Formal gluing along non-linear flags

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.