars (λ_{1}, …, λ_{m}) ∈ k^{m}, let

τ(λ_{1}, …, λ_{m}) = λ_{1}x_{1} + λ_{2}x_{2}g_{1} + ⋯ + λ_{m}g_{1}g_{2} ⋯ g_{m−1}.

It can be shown that τ(λ_{1}, …, λ_{m})^{n} = 0 in A, and that τ(λ_{1}, …, λ_{n}) generates a subalgebra k⟨τ(λ_{1}, …, λ_{m})⟩ of A isomorphic to k[t]/(t^{n}). Define the rank variety of an A-module U to be

(4.5) V_{A}^{r}(U) = {(λ_{1}, …, λ_{m}) ∈ k^{m} : U ↓_{k⟨τ(λ_{1},...,λ_{m})⟩} is not projective}/G,

where the quotient indicates the orbit space under the action of G = ⟨g_{1}, …, g_{m}⟩ on k^{m} by g_{i} · e_{j} = q^{δ_{ij}}e_{j} (the e_{j} are the standard basis vectors). The downarrow notation denotes restriction of a module to a subalgebra. Compare to [19, Definition 3.2], in which the rank variety is defined instead as the corresponding projective variety. We have chosen here to define it as an affine variety instead.

Next we give some examples where the rank variety, which is homeomorphic to the support variety, can be determined fairly quickly.

Example 4.6. We will look more closely at the specific example A = H_{4} ⊗ H_{4}, the tensor product of two copies of the Sweedler Hopf algebra H_{4} = T_{2}. Let G = ℤ_{2} = ⟨h⟩, acting on A by interchanging the generators of the first and second copies of H_{4} in A, that is, h · g_{1} = g_{2}, h · g_{2} = g_{1}, h · x_{1} = x_{2}, and h · x_{2} = x_{1}. Let K = A♮k^{G}.

Consider the A-module U = \overline{A} given by the quotient of the left regular module A by the ideal generated by x_{2} and g_{2} − 1. Then U ≃ H_{4} ⊗ k, the first copy of the Sweedler algebra in A.

We will find the rank variety (4.5) of the A-module U. Recall that the rank variety is homeomorphic to the support variety. The restricted module U ↓_{k⟨x_{1}⟩} is isomorphic to the direct sum of two copies of the right regular module k⟨x_{1}⟩, indexed by 1 and g_{1}, since this is the action of x_{1} on the regular module H_{4}. Therefore U ↓_{k⟨x_{1}⟩} is projective as a k⟨x_{1}⟩-module. In this case, we are letting (λ_{1}, λ_{2}) = (1, 0) in the notation described above. Similarly, whenever λ_{1} ≠ 0, the restricted module U ↓_{k⟨λ_{1}x_{1}+λ_{2}x_{2}⟩} is projective since it is isomorphic to the left regular k⟨λ_{1}x_{1} + λ_{2}x_{2}⟩-module generated by 1 ⊗ 1.

Now, we consider (λ_{1}, λ_{2}) = (0, 1). The restricted module U ↓_{k⟨x_{2}g_{1}⟩} is a trivial module. So U ↓_{k⟨x_{2}g_{1}⟩} is not projective. The rank variety of U is thus

V_{A}^{r}(U) = {(0, λ_{2}) : λ_{2} ∈ k}/G.

The conjugate module ^{h}U, on the other hand, is trivial on restriction to k⟨x_{1}⟩ and free on restriction to k⟨x_{2}g_{1}⟩, and so

V_{A}^{r}(^{h}U) = {(λ_{1}, 0) : λ_{1} ∈ k}/G.

We will next use the A-module U and the trivial A-module k to construct some specific K-modules and study their properties.

• Consider the K-modules M = U ⊗ kp_{1} and N = k ⊗ kp_{h}. We want to compare M ⊗ N with N ⊗ M. First notice that V_{K}(M) ⋂ V_{K}(N) = 0, since M is concentrated on the 1-component while N is concentrated on the h-component.

