Equivariant Functors and Sheaves

Abstract

In this thesis we study two main topics which culminate in a proof that four distinct definitions of the equivariant derived category of l-adic sheaves on a variety X carrying an action by a smooth affine algebraic group G are in fact equivalent. In the first part of this thesis we introduce and study equivariant categories on a quasi-projective variety X. These equivariant categories are a generalization of the equivariant derived category and are indexed by certain pseudofunctors defined on a category of smooth free resolutions of X that take values in the 2-category of categories. This 2-categorical generalization allow us to prove rigorously and carefully when such categories are additive, monoidal, triangulated, and admit t-structures, among other properties. We also define equivariant functors and natural transformations before using these to prove how to lift adjoints to the equivariant setting. We also give a careful foundation of how to manipulate t-structures on these equivariant categories for future use and with an eye towards future applications. In the final part of this thesis we assume that G is an affine algebraic group and prove a four-way equivalence between the different formulations of the equivariant derived category of l-adic sheaves on a quasi-projective variety X. More explicitly, we show that the equivariant derived category of Lusztig is equivalent to the equivariant derived category of Bernstein-Lunts before showing that these are equivalent to the simplicial equivariant derived category. As a final step we show that these equivariant derived categories are equivalent to the derived l-adic category on the algebraic stack [G\X] of Behrend. In the course of showing these equivalences, we provide an isomorphism of the simplicial equivariant derived category on the variety X with the simplicial equivariant derived category on the simplicial presentation of the algebraic stack [G\X], as well as prove explicit equivalences between the categories of equivariant l-adic sheaves and l-adic local systems with the classical incarnations of such equivariant categories.