Macaulayfication of Noetherian schemes

Abstract

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 $\mathrm{CM}(X)\subset X$, where X is already Cohen–Macaulay. We extend Kawasaki’s methods to show that every quasi-excellent, Noetherian scheme X has a Cohen–Macaulay $\tilde{X}$ with a proper map $\tilde{X}\to X$ that is an isomorphism over $\mathrm{CM}(X)$. 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.