Theory Of Support Varieties Research Articles

New approaches to finite generation of cohomology rings

In support variety theory, representations of a finite dimensional (Hopf) algebra A can be studied geometrically by associating any representation of A to an algebraic variety using the cohomology ring of A. An essential assumption in this theory is the finite generation condition for the cohomology ring of A and that for the corresponding modules.In this paper, we introduce various approaches to study the finite generation condition. First, for any finite dimensional Hopf algebra A, we show that the finite generation condition on A-modules can be replaced by a condition on any affine commutative A-module algebra R under the assumption that R is integral over its invariant subring RA. Next, we use a spectral sequence argument to show that a finite generation condition holds for certain filtered, smash and crossed product algebras in positive characteristic if the related spectral sequences collapse. Finally, if A is defined over a number field over the rationals, we construct another finite dimensional Hopf algebra A′ over a finite field, where A can be viewed as a deformation of A′, and prove that if the finite generation condition holds for A′, then the same condition holds for A.

Endo-trivial modules for cyclic p-groups and generalized quaternion groups via Galois descent

In this paper, we investigate the group of endotrivial modules for certain $p$-groups. Such groups were already been computed by Carlson-Thevenaz using the theory of support varieties; however, we provide novel homotopical proofs of their results for cyclic $p$-groups, the quaternion group of order 8, and for generalized quaternion groups using Galois descent and Picard spectral sequences, building on results of Mathew and Stojanoska. Our computations provide conceptual insights into the classical work of Carlson-Thevenaz.

Support varieties for finite tensor categories: Complexity, realization, and connectedness

We advance support variety theory for finite tensor categories. First we show that the dimension of the support variety of an object equals the rate of growth of a minimal projective resolution as measured by the Frobenius-Perron dimension. Then we show that every conical subvariety of the support variety of the unit object may be realized as the support variety of an object. Finally, we show that the support variety of an indecomposable object is connected.

Derived invariance of support varieties

The (Fg) condition on Hochschild cohomology as well as the sup-port variety theory are shown to be invariant under derived equivalence.

Support variety theory is invariant under singular equivalences of Morita type

Using a new equivalent definition of support varieties in the sense of Snashall and Solberg [23], we show that both the (Fg) condition and support varieties are preserved under singular equivalences of Morita type. In particular, support variety theory is invariant under stable equivalences of Morita type.

Tensor Products and Support Varieties for Some Noncocommutative Hopf Algebras

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 cocycles. We apply support variety theory for these Hopf algebras as a tool in our investigations.

Matrix factorizations in higher codimension

We observe that there is an equivalence between the singularity category of an affine complete intersection and the homotopy category of matrix factorizations over a related scheme. This relies in part on a theorem of Orlov. Using this equivalence, we give a geometric construction of the ring of cohomology operators, and a generalization of the theory of support varieties, which we call stable support sets. We settle a question of Avramov about which stable support sets can arise for a given complete intersection ring. We also use the equivalence to construct a projective resolution of a module over a complete intersection ring from a matrix factorization, generalizing the well-known result in the hypersurface case.

Erratum to “Support Varieties and Representation Type of Small Quantum Groups”

