Flat Cohomological Purity for Syntomic Schemes over Valuation Rings

TL;DR

This paper extends Grothendieck's cohomological purity to flat schemes over arbitrary valuation rings, establishing invariance of cohomology under certain closed subscheme removals in a non-noetherian setting.

Contribution

It proves a form of flat cohomological purity over general valuation rings, extending previous results to non-noetherian and mixed-characteristic cases.

Findings

Cohomology remains unchanged after removing certain closed subschemes.

Vanishing in low degrees and injectivity in the critical degree are established.

Sharper bounds in higher rank than previous works by Bhatt--Lurie and Madapusi--Mondal.

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 \v{C}esnavi\v{c}ius--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.