arXiv math/00122192000-12math.AGmath.ATmath.CT

仿射堆栈（Champs affines）

Affine stacks (Champs affines)

B. Toen

备注：: French, 112 pages. New and expended version. Several additional applications to algebraic geometry and algebraic topology are included. Final version, to appear in Selecta

摘要：

这是预印本《同伦类型的图式化》的扩充和修订版。这项工作的目的是介绍一个概念的\emph{affine stacks}，这是一个同伦版本的概念的仿射计划，并给出几个应用的背景下，代数拓扑和代数几何。作为第一个应用程序，我们展示了如何仿射栈可以使用，以给出一个新的观点（和新的证明）的理性和$p$进同伦理论。这给出了A的第一个解。\cite{gr}中描述的Grothendieck \emph{schematization problem}。我们还使用仿射栈来引入\emph{schematic homotopy types}的概念。我们表明，示意同伦类型给出了第二个解决方案的图式化问题，这也使我们能够超越理性和$p$ -进同伦理论的空间与任意基本群。图式同伦类型的概念也用于构造对应于各种上同调理论的代数簇的各种同伦类型（Betti，de Rham，$l$ -adic，.），扩展了各种基本群的公知结构。最后，由于代数栈是通过胶合仿射方案得到的，我们将\emph{$\infty$-geometric stacks}定义为仿射栈的某种胶合。$\infty$ -几何堆栈的背景下，代数拓扑（模空间的dga结构的准同构）和霍奇理论（非阿贝尔周期）的例子。

英文摘要：

This is an expended and revised version of the preprint "Schematization of homotopy types". The purpose of this work is to introduce a notion of \emph{affine stacks}, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and $p$-adic homotopy theory. This gives a first solution to A. Grothendieck's \emph{schematization problem} described in \cite{gr}. We also use affine stacks in order to introduce a notion of \emph{schematic homotopy types}. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyound rational and $p$-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various cohomology theories (Betti, de Rham, $l$-adic, ...), extending the well known constructions of the various fundamental groups. Finally, as algebraic stacks are obtained by gluing affine schemes we define \emph{$\infty$-geometric stacks} as a certain gluing of affine stacks. Example of $\infty$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.

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