Actions of Totally Disconnected Groups and Equivariant Singular Cohomology

In this work, we study the special properties of the equivariant singular cohomology of a G-space X, where G is a totally disconnected, locally compact group. We prove that any short exact sequence of coefficient systems for G, over a ring R, gives a long exact sequence of the associated equivariant singular cohomology modules. We establish the relationship between the ordinary singular cohomology modules and the equivariant singular cohomology modules with the natural contravariant coefficient system. Moreover, under some conditions, we give an isomorphism of the equivariant singular cohomology modules of the G-space X onto the ordinary singular cohomology modules of the orbit space X/G.     

Integral closure of invariant ideals, toroidal resolution, and equivariant vector bundles

We establish a connection between equivariant integrally closed ideal sheaves on a G-fibration Y over a G-spherical variety X with an affine fiber V and equivariant vector bundles on the universal toroidal resolution of X. As an application, we reduce the study of invariant integrally closed ideals of V×X to that of some smaller variety in the case of X=M_n,m. Moreover, we present an affirmative answer to a problem raised by Michel Brion [Comment. Math. Helv. 66 (1991) 237-262] for two special infinite series.     

The hypertoric intersection cohomology ring

We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset. T. Braden’s research was supported in part by NSF grant DMS-0201823. N. Proudfoot’s research was supported in part by an NSF Postdoctoral Research Fellowship and NSF grant DMS-0738335.     

Staggered t-structures on toric varieties

Achar has recently introduced a family of t-structures on the derived category of equivariant coherent sheaves on a G-scheme, generalizing the perverse coherent t-structures of Bezrukavnikov and Deligne. They are called staggered t-structures, and one of their points of interest is that they are more often self-dual. In this paper we investigate these t-structures on the T-equivariant derived category of a toric variety.     

On a sheaf-theoretic version of the Witt’s decomposition theorem. A Lagrangian perspective

The approach to a counterpart, in Abstract Geometric Algebra, that is, Geometric Algebra via sheaves of modules, of the classical Witt’s decomposition theoremis based on the axiomatization of the classical context, which however leads to the formulation of a specific subcategory of the category of sheaves of modules: the full subcategory of convenient sheaves of modules. Convenient sheaves of modules turn out, by the very essence of the matter at hand, to be of further importance as far as the setting of results leading to the sheaf-theoretic aspect of several forms of the Witt’s theorem is concerned. Further versions of the Witt’s theorem are still to be treated elsewhere.      

Dominant K-theory and integrable highest weight representations of Kac-Moody groups

We give a topological interpretation of the highest weight representations of Kac-Moody groups. Given the unitary form G of a Kac-Moody group (over C), we define a version of equivariant K-theory, KG on the category of proper G-CW complexes. We then study Kac-Moody groups of compact type in detail (see Section 2 for definitions). In particular, we show that the Grothendieck group of integrable highest weight representations of a Kac-Moody group G of compact type, maps isomorphically onto , where EG is the classifying space of proper G-actions. For the affine case, this agrees very well with recent results of Freed-Hopkins-Teleman. We also explicitly compute for Kac-Moody groups of extended compact type, which includes the Kac-Moody group E₁₀.     

Modules de cycles et motifs mixtes

For a perfect field k, we give a relation between the category of homotopy invariant sheaves with transfers defined by Voevodsky and the category of cycle modules defined by Rost. More precisely, the category of cycle modules over k is equivalent to the category obtained from the homotopy invariant sheaves with transfers by formally inverting the sheaf represented by $\text{G}{}_{\mathrm{m}}$ with its canonical structure of a presheaf with transfers. This gives a canonical monoidal structure on the category of cycle modules over k, and shows that it is Abelian. To cite this article: F. Déglise, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Staggered sheaves on partial flag varieties

Staggered t-structures are a class of t-structures on derived categories of equivariant coherent sheaves. In this Note, we show that the derived category of coherent sheaves on a partial flag variety, equivariant for a Borel subgroup, admits a staggered t-structure with the property that all objects in its heart have finite length. As a consequence, we obtain a basis for its equivariant K-theory consisting of simple staggered sheaves. To cite this article: P.N. Achar, D.S. Sage, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Integral approximation of the characteristic function of an interval and the Jackson inequality in C($$ \mathbb{T} $$)

We give an analog of D.O. Orlov’s theorem on semiorthogonal decompositions of the derived category of projective bundles for the case of equivariant derived categories. Under the condition that the action of a finite group on the projectivization X of a vector bundle E is compatible with the twisted action of the group on the bundle E, we construct a semiorthogonal decomposition of the derived category of equivariant coherent sheaves on X into subcategories equivalent to the derived categories of twisted sheaves on the base scheme.     

