A theorem on meromorphic descent and the specialization of the proétale fundamental group
Abstract
Given a Noetherian formal scheme $\hat X$ over ${\rm Spf}(R)$, where $R$ is a complete DVR, we first prove a theorem of meromorphic descent along a possibly infinite cover of $\hat{X}$. Using this we construct a specialization functor from the category of continuous representations of the proétale fundamental group of the special fiber to the category of $F$divided sheaves on the generic fiber. This specialization functor partially recovers the specialization functor of the étale fundamental groups. We also express the proétale fundamental group of a connected scheme $X$ of finite type over a field as coproducts and quotients of the free group and the étale fundamental groups of the normalizations of the irreducible components of $X$ and those of its singular loci.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 arXiv:
 arXiv:2103.11543
 Bibcode:
 2021arXiv210311543L
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  Category Theory;
 Mathematics  Number Theory