Fundamental Groups in Arithmetic and Algebraic Geometry

# The homotopy exact sequence for the fundamental group scheme defined by stratified sheaves

speaker: João Pedro dos Santos (Institut de Mathématiques de Jussieu)

abstract: Let $X$ be a smooth and connected scheme over an algebraically closed field $k$. The category of $\mathcal DX$-modules ($\mathcal DX$=all differential operators as in EGA IV-4) which are $\mathcal OX$-coherent is equivalent to the category of representations of an affine group scheme $\Pi(X)$. The goal of the work on which this talk is based is the study of the kernel of $\Pi(X)\to\Pi(S)$. By making the assumption that $f:X\to S$ is proper, smooth, and geometrically connected, we show that the kernel is precisely $\Pi(f{-1}(s))$. This is an analogue of a result in SGA1 (the homotopy exact sequence''). The main difficulty in the proof comes from the fact that the Tannakian theory is not well suited to non affine objects; this makes it hard to give a Tannakian sense to homogeneous spaces $GH$. To remedy, we use the notion of stratified scheme and infinitesimal equivalence relations, which is simply how C. Ehresmann introduced connections.

timetable:
Wed 18 Dec, 10:30 - 11:30, Aula Dini
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