Finite quotients of sousperfectoid adic spaces need not be adic

Abstract

For every prime $p$ and every perfectoid field $K$ of characteristic $p$, we construct a geometrically normal sousperfectoid affinoid adic space with a $C_p$-action whose quotient in the category of $v$-ringed spaces is not adic. The source is stably uniform and sheafy, while a cofinal sequence of rational neighborhoods at a fixed point acquires new degree-one invariant sections that do not descend by completed rational localization. Consequently, finite quotient stability for affinoid perfectoid spaces does not extend to sousperfectoid spaces.

12 pages. Revised the introduction to clarify the historical context and the relation to Hansen's 2016 quotient question and Theorem 1.2; results and proofs unchanged