Notions of purity and the cohomology of quiver moduli

TL;DR

This paper investigates various notions of purity related to Frobenius actions on schemes over finite fields, and applies these concepts to prove the strong purity of cohomology groups of quiver moduli spaces.

Contribution

It introduces new notions of purity, studies their behavior under natural geometric operations, and proves strong purity results for the cohomology of quiver moduli.

Findings

A natural stratification of quiver representation spaces is equivariantly perfect.

Each l-adic cohomology group of quiver moduli is strongly pure.

Purity notions are preserved under quotients for principal bundles and geometric quotients.

Abstract

We explore several variations of the notion of purity for the action of Frobenius on schemes defined over finite fields. In particular, we study how these notions are preserved under certain natural operations like quotients for principal bundles and also geometric quotients for reductive group actions. We then apply these results to study the cohomology of quiver moduli. We prove that a natural stratification of the space of representations of a quiver with a fixed dimension vector is equivariantly perfect and from it deduce that each of the $l$-adic cohomology groups of the quiver moduli space is strongly pure.