FUNDAMENTAL GROUPS OF ALGEBRAIC STACKS

By B. Noohi. Journal of the Institute of Mathematics of Jussieu, 2002

107 citations recorded by the source corpus.

We study the fundamental groups of algebraic stacks. We show that these fundamental groups carry an additional structure coming from the inertia groups. We use this additional structure to analyse geometric/topological properties of stacks. We give an explicit formula for the fundamental group of the coarse moduli space. As an application, we find an explicit formula for the fundamental group of the geometric quotient of an arbitrary algebraic group action. Also, we use these additional structures to give a necessary and sufficient for an algebraic stack to be uniformizable (i.e. quotient of an algebraic space by a finite group action). AMS 2000 Mathematics subject classification: Primary 14A20. Secondary 14F35; 14L30

Subfields: Computer Vision and Pattern Recognition, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a755299becbb790d5da45907ea833534a58ff320

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Citations in the corpus

The tropicalization of the moduli space of curves (2012, unknown): “See [Noo04] for details on ´etale fundamental groups of Deligne–Mumford stacks.”
On the geometry of Deligne-Mumford stacks (2009, methodology): “The orbifold fundamental group, or fundamental group of a DM stack, can be defined in algebraic geometry by imitating the treatment of algebraic fundamental group given by Grothendieck [SGA1]. For details, see [Z] and [ No ].”
Differentiable stacks and gerbes (2006, background): “cal charts” on a diﬀerentiable stack. Establishing such a dictionary consists of the ﬁrst part of the paper. We note that this viewpoint of connecting stacks with groupoids is somehow 2 folklore (see [15, 34, 39, 40]). However, we feel that it is useful to spell it out in detail in the diﬀerentiable”
Local Systems on Proper Algebraic V -manifolds (2010, background): “Local systems onX may be identified with representations of Noohi’s fundamental group π1(X, x) defined in [78], which is the same as the fundamental group of the topological realization |X|.”
From double Lie groupoids to local Lie 2-groupoids (2010, methodology): “q), where xi ∈ Xq−1, satisfying the horn compatibility equations (4.1) fq−1 i xj = f q−1 j−1xi 3One should compare this computation with the deﬁnition of fundamental group of a diﬀerentiable stack in [Noo04] and [LGTX07]. FROM DOUBLE LIE GROUPOIDS TO LOCAL LIE 2-GROUPOIDS 11 for i < j. The space ”
Odds and Ends on Finite Group Actions and Traces (2010, background): “nd integral. If ¯ηis a geometric point above the generic point ηof Y, the category of lisse E-sheaves on Y is then equivalent to the category of (ﬁnite-dimensional) E-representations of π1(Y,η¯) (see [Noo04, 4] for a deﬁnition of π1(Y,¯η)). Let Klisse(Y,E) denote the corresponding Grothendieck group”
On the classification of rank-two representations of quasiprojective fundamental groups (2007, background): “The paper of Behrend and Noohi [BN06] is a very complete description of the possibilities for DM-curves and orbicurves, and the reader is referred there.”
ON ARITHMETIC CURVES IN THE MODULI SPACES OF CURVES (2005, background): “The theory of the fundamental group for stacks parallels, but with interesting differences, Grothendieck’s classical [22] theory for schemes (see [57] and [45]); for the specific case of the moduli stacks of curves, which are classifying spaces, we also refer to the contribution of T.”