Formal gluing along non-linear flags

TL;DR

This paper establishes formal glueing and decomposition results for stacks of perfect complexes and G-bundles on Artin stacks, extending to derived stacks and connecting to the Geometric Langlands program.

Contribution

It proves formal glueing along arbitrary closed substacks for Artin stacks and extends these results to derived stacks, providing new tools for studying categories of complexes and bundles.

Findings

Formal glueing along arbitrary closed substacks is established.

Decomposition formulas for stacks along nonlinear flags are derived.

Localization theorem for almost perfect complexes on schemes is proved.

Abstract

In this paper we prove formal glueing along an arbitrary closed substack $Z$ of an arbitrary Artin stack $X$ (locally of finite type over a field $k$), for the stacks of (almost) perfect complexes , and of $G$-bundles on $X$ (for $G$ a smooth affine algebraic $k$-group scheme). By iterating this result, we get a decomposition of these stacks along an arbitrary nonlinear flag of closed substacks in $X$. By taking points over the base field, we deduce from this both a formal glueing, and a flag-related decomposition formula for the corresponding symmetric monoidal derived $\infty$-categories of (almost) perfect modules. When $X$ is a quasi-compact and quasi-separated scheme, we also prove a localization theorem for almost perfect complexes on $X$, which parallels Thomason's localization results for perfect complexes. This is one of the main ingredients we need to provide a global characterization of the category of almost perfect complexes on the punctured formal neighbourhood. We then extend all of the previous results - i.e. the formal glueing and flag-decomposition formulas - to the case when $X$ is a derived Artin stack (locally almost of finite type over a field $k$), for the derived versions of the stacks of (almost) perfect modules, and of $G$-bundles on $X$. We close the paper by highlighting some expected progress in the subject matter of this paper, related to a Geometric Langlands program for higher dimensional varieties. In an Appendix (for $X$ a variety), we give a precise comparison between our formal glueing results and the rigid-analytic approach of Ben-Bassat and Temkin.