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.