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Cohen-Macaulayness of absolute integral closures
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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).
Forward citations
Cited by 3 Pith papers
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The Brian\c{c}on-Skoda theorem for pseudo-rational and Du Bois singularities and uniformity in excellent rings
For pseudo-rational and many Du Bois singularities, the full Briançon–Skoda containment J^{n+k-1} ⊆ J^k holds, and quasi-excellent finite-dimensional rings satisfy uniform Briançon–Skoda and uniform Artin–Rees.
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Factoring maps to big Cohen-Macaulay algebras through blowups
Functorial balanced big Cohen-Macaulay algebra assignments factor through RΓ(Y, O_Y) for every proper birational map Y → Spec R.
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A lower bound on levels with applications to Koszul Complexes
For perfect complexes with power torsion homology, the paper proves level_R F ≥ dim R - dim R/I + 1 and shows the bound is optimal for Koszul complexes.
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