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 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 $\widetilde{X}$ with a proper map $\widetilde{X} \rightarrow X$ that is an isomorphism over $\mathrm{CM}(X)$. This completes Faltings' 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.