Cohen-Macaulayness of absolute integral closures

Abstract

We prove that, modulo any power of a prime $p$, the absolute integral closure of an excellent noetherian domain is Cohen-Macaulay. A graded analog is also established, yielding variants of Kodaira vanishing "up to finite covers" in mixed characteristic. Our main tools are (log) prismatic cohomology (which yields a Frobenius action in mixed characteristic) and the $p$-adic Riemann-Hilbert functor for constructible étale ${\mathbf{F}}_p$-sheaves on varieties over a $p$-adic field (which almost controls perfectified prismatic cohomology).

v2: minor updates

References in corpus (2)

• Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic
• Minimal model program for semi-stable threefolds in mixed characteristic

Cited by in corpus (7)

• Globally +-regular varieties and the minimal model program for threefolds in mixed characteristic
• Minimal model program for semi-stable threefolds in mixed characteristic
• An analog of adjoint ideals and PLT singularities in mixed characteristic
• Coherence of absolute integral closures
• The Second Vanishing Theorem for Local Cohomology Modules
• Local cohomology -- an invitation
• On local rings without small Cohen-Macaulay algebras in mixed characteristic