Flat Cohomological Purity for Syntomic Schemes over Valuation Rings

Abstract: Grothendieck's cohomological purity predicts that the cohomology of a scheme is insensitive to removing a closed subscheme of sufficiently high codimension. In this article, we establish a form of flat cohomological purity over arbitrary (possibly infinite-rank) mixed-characteristic valuation rings $V$, thereby extending the theorem of Česnavičius--Scholze to the non-noetherian setting. More precisely, for a flat finite-type scheme over $V$ with local complete intersection fibres, we prove that the cohomology with coefficients in a commutative finite locally free group scheme remains unchanged after removing a closed subscheme satisfying a suitable fibrewise codimension condition; in particular, we obtain vanishing in low degrees and injectivity in the critical degree. As applications, we deduce purity results for local cohomology, for torsion in the Picard group, and for the Brauer group. In higher rank, our results yield sharper bounds than those previously obtained by Bhatt--Lurie and Madapusi--Mondal. The argument rests on recent advances in the structure theory of valuation rings.