15 May 2021 Macaulayfication of Noetherian schemes
Kęstutis Česnavičius
Author Affiliations +
Duke Math. J. 170(7): 1419-1455 (15 May 2021). DOI: 10.1215/00127094-2020-0063


To reduce to resolving Cohen–Macaulay singularities, Faltings initiated the program of “Macaulayfying” a given Noetherian scheme X. For a wide class of X, Kawasaki built the sought-for Cohen–Macaulay modifications, with a crucial drawback that his blowups did not preserve the locus CM(X)X, where X is already Cohen–Macaulay. We extend Kawasaki’s methods to show that every quasi-excellent, Noetherian scheme X has a Cohen–Macaulay X˜ with a proper map X˜X that is an isomorphism over CM(X). This completes Faltings’s program, reduces the conjectural resolution of singularities to the Cohen–Macaulay case, and implies that every proper, smooth scheme over a number field has a proper, flat, Cohen–Macaulay model over the ring of integers.


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Kęstutis Česnavičius. "Macaulayfication of Noetherian schemes." Duke Math. J. 170 (7) 1419 - 1455, 15 May 2021. https://doi.org/10.1215/00127094-2020-0063


Received: 29 November 2018; Revised: 21 June 2020; Published: 15 May 2021
First available in Project Euclid: 23 December 2020

Digital Object Identifier: 10.1215/00127094-2020-0063

Primary: 14E15
Secondary: 13H10 , 14B05 , 14J17 , 14M05

Keywords: Cohen–Macaulay , excellence , Macaulayfication , resolution of singularities

Rights: Copyright © 2021 Duke University Press

Vol.170 • No. 7 • 15 May 2021
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