Equivariant geometry and the cohomology of the moduli space of curves

TL;DR

This paper provides a categorical framework for understanding the integral cohomology of stacks, especially quotient stacks, and relates it to equivariant and coarse moduli space cohomology, focusing on stacks of smooth and stable curves.

Contribution

It introduces a categorical definition of cohomology for stacks and establishes isomorphisms with equivariant and coarse moduli space cohomology for Deligne-Mumford quotient stacks.

Findings

Categorical cohomology of quotient stacks matches equivariant cohomology.

Rational cohomology of Deligne-Mumford quotient stacks is isomorphic to that of the coarse moduli space.

Focus on stacks of smooth and stable curves enhances understanding of their cohomological properties.

Abstract

In this expository article we give a categorical definition of the integral cohomology ring of a stack. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology. Via this identification we show that for Deligne-Mumford quotient stacks this cohomology is rationally isomorphic to the rational cohomology of the coarse moduli space. The theory is presented with a focus on the stacks of smooth and stable curves.