Macaulayfication of Noetherian schemes

TL;DR

This paper extends methods to construct Cohen-Macaulay modifications of Noetherian schemes, ensuring preservation of existing Cohen-Macaulay loci, thus advancing Faltings' program and implications for models over number fields.

Contribution

It generalizes Kawasaki's Cohen-Macaulay modifications to all quasi-excellent Noetherian schemes, preserving the Cohen-Macaulay locus and completing Faltings' program.

Findings

Constructed Cohen-Macaulay modifications for quasi-excellent schemes.

Ensured the modifications are isomorphisms over the Cohen-Macaulay locus.

Implications for models over number fields.

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.