Stably uniform affinoids are sheafy

TL;DR

This paper develops foundational aspects of affinoid pre-adic spaces without relying on Noetherian or finiteness assumptions, providing explicit examples and establishing sheaf properties under uniformity conditions.

Contribution

It introduces a framework for affinoid pre-adic spaces without finiteness hypotheses and proves sheafiness when all affinoid subspaces are uniform, including a new proof for perfectoid spectra.

Findings

Explicit examples of non-adic affinoid pre-adic spaces, including a locally perfectoid one

Proved that uniformity of all affinoid subspaces implies the structure presheaf is a sheaf

Provided a new proof that the spectrum of a perfectoid algebra is an adic space

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.