Cohomological Support Varieties Research Articles

Cohomological varieties associated to vertex operator algebras

Given a vertex operator algebra V, one can attach a graded Poisson algebra called the C2-algebra R(V). The associate Poisson scheme provides an important invariant for V and has been studied by Arakawa as the associated variety. In this article, we define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties and then construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational C2-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. In this paper, we study R(V) as a commutative algebra only and do not use the property of its Poisson structure, which is expected to provide more refined invariants. The goal of this work is to study the cohomological supports of modules for vertex algebras as the cohomological support varieties for finite groups and restricted Lie algebras.

Bounds on cohomological support varieties

Over a local ring R R , the theory of cohomological support varieties attaches to any bounded complex M M of finitely generated R R -modules an algebraic variety V R ( M ) {\mathrm {V}}_R(M) that encodes homological properties of M M . We give lower bounds for the dimension of V R ( M ) {\mathrm {V}}_R(M) in terms of classical invariants of R R . In particular, when R R is Cohen–Macaulay and not complete intersection we find that there are always varieties that cannot be realized as the cohomological support of any complex. When M M has finite projective dimension, we also give an upper bound for dim ⁡ V R ( M ) \dim {\mathrm {V}}_R(M) in terms of the dimension of the radical of the homotopy Lie algebra of R R . This leads to an improvement of a bound due to Avramov, Buchweitz, Iyengar, and Miller on the Loewy lengths of finite free complexes, and it recovers a result of Avramov and Halperin on the homotopy Lie algebra of R R . Finally, we completely classify the varieties that can occur as the cohomological support of a complex over a Golod ring.

Cohomological supports of tensor products of modules over commutative rings

This works concerns cohomological support varieties of modules over commutative local rings. The main result is that the support of a derived tensor product of a pair of differential graded modules over a Koszul complex is the join of the supports of the modules. This generalizes, and gives another proof of, a result of Dao and the third author dealing with Tor-independent modules over complete intersection rings. The result for Koszul complexes has a broader applicability, including to exterior algebras over local rings.

Support varieties and modules of finite projective dimension for modular Lie superalgebras

We investigate cohomological support varieties for finite-dimensional Lie superalgebras defined over fields of odd characteristic. Verifying a conjecture from our previous work, we show the support variety of a finite-dimensional supermodule can be realized as an explicit subset of the odd nullcone of the underlying Lie superalgebra. We also show the support variety of a finite-dimensional supermodule is zero if and only if the supermodule is of finite projective dimension. As a consequence, we obtain a positive characteristic version of a theorem of B{\o}gvad, showing that if a finite-dimensional Lie superalgebra over a field of odd characteristic is absolutely torsion free, then its enveloping algebra is of finite global dimension.

AR-Components of domestic finite group schemes: McKay-Quivers and Ramification

For a domestic finite group scheme, we give a direct description of the Euclidean components in its Auslander-Reiten quiver via the McKay-quiver of a finite linearly reductive subgroup scheme of $SL(2)$. Moreover, for a normal subgroup scheme $\mathcal{N}$ of a finite group scheme $\mathcal{G}$, we show that there is a connection between the ramification indices of the restriction morphism $\mathbb{P}(\mathcal{V}_{\mathcal{N}})\rightarrow\mathbb{P}(\mathcal{V}_{\mathcal{G}})$ between their projectivized cohomological support varieties and the ranks of the tubes in their Auslander-Reiten quivers.

Module invariants and blocks of finite group schemes

We investigate various topological spaces and varieties that can be associated to a block of a finite group scheme G. These spaces come from the theory of cohomological support varieties for modules, as well as from the representation-theoretic constructions of Friedlander and Pevtsova.

Rank varieties for Hopf algebras

We construct rank varieties for the Drinfeld double of the Taft algebra Λ n and for u q ( s l 2 ) . For the Drinfeld double when n = 2 this uses a result which identifies a family of subalgebras that control projectivity of Λ -modules whenever Λ is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext ∗ ( M , M ) is finitely generated over the cohomology ring of the Drinfeld double for any finitely generated module M .

Cohomology and support varieties for Lie superalgebras

Unlike Lie algebras, the finite dimensional complex representations of a simple Lie superalgebra are usually not semisimple. As a consequence, despite over thirty years of study, these remain mysterious objects. In this paper we introduce a new tool: the notion of cohomological support varieties for the finite dimensional supermodules for a classical Lie superalgebra g = g 0 ⊕ g 1 which are completely reducible over g 0 . They allow us to provide a new, functorial description of the previously combinatorial notions of defect and atypicality. We also introduce the detecting subalgebra of g. Its role is analogous to the defect subgroup in the theory of finite groups in positive characteristic. Using invariant theory we prove that there are close connections between the cohomology and support varieties of g and the detecting subalgebra.