On the other hand, by Theorem 4.2, V_{K}(M ⊗ N) = V_{A}(U) × h ≠ 0 and V_{K}(N ⊗M) = ^{h}(V_{A}(U))×h ≠ 0. So neither V_{K}(M ⊗N) nor V_{K}(N ⊗M) is contained in V_{K}(M) ⋂ V_{K}(N) = 0. As a consequence, K does not inherit the tensor product property from A. (See the proof of Theorem 4.3 for a different example illustrating this phenomenon.)

Moreover, since V_{A}(U) ≠ ^{h}(V_{A}(U)), we see that

V_{K}(M ⊗ N) ≠ V_{K}(N ⊗ M).

Then, the order of the tensor product matters in computing the varieties.

• Consider the K-module M = U ⊗ kp_{h}. We claim V_{K}(M) ≠ V_{K}(M^{∗}): In [20], to prove the tensor product property for A, the authors use the fact that V_{A}(U) = V_{A}(U^{∗}), for every finite dimensional A-module U. But this is not true for K. For example, since h^{2} = 1, it follows from Theorem 2.3(ii) and this fact about varieties of A-modules that V_{K}(M^{∗}) = V_{A}(^{h}U^{∗}) × h = hV_{A}(U^{∗}) × h = ^{h}V_{A}(U) × h while V_{K}(M) = V_{A}(U) × h, and these two varieties are different.

• Consider the K-modules M = U ⊗ kp_{h} and N = U ⊗ kp_{1}. We will show

that M ⊗ N is projective while N ⊗ M is not.

Since V_{K}(M ⊗ N) = V_{A}(U ⊗ ^{h}U) × h and A has the tensor product property, it follows that V_{K}(M ⊗ N) = 0. Then M ⊗ N is a projective K-module by Theorem 3.1(i).

On the other hand, N ⊗ M is not projective since

V_{K}(N ⊗ M) = V_{A}(U ⊗ U) × h = V_{A}(U) × h,

and the dimension of this variety is not 0.

• Consider the K-module M = U ⊗ kp_{h}. We will show that M ⊗ M is projective while M is not: V_{K}(M ⊗ M) = V_{A}(U ⊗^{h} U) × 1 = 0 since A has the tensor product property. Then M ⊗ M is a projective K-module as a consequence of Theorem 3.1(i).

Example 4.7. Let A = H_{4}^{⊗m}, that is, m copies of the Sweedler Hopf algebra H_{4}. Let G = ℤ_{m} = ⟨h⟩, with h acting on A by cyclically permuting the tensor factors. Let U = H_{4} ⊗ k ⊗ ⋯ ⊗ k, so that the first copy of H_{4} in A acts on U as on the left regular module, while each of the others acts as on a trivial module. Let M = U ⊗ kp_{h}. Then by reasoning similar to that in Example 4.6,

V_{k}(M^{⊗m}) = V_{A}(U ⊗^{h}U ⊗ ⋯ ⊗^{h^{m−1}}U) × h^{m} = ( ⋂_{i=0}^{m−1}^{h^{i}}(V_{A}(U))) × h^{m} = 0,

while dim(V_{K}(M^{⊗(m−1)})) ≠ 0. Therefore M^{⊗m} is projective and M^{⊗(m−1)} is not projective.

5. Properties of modules for crossed coproducts

Let k have positive characteristic p. In this section we consider modules for K = kL♮_{σ}^{τ}k^{G}, the crossed coproduct algebra defined in Section 2, when the group L has order divisible by p. We showed in Section 2 that as an algebra, K ≃ ⊕_{x∈G}k^{σ_{x}}L, a direct sum of the twisted group algebras k^{σ_{x}}L. We first find a description of the Hochschild cohomology of twisted group algebras.

Hochschild cohomology of twisted group algebras. Let α be a normalized 2-cocycle on the group L with values in k^{×}.