Some of the general results in the paper require an additional hypothesis, such as quasitriangularity. Applications to specific types of Hopf algebras are correct, as some of these are quasitriangular, and for those that are not, the Hochschild support variety theory may be applied instead. Let A be a finite dimensional Hopf algebra over a field k. The vector space H∗(A, k) := ExtA(k, k) is an associative, graded commutative k-algebra under the cup product, or equivalently under Yoneda composition. If M and N are finitely generated left A-modules, then H∗(A, k) acts on ExtA(M,N) via the cup product, or equivalently by −⊗N followed by Yoneda composition. Let S be the composition inverse of the antipode S. The isomorphism towards the top of p. 1350 in [3], Ext q A(M,N) ∼= Ext q A(k,M ⊗N), assumes the following A-module structure on M∗ = Homk(M,k), which was not stated explicitly in the paper: (a · f)(m) = ∑ a2f(S(a1)m) for all a ∈ A, f ∈ Homk(M,k), and m ∈M . Under some finiteness assumptions as in [3], we recall the definition of support variety: Let M be a finitely generated left A-module. Let IA(M) be the annihilator of the action of H(A, k) on Ext q A(M,M), and let VA(M) denote the maximal ideal spectrum of the finitely generated commutative k-algebra H(A, k)/IA(M). In part (5) of [3, Proposition 2.4], it is stated that (*) VA(M ⊗N) ⊆ VA(M) ∩ VA(N). Our proof of this statement requires an additional hypothesis such as quasitriangularity of A, as we will explain. However, (*) holds under other hypotheses, for example, whenever VA(M) = VA(M) for all finitely generated A-modules M , as follows from the fact that dualization reverses the order of tensor products. For a counterexample to the general statement (*), see [1]. In the proof of (*) given in [3], the action of H(A, k) on Ext q A(M ⊗N,M ⊗N) does indeed factor through its action on Ext q A(M,M) as stated, since we may first apply − ⊗M , then − ⊗ N , then Yoneda composition. If A is quasitriangular, then M ⊗N ∼= N ⊗M , so the action factors through that on Ext q A(N,N) as well, Date: August 22, 2013. 1 2 JORG FELDVOSS AND SARAH WITHERSPOON and consequently (*) holds. However, it does not necessarily factor through its action on Ext q A(N,N) in general: Under the isomorphism on Ext that is used in the proof, Ext q A(M⊗N,M⊗N) ∼= Ext q A(N,M ∗⊗M⊗N), the actions of H(A, k) do not always coincide. The order of the tensor product affects the action. Thus we obtain VA(M ⊗N) ⊆ VA(M) in general, but not necessarily (*). Theorem 2.5, Corollary 2.6, Theorem 3.1, and Corollary 3.2 of [3] require an additional hypothesis such as quasitriangularity of A, since they use [3, Proposition 2.4(5)]. The proof of [3, Theorem 4.3] is incorrect, or at least incomplete; the Hopf algebras uq (g) are not in general quasitriangular. However, Hochschild support variety theory [2, 4] provides an alternative proof of this statement: By [4, Lemma 6.3] or [5, §2], the representation type of uq (g) is the same as that of u q (g), and we may apply [4, Theorem 5.4] to the latter algebra. The remainder of the proofs of the statements in [3, Section 4] are correct, as the Hopf algebras uq(g) are quasitriangular. We note that our results in [4] almost exclusively use the Hochschild support variety theory, and do not require any additional hypotheses. This theory is more general, and does not take advantage of the tensor products of modules one has at hand for a Hopf algebra. One may obtain stronger results by taking advantage of tensor products of modules, yet the constructions are necessarily one-sided in general. The one-sidedness affects results if the tensor product is not commutative up to isomorphism.

Support varieties for transporter category algebras

Let G be a finite group. Over any finite G-poset P we may define a transporter category as the corresponding Grothendieck construction. The classifying space of the transporter category is the Borel construction on the G-space BP, while the k-category algebra of the transporter category is the (Gorenstein) skew group algebra on the G-algebra kP.We introduce a support variety theory for the category algebras of transporter categories. It extends Carlson’s support variety theory on group cohomology rings to equivariant cohomology rings. In the mean time it provides a class of (usually non selfinjective) algebras to which Snashall–Solberg’s (Hochschild) support variety theory applies. Various properties will be developed. Particularly we establish a Quillen stratification for modules.

A Spectral Sequence for the Hochschild Cohomology of a Coconnective DGA

A spectral sequence for the computation of the Hochschild cohomology of a coconnective dga over a field is presented. This spectral sequence has a similar flavour to the spectral sequence presented in [7] for the computation of the loop homology of a closed orientable manifold. Using this spectral sequence we identify a class of spaces for which the Hochschild cohomology of their mod-$p$ cochain algebra is Noetherian. This implies, among other things, that for such a space the derived category of mod-$p$ chains on its loop-space carries a theory of support varieties.

Cohomology and Support Varieties for Restricted Lie Superalgebras

Let \((\mathfrak{g}, [p]) \) be a restricted Lie superalgebra over an algebraically closed field k of characteristic p > 2. Let \(\mathfrak{u}(\mathfrak{g})\) denote the restricted enveloping algebra of \(\mathfrak{g}\). In this paper we prove that the cohomology ring \(\operatorname{H}^\bullet(\mathfrak{u}(\mathfrak{g}), k)\) is finitely generated. This allows one to define support varieties for finite dimensional \(\mathfrak{u}(\mathfrak{g})\)-supermodules. We also show that support varieties for finite dimensional \(\mathfrak{u}(\mathfrak{g})\)- supermodules satisfy the desirable properties of a support variety theory.

