Stably uniform affinoids are sheafy
Number Theory
2015-09-15 v2 Algebraic Geometry
Abstract
We develop some of the foundations of affinoid pre-adic spaces without Noetherian or finiteness hypotheses. We give some explicit examples of non-adic affinoid pre-adic spaces (including a locally perfectoid one). On the positive side, we also show that if every affinoid subspace of an affinoid pre-adic space is uniform, then the structure presheaf is a sheaf; note in particular that we assume no finiteness hypotheses on our rings here. One can use our result to give a new proof that the spectrum of a perfectoid algebra is an adic space.
Cite
@article{arxiv.1404.7020,
title = {Stably uniform affinoids are sheafy},
author = {Kevin Buzzard and Alain Verberkmoes},
journal= {arXiv preprint arXiv:1404.7020},
year = {2015}
}
Comments
Version 2 of the manuscript -- the arguments are now presented for general f-adic rings with a topologically nilpotent unit (the original proofs still go through in this generality)