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Fundamental groups of algebraic stacks. (English) Zbl 1052.14001
In this paper, organized into two parts with an appendix on fibrations, the author studies the fundamental groups of algebraic stacks. Part I consists of the main construction and the main results. Part II is a technical companion to Part I where the author strengthens the results of Part I by introducing some more elaborate techniques.
Proving that these fundamental groups carry an additional structure coming from the inertia groups, the author uses this to analyze geometric/topological properties of stacks. He gives an explicit formula for the fundamental group of the coarse moduli space. As an application, he finds an explicit formula for the fundamental group of the geometric quotient of an arbitrary algebraic group action. Also, using additional structures the author proves a necessary and sufficient condition for an algebraic stack to be uniformizable.
The paper also contains results on Galois category of an algebraic stack, basic properties of hidden fundamental groups, classification of uniformizable stacks, fundamental group of the moduli space.

MSC:
14A20 Generalizations (algebraic spaces, stacks)
14F35 Homotopy theory and fundamental groups in algebraic geometry
14L30 Group actions on varieties or schemes (quotients)
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