4
$\begingroup$

I have two questions about v-covers and the h-topology (as defined by Voevodsky) which arose when reading Bhatt-Scholze's "Projectivity of the Witt vector affine Grassmannian" available here https://arxiv.org/abs/1507.06490

Question 1. In Definition 2.1 of Bhatt-Scholze's paper, a v-cover is defined by using valuation rings. Does it suffice to use discrete valuation rings in this definition?

Question 2. How does one show that the h-topology as defined by Bhatt-Scholze is the same as the h-topology defined by Voevodsky? I did not find a detailed proof of this in their paper (because it is presumably very simple...).

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ The answer to question 1 is no. For question 2, Bhatt and Scholze do not claim any thing being easy but rather explcitly rely on the beautiful paper by David Rydh, Submersions and effective descent of ́etale morphisms. Bull. Soc. Math. France, 138(2):181–230, 2010 (freely available on NUMDAM), which very fine to read. This paper of Gabber and Kelly might be enlightening: arXiv:1407.5782 as well, at least to forge an intuition within noetherian schemes. $\endgroup$
    – D.-C. Cisinski
    Commented Sep 27, 2021 at 11:20
  • $\begingroup$ @Denis-CharlesCisinski Thank you for this. I will read the paper by Rydh, and also have a look at Gabber-Kelly. About question 1: I presume this is a standard argument. Is the answer to question 1 yes if one sticks to the realm of noetherian schemes and/or finitely presented morphisms? $\endgroup$
    – Hinter
    Commented Sep 27, 2021 at 12:05
  • 3
    $\begingroup$ You could also consult Bhatt-Mathew, the arc-topology: arxiv.org/abs/1807.04725 $\endgroup$
    – Z. M
    Commented Sep 27, 2021 at 16:53

0

Reset to default

You must log in to answer this question.