Radical cube zero selfinjective algebras of finite complexity

One of our main results is a classification of all the possible quivers of selfinjective radical cube zero finite-dimensional algebras over an algebraically closed field having finite complexity. In the paper (Erdmann and Solberg, 2011) [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite-dimensional algebra to have such a theory of support varieties implies that the algebra has finite complexity. Hence this paper is a partial extension of [5].

Cohomology and Support Varieties for Lie Superalgebras of Type W(n)

Boe, Kujawa and Nakano [3; 4] recently investigated relative cohomology for classical Lie superalgebras and developed a theory of support varieties. The dimensions of these support varieties give a geometric interpretation of the combinatorial notions of defect and atypicality due to Kac, Wakimoto, and Serganova. In this paper, we calculate the cohomology ring of the Cartan-type Lie superalgebra W(n) relative to the degree zero component W(n)0 and show that this ring is a finitely generated polynomial ring. This allows one to define support varieties for finite-dimensional W(n)-supermodules, which are completely reducible over W(n)0. We calculate the support varieties of all simple supermodules in this category. The outcome of our computations naturally divides the simple supermodules into two families. Remarkably, this partition coincides with the one based on Serganova's prior notion of atypicality for Cartan-type superalgebras. In this way the support variety construction gives a geometric interpretation of atypicality. We also present new results on the realizability of support varieties, which hold for both classical and Cartan-type superalgebras.

Filtrations in Modular Representations of Reductive Lie Algebras

Let G be a connected reductive algebraic group over an algebraically closed field k of prime characteristic p, and 𝔤 = Lie (G). In this paper, we study representations of the reductive Lie algebra 𝔤 with p-character χ of standard Levi form associated with an index subset I of simple roots. With aid of the support variety theory, we prove that a Uχ(𝔤)-module is projective if and only if it is a strong "tilting" module, i.e., admitting both [Formula: see text]- and [Formula: see text]-filtrations. Then by an analogy of the arguments in [2] for G1T-modules, we construct so-called Andersen–Kaneda filtrations associated with each projective 𝔤-module of p-character χ, and finally obtain sum formulas from those filtrations.

Support Varieties and Representation Type of Small Quantum Groups

In this paper, we provide a wildness criterion for any finite dimensional Hopf algebra with finitely generated cohomology. This generalizes a result of Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields of arbitrary characteristic. Our proof uses the theory of support varieties for modules, one of the crucial ingredients being a tensor product property for some special modules. As an application, we prove a conjecture of Cibils stating that small quantum groups of rank at least two are wild.

Representation Theory of Finite Dimensional Algebras

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Cohomology and support varieties for Lie superalgebras II

In [2] (Preprint, 2006, arXiv:math.RT/0609363) the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra, and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties of Kac supermodules for Type-I Lie superalgebras and of the simple supermodules for g l(m|n). The latter result verifies our earlier conjecture for g l(m|n). In our investigation we also delineate several of the major differences between Type-I versus Type-II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.

Complexes of injectivekG-modules

Let G be a finite group and k be a field of characteristic p. We investigate the homotopy category K(InjkG) of the category C(InjkG) of complexes of injective (= projective) kG-modules. If G is a p-group, this category is equivalent to the derived category Ddg(C∗(BG;k)) of the cochains on the classifying space; if G is not a p-group, it has better properties than this derived category. The ordinary tensor product in K(InjkG) with diagonal G-action corresponds to the E∞ tensor product on Ddg(C∗(BG;k)). We show that K(InjkG) can be regarded as a slight enlargement of the stable module category StModkG. It has better formal properties inasmuch as the ordinary cohomology ring H∗(G,k) is better behaved than the Tate cohomology ring Ĥ∗(G,k). It is also better than the derived category D(ModkG), because the compact objects in K(InjkG) form a copy of the bounded derived category Db(modkG), whereas the compact objects in D(ModkG) consist of just the perfect complexes. Finally, we develop the theory of support varieties and homotopy colimits in K(InjkG).

Support Varieties for Selfinjective Algebras

Support varieties for any finite dimensional algebra over a field were introduced by Snashall-Solberg using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb's theorem is true.

Cohomology of Quantum Groups: the Quantum Dimension

AbstractThis paper uses the notion of the quantum dimension to obtain new results on the cohomology and representation theory of quantum groups at a root of unity. In particular, we consider the elementary theory of support varieties for quantum groups.