Six operations and Lefschetz-Verdier formula for Deligne-Mumford stacks

TL;DR

This paper develops a direct construction of Grothendieck's six operations for complexes on Deligne-Mumford stacks without finiteness assumptions and proves a Lefschetz-Verdier formula, extending previous work on Artin stacks.

Contribution

It provides a new, more direct method for constructing six operations on Deligne-Mumford stacks and establishes base change theorems in derived categories without finiteness constraints.

Findings

Constructed six operations for Deligne-Mumford stacks without finiteness assumptions.

Proved base change theorems in derived categories for these stacks.

Established a Lefschetz-Verdier formula applicable to Deligne-Mumford stacks.

Abstract

Laszlo and Olsson constructed Grothendieck's six operations for constructible complexes on Artin stacks in \'etale cohomology under an assumption of finite cohomological dimension, with base change established on the level of sheaves. In this article we give a more direct construction of the six operations for complexes on Deligne-Mumford stacks without the finiteness assumption and establish base change theorems in derived categories. One key tool in our construction is the theory of gluing finitely many pseudofunctors developed in arXiv:1211.1877. As an application, we prove a Lefschetz-Verdier formula for Deligne-Mumford stacks. We include both torsion and $\ell$-adic coefficients.