First note that L acts on the twisted group algebra k^{α}L by automorphisms:

l(\overline{m}) = \overline{l} · \overline{m} · (\overline{l})^{−1} = α^{−1}(l^{−1}, l)\overline{l} · \overline{m} · \overline{l^{−1}}

for all l, m ∈ L. We call this the adjoint action and denote by (k^{α}L)^{ad} the corresponding kL-module. One may check, using the cocycle condition, that this is indeed an action. A similar calculation shows that there is an injective algebra homomorphism from the group algebra kL to the enveloping algebra (k^{α}L)^{e} of the twisted group algebra k^{α}L, given by

δ : kL ⟶ k^{α}L ⊗ (k^{α}L)^{op},
l ↦ \overline{l} ⊗ (\overline{l})^{−1},

for all l ∈ L. Identify kL with its image δ(kL) in (k^{α}L)^{e}. In the following statement, the uparrow denotes tensor induction, that is, M ↑_{A}^{B}= B ⊗_{A} M for all A-modules M, where A is a subalgebra of B. The action of B on M ↑_{A}^{B} is by multiplication on the left tensor factor.

Lemma 5.1. As a (k^{α}L)^{e}-module, k^{α}L ≃ k ↑_{kL}^{(k^{α}L)^{e}}.

Proof. Let f : k^{α}L → k ↑_{kL}^{(k^{α}L)^{e}}= (k^{α}L)^{e} ⊗_{δ(kL)} k be the k-linear function defined by f(\overline{l}) = (\overline{l} ⊗ \overline{1}) ⊗_{δ(kL)} 1 for all l ∈ L. Clearly f is bijective. We check that it is a (k^{α}L)^{e}-module homomorphism: Let l, m, n ∈ L. Then

f((\overline{m} ⊗ \overline{n}) · \overline{l}) = f(\overline{m} · \overline{l} · \overline{n})
= α(m, l)α(ml, n)f(\overline{mln})
= α(m, l)α(ml, n)(\overline{mln} ⊗ \overline{1}) ⊗_{δ(kL)} 1,

and on the other hand, since the tensor product is taken over δ(kL),

(\overline{m} ⊗ \overline{n}) · f(\overline{l}) = (\overline{m} ⊗ \overline{n}) · (\overline{l} ⊗ \overline{1}) ⊗_{δ(kL)} 1
= α(m, l)(\overline{ml} ⊗ \overline{n}) ⊗_{δ(kL)} 1
= α(m, l)(\overline{ml} ⊗ \overline{n})(\overline{n} ⊗ (\overline{n})^{−1}) ⊗_{δ(kL)} 1
= α(m, l)α(ml, n)(\overline{mln} ⊗ \overline{1}) ⊗_{δ(kL)} 1.
□

As a consequence of Lemma 5.1, we may apply the Eckmann-Shapiro Lemma [5, Corollary 2.8.4] to obtain a result on Hochschild cohomology next. For a kLmodule M, the group cohomology of L with coefficients in M is

H^{}(L, M) = Ext_{kL}(k, M).

If M is in addition an algebra and the action of L is by algebra automorphisms, then H^{}(L, M) is an algebra under cup product (corresponding to tensor product of generalized extensions) followed by the multiplication map M ⊗ M → M. In this way, H^{}(L, (k^{α}L)^{ad}) is itself an algebra. We show that this is none other than the Hochschild cohomology of the twisted group algebra k^{α}L, generalizing the well known result for the group algebra kL (see, e.g., [21, Proposition 3.1]).

Theorem 5.2. Let L be a finite group and let α : L × L → k^{×} be a 2-cocycle. There is an isomorphism of algebras, HH^{}(k^{α}L) ≃ H^{}(L, (k^{α}L)^{ad}).

Proof. The enveloping algebra (k^{α}L)^{e} of k^{α}L is free as a right kL-module where kL acts via the embedding δ: Elements of the form 1 ⊗ \overline{l} (l ∈ L) constitute a set of free generators.

