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 , where X is already Cohen–Macaulay. We extend Kawasaki’s methods to show that every quasi-excellent, Noetherian scheme X has a Cohen–Macaulay with a proper map that is an isomorphism over . 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.
"Macaulayfication of Noetherian schemes." Duke Math. J. 170 (7) 1419 - 1455, 15 May 2021. https://doi.org/10.1215/00127094-2020-0063