Integral Motivic Sheaves And Geometric Representation Theory

TL;DR

This paper develops a formalism of reduced motivic sheaves with integral coefficients, unifying various approaches to mixed sheaves in representation theory and exploring their structural properties.

Contribution

It introduces reduced motives with integral coefficients and demonstrates their favorable properties, unifying multiple existing frameworks in geometric representation theory.

Findings

Reduced stratified Tate motives satisfy weight and t-structures.

Reduced motives on cellular schemes unify various approaches to mixed sheaves.

The formalism facilitates applications in representation theory.

Abstract

With representation-theoretic applications in mind, we construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. We show that reduced stratified Tate motives satisfy favorable properties including weight and t-structures. We also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory, such as Soergel-Wendt's semisimplified Hodge motives, Achar-Riche's complexes of parity sheaves, as well as Ho-Li's recent category of graded $\ell$-adic sheaves.