Since (k^{α}L)^{e} is flat as a kL-module, we may apply the Eckmann-Shapiro Lemma in combination with Lemma 5.1 to obtain an isomorphism of graded vector spaces,

HH^{}(k^{α}L) = Ext_{(k^{α}L)^{e}}(k^{α}L, k^{α}L)
≃ Ext_{(k^{α}L)^{e}}(k ↑_{kL}^{(k^{α}L)^{e}}, k^{α}L)
≃ Ext_{kL}(k, k^{α}L).

Note that the corresponding action of kL on k^{α}L is precisely the adjoint action, since kL embeds into (k^{α}L)^{e} as δ(kL). Finally, one may check that this is in fact an isomorphism of algebras, using essentially the same proof as [21, Proposition 3.1].

□

Corollary 5.3. Let L be a finite group and let α : L × L → k^{×} be a 2-cocycle. The group cohomology ring H^{}(L, k) embeds as a subalgebra of the Hochschild cohomology ring HH^{}(k^{α}L) of the twisted group algebra k^{α}L.

Proof. By Theorem 5.2, HH^{}(k^{α}L) ≃ H^{}(L, (k^{α}L)^{ad}). As a kL-module, (k^{α}L)^{ad} decomposes into a direct sum of submodules indexed by conjugacy classes of L. In particular, the one-dimensional direct summand k · \overline{1} of (k^{α}L)^{ad} is a trivial kLmodule. The corresponding direct summand Ext_{kL}(k, k · \overline{1}) of Ext_{kL}(k, (k^{α}L)^{ad}) is a subalgebra under cup product, since k · \overline{1} is a subalgebra of k^{α}L. This summand is precisely the group cohomology H^{}(L, k) = Ext_{kL}(k, k). □

Crossed coproducts and finite generation assumptions. We now show that the crossed coproducts kL♮_{σ}^{τ}k^{G} satisfy our finite generation assumptions.

Theorem 5.4. Let K = kL♮_{σ}^{τ}k^{G} be a crossed coproduct Hopf algebra as defined in Section 2. Then K satisfies conditions (fg1) and (fg2) of Section 3.

Proof. As conditions (fg1) and (fg2) depend only on the algebra structure of K, the theorem statement does not depend on τ. We view K as a direct sum of the twisted group algebras k^{σ_{x}}L, where x ranges over G. Choose

(5.5)  H_{K} = H^{ev}(L, k) · HH^{0}(K),

where the superscript ev indicates that we take only the evenly graded part if the characteristic of k is not 2 (to ensure commutativity). Then H_{K}^{0} = HH^{0}(K) by design. The subalgebra H^{ev}(L, k) is finitely generated (see, e.g., [4, Corollary 4.2.2]), and since HH^{0}(K) is finite dimensional, we see that H_{K} is finitely generated, and so (fg1) holds.

Next we explain that HH^{}(K, M) is finitely generated over H_{K} for all finite dimensional K-bimodules M. It suffices to show this for bimodules of the form M = _{x}M_{x} = p_{x}·M·p_{x}, as these are the bimodules that contribute to the Hochschild cohomology HH^{}(K, M). By Lemma 5.1 and the Eckmann-Shapiro Lemma, for such a bimodule M,

HH^{}(K, M) = Ext_{(k^{σx}L)^{e}}(k^{σ_{x}}L, M) ≃ Ext_{L}(k, M^{ad}).

Now consider the action of H^{}(L, k) on HH^{}(K, M) induced by the inclusion of H^{}(L, k) into HH^{}(K) given by Corollary 5.3. We claim that this action corresponds to that of H^{}(L, k) on H^{}(L, M^{ad}) = Ext_{L}(k, M^{ad}) by cup product. Again this is essentially the same proof as [21, Proposition 3.1]. Since H^{}(L, M^{ad}) is finitely generated over H^{}(L, k) (see, e.g., [4, Theorem 4.2.1]), we conclude that H^{}(K, M) is finitely generated over H_{K}. So (fg2) holds. □

Since kL♮_{σ}^{τ}k^{G} satisfies (fg1) and (fg2) by the theorem, support varieties for kL♮_{σ}^{τ}k^{G} may be defined as in Section 3. We next examine these varieties and some modules for a specific example.

