沿非线性标志的形式粘合
Formal gluing along non-linear flags
摘要：
在本文中，我们证明了形式胶合沿任意封闭子栈$Z$的任意Artin堆栈$X$（局部有限型域$k$），（几乎）完美复形的堆栈，和$G$ -丛$X$（$G$的光滑仿射代数$k$ -群计划）。通过迭代这个结果，我们得到这些堆栈沿着$X$中的封闭子堆栈的任意非线性标志的分解。通过采取点的基础领域，我们推导出这两个正式的胶合，和标志相关的分解公式相应的对称monoidal导出$\infty$类（几乎）完美的模块。当$X$是一个拟紧的拟分离的方案时，我们还证明了几乎完全复形在$X$上的一个局部化定理，它平行于Mesquason关于完全复形的局部化结果.这是一个主要的成分，我们需要提供一个全球性的性质的范畴几乎完美的复合物的穿孔正式附近。然后，我们扩展了所有以前的结果-即正式的胶合和标志分解公式-的情况下，当$X$是一个派生的Artin堆栈（局部几乎有限型域$k$），（几乎）完美的模块的堆栈的派生版本，和$X$上的$G$ -丛。我们关闭的文件，突出了一些预期的进展，本文的主题，涉及到几何Langlands程序高维品种。在附录（$X$的品种），我们给出了一个精确的比较，我们的正式胶合结果和刚性分析方法的Ben-Bassat和Temkin。
英文摘要：
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.
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