Specialization for the proétale fundamental group
Abstract
For a formal scheme $\mathfrak{X}$ of finite type over a complete rank one valuation ring, we construct a specialization morphism \[ \pi^{\rm dJ}_1(\mathfrak{X}_\eta) \to \pi^{\rm proet}_1(\mathfrak{X}_k) \] from the de Jong fundamental group of the rigid generic fiber to the BhattScholze proétale fundamental group of the special fiber. The construction relies on an interplay between admissible blowups of $\mathfrak{X}$ and normalizations of the irreducible components of $\mathfrak{X}_k$, and employs the Berthelot tubes of these irreducible components in an essential way. Using related techniques, we show that under certain smoothness and semistability assumptions, covering spaces in the sense of de Jong of a smooth rigid space which are tame satisfy étale descent.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.06761
 Bibcode:
 2021arXiv210706761A
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 14G22 (Primary) 14F20;
 14F35 (Secondary)
 EPrint:
 29 pages, comments welcome!