Enhanced six operations and base change theorem for higher Artin stacks

TL;DR

This paper develops a comprehensive theory of Grothendieck's six operations for derived categories in étale cohomology of higher Artin stacks, extending previous work with new methods and broader applicability.

Contribution

It introduces a new approach using stable ∞-categories to extend six operations and base change theorems to higher Artin stacks, surpassing prior assumptions.

Findings

Established six operations for higher Artin stacks.

Proved base change theorem in derived categories.

Extended perverse t-structures to higher stacks.

Abstract

In this article, we develop a theory of Grothendieck's six operations for derived categories in \'etale cohomology of Artin stacks, for both torsion and adic coefficients. We prove several desired properties of the operations, including the base change theorem in derived categories. This extends many previous theories on this subject, including the one developed by Laszlo and Olsson, in which the operations are subject to more assumptions and the base change isomorphism is only constructed on the level of sheaves. Moreover, our theory works for higher Artin stacks as well. In addition, we define perverse t-structures on higher Artin stacks for general perversity, extending Gabber's work on schemes. Our method differs from previous approaches, as we exploit the theory of stable $\infty$-categories developed by Lurie. We enhance derived categories, functors, and natural isomorphisms to the level of $\infty$-categories and introduce $\infty$-categorical (co)homological descent. To handle the issue of ``homotopy coherence'', we develop a general technique for gluing subcategories of $\infty$-categories and several other $\infty$-categorical techniques. We obtain categorical equivalences between simplicial sets associated to certain multisimplicial sets. Such equivalences can be used to construct functors in different contexts. One of our category-theoretical results generalizes Deligne's gluing theory developed in the construction of the extraordinary pushforward operation in \'etale cohomology of schemes.