Example 5.6. Let k be a field of characteristic 3. Consider the groups

G = ℤ_{2} = ⟨h⟩ and L = ℤ_{2} × ℤ_{2} × ℤ_{3} × ℤ_{3} = ⟨a⟩ × ⟨b⟩ × ⟨c⟩ × ⟨d⟩.

Let the generator h of G act on L by interchanging the generators c and d (and leaving alone the generators a and b).

A projective representation ρ of the Klein four group ⟨a⟩ × ⟨b⟩ may be given by

ρ(a) = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) , ρ(b) = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , ρ(ab) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

There is an associated nontrivial 2-cocycle α, defined by

α(l, m) = ρ(l)ρ(m)(ρ(lm))^{−1}

for all l, m ∈ ⟨a⟩ × ⟨b⟩. We will also denote by α the trivial extension of this cocycle to L, that is

α(a^{n_{1}}b^{m_{1}}c^{s_{1}}d^{t_{1}}, a^{n_{2}}b^{m_{2}}c^{s_{2}}d^{t_{2}}) = α(a^{n_{1}}b^{m_{1}}, a^{n_{2}}b^{m_{2}})

for all integers m_{1}, m_{2}, n_{1}, n_{2}, s_{1}, s_{2}, t_{1}, t_{2}. This gives rise to a 2-cocycle σ : L×L → (k^{G})^{×} in the following way. Let

σ(l, m) = p_{1} + α(l, m)p_{h}

for all l, m ∈ L. Thus σ_{h} = α and σ_{1} is trivial. We take τ : L → k^{G} ⊗ k^{G} to be trivial, that is, τ(l) = 1 ⊗ 1 for all l ∈ L. Thus we leave τ out of the notation, writing kL♮_{σ}k^{G} for the resulting crossed coproduct.

One checks that the center of K = kL♮_{σ}k^{G} is spanned by 1♮p_{h} together with all x♮p_{1}, x ∈ L. So HH^{0}(K) ≃ Z(K) is spanned by these elements. Letting H_{K} be as in (5.5), V_{K} = (∪_{e}V_{kL}) × G where e ranges over a set of primitive central idempotents in kL. It follows that K-modules of the form M = M_{h} have support varieties of the form V_{kL}(M_{h})×h, where V_{kL}(M_{h}) is the maximal ideal spectrum of the quotient of H^{ev}(L, k) by the annihilator of Ext_{k^{σ_{h}}L}(M_{h}, M_{h}) under the action given by − ⊗ M_{h} (which takes kL-modules to k^{σ_{h}}L-modules) followed by Yoneda composition. (The notation is not meant to indicate that this is a variety of a kLmodule.) Indecomposable K-modules of the form M = M_{1} have support varieties of the form V_{kL}(M_{1})_{e} × 1, where the subscript e is meant to indicate the block in which M_{1} lies. As M_{1} is a kL-module, we may view V_{kL}(M_{1}) as the usual variety of a kL-module.

Let U_{ρ} be the k^{α}(⟨a⟩ × ⟨b⟩)-module corresponding to the projective representation ρ described above. Let U = U_{ρ} ↑_{k^{α}(⟨a⟩×⟨b⟩)}^{k^{α}(⟨a⟩×⟨b⟩×⟨c⟩)}, and let d act trivially on U so that U is a k^{α}L-module. Note that U is not projective: If it were, then restricting to k⟨d⟩ would produce a projective k⟨d⟩-module since k^{α}L is free over k⟨d⟩. However the action of d on U is trivial, and the characteristic of k is the order of d, so by Maschke’s Theorem, the restriction of U to k⟨d⟩ is not projective. We will use U to construct nonprojective K-modules with similar properties to those in the last section and in [6]. For example, setting M = U ⊗ kp_{h}, a nonprojective K-module, we claim that M ⊗ M is projective: By Theorem 2.11, M ⊗ M ≃ U ⊗ (^{h}U) ⊗ kp_{1}. Restricting to kL, we have U ⊗ (^{h}U), a projective kL-module since its restriction to the Sylow 3-subgroup ⟨c⟩ × ⟨d⟩ of L is isomorphic to a direct sum of copies of k⟨c⟩ ⊗ k⟨d⟩, which is free. It follows that M ⊗ M is projective as a K-module. Note that here we did not use varieties in our argument, as the tensor product property is known for kL-modules but we are dealing with the tensor product of a k^{σ_{h}}L-module and a k^{σ_{h}^{−1}}L-module.

