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arXiv · 2207.01563

Finite generation of nilpotent quotients of fundamental groups of punctured spectra

Takashi Suzuki
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Abstract

In SGA 2, Grothendieck conjectures that the étale fundamental group of the punctured spectrum of a complete noetherian local domain of dimension at least two with algebraically closed residue field is topologically finitely generated. In this paper, we prove a weaker statement, namely that the maximal pro-nilpotent quotient of the fundamental group is topologically finitely generated. The proof uses $p$-adic nearby cycles and negative definiteness of intersection pairings over resolutions of singularities as well as some analysis of Lie algebras of certain algebraic group structures on deformation cohomology.

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BibTeXRIS

Takashi Suzuki. 2023-07-04. Finite generation of nilpotent quotients of fundamental groups of punctured spectra. https://doi.org/10.1016/j.jalgebra.2023.07.005

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