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arXiv · 2008.08070

Cohen-Macaulayness of absolute integral closures

Bhargav Bhatt
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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 \'etale $\mathbf{F}_p$-sheaves on varieties over a $p$-adic field (which almost controls perfectified prismatic cohomology).

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Bhargav Bhatt. 2020-08-18. Cohen-Macaulayness of absolute integral closures. https://arxiv.org/abs/2008.08070

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