6. Projective modules of quasitriangular Hopf algebras

In this section we show that the behavior of projective and nonprojective modules occurring in the last two sections does not happen when the Hopf algebra is quasitriangular. More generally, we consider a finite dimensional Hopf algebra A and a finite dimensional A-module M for which M ⊗ M^{∗} ≃ M^{∗} ⊗ M.

The category A-mod of finite dimensional (left) A-modules is a rigid monoidal category. In particular for every finite dimensional A-module M, the composition

(6.1)  M ⟶^{coev_{M} ⊗id_{M}} M ⊗ M^{∗} ⊗ M ⟶^{id_{M} ⊗ev_{M}} M

is the identity map on M, where ev_{M} and coev_{M} denote the canonical evaluation and coevaluation maps. See, for example, [3, Section 2.1] for details.

The tensor product of a projective A-module with another module is projective (see, for example, [5, Proposition 3.1.5]). The tensor product of two nonprojective modules can be projective, and when A has the tensor product property, there are necessary and sufficient conditions for this projectivity as we saw in Theorem 3.1(i): The dimension of the variety of a module is 0 precisely when the module is projective. Under a further assumption on the dual module, an elementary argument shows that tensor powers of a nonprojective module are nonprojective:

Theorem 6.2. Let A be a finite dimensional Hopf algebra, and let n be a positive integer. Let M be a finite dimensional A-module for which M ⊗ M^{∗} ≃ M^{∗} ⊗ M. Then M is projective if and only if M^{⊗n} is projective.

Proof. The tensor product of a projective module with any module is projective (see, e.g., [5, Proposition 3.1.5]), so if M is a projective module, then M^{⊗n} is projective.

For the converse, first consider the case n = 2. Since the composition of functions in (6.1) is the identity map, M is a direct summand of M ⊗ M^{∗} ⊗ M. By hypothesis, M ⊗M^{∗}⊗M ≃ M ⊗M ⊗M^{∗}. If M ⊗M is projective, then M ⊗M ⊗M^{∗} is projective, and so M is a direct summand of a projective module and thus is projective itself.

We will prove the general statement of the converse by induction on n. Assume that for some n ≥ 2, if M^{⊗n} is projective, then M is projective. Apply M^{⊗(n−1)}⊗− to (6.1) to see that M^{⊗n} is a direct summand of M^{⊗n} ⊗M^{∗} ⊗M ≃ M^{⊗(n+1)} ⊗M^{∗}. Therefore if M^{⊗(n+1)} is projective, then M^{⊗n} is projective. By the inductive hypothesis, M is projective. □

Remark 6.3. Let A-stmod be the stable module category of A, that is, objects are finite dimensional A-modules, and for any two objects M, N, morphisms are

\underline{Hom}_{A}(M, N) = Hom_{A}(M, N)/PHom_{A}(M, N)

where PHom_{A}(M, N) is the subspace consisting of each A-module homomorphism from M to N that factors through a projective module. Thus each projective module is isomorphic to the 0 object in A-stmod. If A is quasitriangular, we may view the above theorem as stating that there are no nilpotent objects in the category A-stmod.

References

[1] Nicolás Andruskiewitsch. “Notes on extensions of Hopf algebras”. In: Can. J. Math. 48.1 (1996), pp. 3–42.