Weil restriction and support varieties

This paper extends the comparison results for cohomological support varieties for Chevalley groups of the form G(𝔽p) established by J. Carlson, Z. Lin, and D. Nakano to Chevalley groups of the form G(𝔽pd). Other topics considered are complexity and non-maximal support varieties for modules. Techniques developed by J. Pevtsova and the author, together with the use of Weil restriction functors, are employed.

The ring of reciprocal polynomials and rank varieties

Let p be a prime and let G be a finite p-group. In a recent paper we introduced a commutative graded ℤ-algebra RG (which classifies the so-called convolutions on G). Now let K be an algebraically closed field of characteristic p and let M be a non-zero finitely generated K[G]-module. A general rank variety WG(M) is constructed quite explicitly as a determinantal subvariety of the variety of K-valued points of the spectrum of RG. Further, it is shown that the quotient variety WG(M)/G is inseparably isogenous to the usual cohomological support variety VG(M).

On support varieties for modules over complete intersections

Let ( A , m , k ) (A, \mathfrak {m}, k) be a complete intersection of codimension c c , and let k ~ \tilde {k} be the algebraic closure of k k . We show that every homogeneous algebraic subset of k ~ c \tilde {k}^c is the cohomological support variety of an A A -module, and that the projective variety of a complete indecomposable maximal Cohen–Macaulay A A -module is connected.

Π-supports for modules for finite group schemes

We introduce the space Π(G) of equivalence classes of π-points of a finite group scheme G and associate a subspace Π(G)M to any G-module M. Our results extend to arbitrary finite group schemes G over arbitrary fields k of positive characteristic and to arbitrarily large G-modules, the basic results about “cohomological support varieties” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite-dimensional) G-module can be detected by its restriction along π-points of G. Unlike the cohomological support variety of a G-module M, the invariant M↦Π(G)M satisfies good properties for all modules, thereby enabling us to determine the thick, tensor-ideal subcategories of the stable module category of finite-dimensional G-modules. Finally, using the stable module category of G, we provide Π(G) with the structure of a ringed space which we show to be isomorphic to the scheme ProjH•(G,k)

Representation-theoretic support spaces for finite group schemes

We introduce the space P ( G ) of abelian p -points of a finite group scheme over an algebraically closed field of characteristic p > 0. We construct a homeomorphism Ψ G : P ( G ) → Proj | G | from P ( G ) to the projectivization of the cohomology variety for any finite group G . For an elementary abelian p -group (respectively, an infinitesimal group scheme), P ( G ) can be identified with the projectivization of the variety of cyclic shifted subgroups (resp., variety of 1-parameter subgroups). For a finite dimensional G -module M , Ψ G restricts to a homeomorphism P ( G ) M → Proj | G | M , thereby giving a representation-theoretic interpretation of the cohomological support variety.

SCHEMES OF TORI AND THE STRUCTURE OF TAME RESTRICTED LIE ALGEBRAS

Much of the recent progress in the representation theory of infinitesimal group schemes rests on the application of algebro-geometric techniques related to the notion of cohomological support varieties (cf. [6, 8–10]). The noncohomological characterization of these varieties via the so-called rank varieties (see [21, 22]) involves schemes of additive subgroups that are the infinitesimal counterparts of the elementary abelian groups. In this note we introduce another geometric tool by considering schemes of tori of restricted Lie algebras. Our interest in these derives from the study of infinitesimal groups of tame representation type, whose determination [12] necessitates the results to be presented in §4 and §5 as well as techniques from abstract representation theory.In contrast to the classical case of complex Lie algebras, the information on the structure of a restricted Lie algebra that can be extracted from its root systems is highly sensitive to the choice of the underlying maximal torus. Schemes of tori obviate this defect by allowing us to study algebraic families of root spaces. Accordingly, these schemes may also shed new light on various aspects of the structure theory of restricted Lie algebras. We intend to pursue these questions in a forthcoming paper [13], and focus here on first applications within representation theory.

Varieties in finite transformation groups

The equivariant cohomology ring of a G-space X defines a homogeneous affine variety. Quillen [Q] and W. Y. Hsiang [Hs] have determined the relation between such varieties and the family of isotropy subgroups as well as their fixed point sets when dim X < oo. In modular representation theory, J. Carlson [Cj] introduced cohomological support varieties and rank varieties (the latter depending on the group algebra) and explored their relationship. We define rank and support varieties for G-spaces and G-chain complexes and apply them to cohomological problems in transformation groups. As a corollary, a useful criterion for ZG-projectivity of the reduced total homology of certain G-spaces is obtained, which improves the projectivity criteria of Rim [R], Chouinard [Ch], and Dade [D].