Quasicoherent sheaves on complex noncommutative two-tori

We introduce the notion of a quasicoherent sheaf on a complex noncommutative two-torus T as an ind-object in the category of holomorphic vector bundles on T. Extending the results of [10] and [9] we prove that the derived category of quasicoherent sheaves on T is equivalent to the derived category of usual quasicoherent sheaves on the corresponding elliptic curve. We define the rank of a quasicoherent sheaf on T that can take arbitrary nonnegative real values. We study the category Qcoh(η _T ) obtained by taking the quotient of the category of quasicoherent sheaves by the subcategory of objects of rank zero (called torsion sheaves). We show that projective objects of finite rank in Qcoh(η _T ) are classified up to an isomorphism by their rank. We also prove that the subcategory of objects of finite rank in Qcoh(η _T ) is equivalent to the category of finitely presented modules over a semihereditary algebra.     

Grade zero part of forced graded algebras

This paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arises from the semisimple modules for the corresponding quantum group at a p-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for G when the Lusztig character formula does not hold.     

Subcategories of fixed points of mutations by exceptional objects in triangulated categories

We first prove that the subcategory of fixed points of mutation determined by an exceptional object E in a triangulated category coincide with the perpendicular category of E. Based on this characterisation, we prove that the subcategory of fixed points of mutation in the derived category of the coherent sheaves on weighted projective line with genus one is equivalent to the derived category of a hereditary algebra. Meanwhile, we induce two new recollements by left and right mutations from a given recollement.     

Quotient categories,stability conditions,and birational geometry

This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. We prove that this category has homological dimension c. As an application, we describe the space of stability conditions on its derived category in the case c  \(=\) 1. Moreover, we describe all exact equivalences between these quotient categories in this particular case, which is closely related to classification problems in birational geometry.     

An algebraic model for free rational G-spectra for connected compact Lie groups G

We show that the category of free rational G-spectra for a connected compact Lie group G is Quillen equivalent to the category of torsion differential graded modules over the polynomial ring H*(BG). The ingredients are the enriched Morita equivalences of Schwede and Shipley (Topology 42(1):103–153, 2003), the functors of Shipley (Am J Math 129:351–379, 2007) making rational spectra algebraic, Koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring.     

A van Kampen theorem for equivariant fundamental groupoids

The equivariant fundamental groupoid of a G-space X is a category which generalizes the fundamental groupoid of a space to the equivariant setting. In this paper, we prove a van Kampen theorem for these categories: the equivariant fundamental groupoid of X can be obtained as a pushout of the categories associated to two open G-subsets covering X. This is proved by interpreting the equivariant fundamental groupoid as a Grothendieck semidirect product construction, and combining general properties of this construction with the ordinary (non-equivariant) van Kampen theorem. We then illustrate applications of this theorem by showing that the equivariant fundamental groupoid of a G-CW complex only depends on the 2-skeleton and also by using the theorem to compute an example.     

Equivariant sheaves on flag varieties

We show that the Borel-equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves. This proves a conjecture of Soergel and Lunts in the case of flag varieties.     

Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves

The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. It turns out to be related to irreducible objects in the heart of a certain t-structure on the derived category of equivariant coherent sheaves on the Springer resolution, and to equivariant coherent IC sheaves on the nil-cone. The support of the cohomology is described in terms of cells in affine Weyl groups. The basis in the Grothendieck group provided by the cohomology modules is shown to coincide with the Kazhdan-Lusztig basis, as predicted by J. Humphreys and V. Ostrik. The proof is based on the results of [ABG ], [AB] and [B], which allow us to reduce the question to purity of IC sheaves on affine flag varieties. To the memory of my father     

On sheaves in finite group representations

Given a general finite group G, we consider several categories built on it, their Grothendieck topologies and resulting sheaf categories. For a certain class of transporter categories and their quotients, equipped with atomic topology, we explicitly compute their sheaf categories via sheafification. This enables us to identify G-representations with various fixed-point sheaves. As a consequence, it provides an intrinsic new proof to the equivalence of M. Artin between the category of sheaves on the orbit category and that of group representations.     

On Frobenius (Completed) Orbit Categories

Let ?? be a Frobenius category, \({\mathcal P}\) its subcategory of projective objects and F : ?? → ?? an exact automorphism. We prove that there is a fully faithful functor from the orbit category ??/F into \(\operatorname {gpr}({\mathcal P}/F)\), the category of finitely-generated Gorenstein-projective modules over \({\mathcal P}/F\). We give sufficient conditions to ensure that the essential image of ??/F is an extension-closed subcategory of \(\operatorname {gpr}({\mathcal P}/F)\). If ?? is in addition Krull-Schmidt, we give sufficient conditions to ensure that the completed orbit category \({\mathcal E} \ \widehat {\!\! /} F\) is a Krull-Schmidt Frobenius category. Finally, we apply our results on completed orbit categories to the context of Nakajima categories associated to Dynkin quivers and sketch applications to cluster algebras.