[2] Nicolás Andruskiewitsch and Jorge Devoto. “Extensions of Hopf algebras”. In: Algebra i Analiz 7.1 (1995), pp. 22–61.

[3] Bojko Bakalov and Alexander Kirillov, Jr. Lectures on Tensor Categories and Modular Functors. Vol. 21. University Lecture Series. American Mathematical Society, Providence, RI, 2001.

[4] Dave Benson. Representations and Cohomology II: Cohomology of groups and modules. Vol. 31. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1991.

[5] Dave Benson. Representations and Cohomology I: Basic representation theory of finite groups and associative algebras. Vol. 30. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1995.

[6] Dave Benson and Sarah Witherspoon. “Examples of support varieties for Hopf algebras with noncommutative tensor products”. In: Arch. Math. (Basel) 102.6 (2014), pp. 513–520.

[7] Karin Erdmann, Miles Holloway, Nicole Snashall, Øyvind Solberg, and Rachel Taillefer. “Support varieties for selfinjective algebras”. In: K-Theory 33.1 (2004), pp. 67–87.

[8] Pavel Etingof and Victor Ostrik. “Finite tensor categories”. In: Mosc. Math. J. 4.3 (2004), pp. 627–654, 782–783.

[9] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik. Tensor categories. Vol. 205. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2015, pp. xvi+343.

[10] Eric Friedlander and Julia Pevtsova. “Π-supports for modules for finite group schemes over a field”. In: Duke Math. J. 139.2 (2007), pp. 317–368.

[11] Eric Friedlander and Andrei Suslin. “Cohomology of finite group schemes over a field”. In: Invent. Math. 127.2 (1997), pp. 209–270.

[12] Victor Ginzburg and Shrawan Kumar. “Cohomology of quantum groups at roots of unity”. In: Duke Math. J. 69.1 (1993), pp. 179–198.

[13] Gregory Karpilovsky. Group representations. Vol. 2. Vol. 177. North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1993.

[14] Yevgenia Kashina, Geoffrey Mason, and Susan Montgomery. “Computing the Frobenius-Schur indicator for abelian extensions of Hopf algebras”. In: J. Algebra 251.2 (2002), pp. 888–913.

[15] Shahn Majid. “More examples of bicrossproduct and double crossproduct Hopf algebras”. In: Israel J. Math. 72 (1990), pp. 133–148.

[16] Richard K. Molnar. “Semi-direct products of Hopf algebras”. In: J. Algebra 47.1 (1977), pp. 29–51.

[17] Susan Montgomery. Hopf Algebras and Their Actions on Rings. Vol. 82. CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence, RI, 1993.

[18] Susan Montgomery, Maria D. Vega, and Sarah Witherspoon. “Hopf automorphisms and twisted extensions”. In: J. Algebra Appl. 15.6 (2016), p. 14.

[19] Julia Pevtsova and Sarah Witherspoon. “Varieties for modules of quantum elementary abelian groups”. In: Algebr. Represent. Theory 12.6 (2009), pp. 567–595.

[20] Julia Pevtsova and Sarah Witherspoon. “Tensor ideals and varieties for modules of quantum elementary abelian groups”. In: Proc. Amer. Math. Soc. 143.9 (2015), pp. 3727–3741.

[21] Stephen Siegel and Sarah Witherspoon. “The Hochschild cohomology ring of a group algebra”. In: Proc. London Math. Soc. 79 (1999), pp. 131–157.

[22] Nicole Snashall and Øyvind Solberg. “Support varieties and Hochschild cohomology rings”. In: Proc. London Math. Soc. (3) 88.3 (2004), pp. 705–732.

[23] Sarah Witherspoon. “Products in Hochschild cohomology and Grothendieck

rings of group crossed products”. In: Adv. Math. 185.1 (2004), pp. 136–158.

E-mail address: julia@math.tamu.edu
E-mail address: sjw@math.tamu.edu

Department of Mathematics, Texas A&M University, College Station, Texas 77843, U.S.A