On the bumpy fundamental group scheme

Marco Antei

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arXiv · 1511.07331

On the bumpy fundamental group scheme

Marco Antei

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Abstract

In this short paper we first recall the definition and the construction of the fundamental group scheme of a scheme $X$ in the known cases: when it is defined over a field and when it is defined over a Dedekind scheme. It classifies all the finite (or quasi-finite) fpqc torsors over $X$. When $X$ is defined over a noetherian regular scheme $S$ of any dimension we do not know if such an object can be constructed. This is why we introduce a new category, containing the fpqc torsors, whose objects are torsors for a new topology. We prove that this new category is cofiltered thus generating a fundamental group scheme over $S$, said \textit{bumpy} as it may not be flat in general. We prove that it is flat when $S$ is a Dedekind scheme, thus coinciding with the \textit{classical} one.

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Marco Antei. 2015-11-23. On the bumpy fundamental group scheme. https://arxiv.org/abs/1511.07331

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