Macaulayfication of Noetherian schemes (Q2037860)

A scheme \(X\) is taid to be \textit{CM-quasi-excellent} if it is locally Noetherian and: (1) the formal fibers of the local rings of \(X\) are Cohen-Macaulay; (2) every integral, closed subscheme \(X'\subset X\) has a non-empty, Cohen-Macaulay open subscheme. A CM-quasi-excellent \(X\) is \textit{CM-excellent} if in addition it is universally catenary. \medskip For each coherent \({\mathcal O}_X\)-module \({\mathcal M}\), denote by CM\((M)\) the subset of points of \(X\) where the stack of \({\mathcal M}\) is Cohen-Macaulay, and by \(U_{(S_2)}({\mathcal M})\) the subset of points of \(X\) where the stack of \({\mathcal M}\) is \(S_2\) (both can be shown to be open if, for instance, \(X\) is CM-quasi-excellent). The following main result is proven: Theorem. For every CM-excellent, Noetherian scheme \(X\) equiped with finitely many cocherent \({\mathcal O}_X\)-modules \({\mathcal M}\) with \(|\text{Supp}({\mathcal M})|=|X|\), there are a composition \[\widetilde{X}: \mathrm{Bl}_Z(X')\to X' \stackrel{\pi'}{\to} X\] and for each \(\mathcal M\) a coherent \(\mathcal O_{X'}\)-module \(\mathcal M'\) for which \(|\mathrm{Supp}(\mathcal M')|=|X'|\) such that \(\widetilde{X}\) is Cohen-Macaulay, its coherent modules \(\mathrm{Bl}_Z({\mathcal M}')\) are also all Cohen-Macaulay, and (1) \(X'\) is CM-excellent and locally equidimensional; (2) \(\pi'\) is finite, birational, and is an isomorphism over the open \[U:=U_{(S_2)}(X)\cap \left(\bigcap_{\mathcal M}U_{(S_2)}({\mathcal M})\right)\subset X\] that is dense in both \(X\) and \(X'\) and for which \({\mathcal M}'|_U\simeq {\mathcal M}|_{U}\); (3) \(Z\subset X'\) is a closed subscheme that is disjoint from the dense open \[U':=CM(X')\cap \left( \bigcap_{\mathcal M}CM({\mathcal M'}) \right);\] (4) \(U'\) is also dense in \(\mathrm{Bl}_Z(X')\), so that, in particular, the map \(\widetilde{X}\to X\) is birational. If \(X\) itself is CM-excellent and locally equidimensional, then we may choose \[X'=X \ \ \text{ and } \ \ \mathcal M'=\mathcal M.\] This main result in particular says that for every CM-quasi-excellent, Noetherian scheme \(X\) there are a Cohen-Macaulay scheme \(\widetilde{X}\) and a birational, projective morphism \[\pi: \widetilde{X}\to X\] that is an isomorphism over the Cohen-Macaulay locus \(CM(X)\subset X\). The following corollaries follow: Corollary 1. For every integral Dedekind scheme \(S\) with the function field \(K\) and every proper, Cohen-Macaulay \(K\)-scheme \(X\), there is a proper, flat, Cohen-Macaulay \(S\)-scheme \(\mathcal X\) with \(\mathcal X_K\simeq X\). If \(X\) is projective over \(K\), then one may choose \(\mathcal X\) to be projective over \(S\). Corollary 2. For a Noetherian, Cohen-Macaulay scheme \(X\) and a closed subscheme \(Z\subset X\), there are a Cohen-Macaulay scheme \(\widetilde{X}\) and a projective morphism \(\widetilde{X}\to X\) such that the (scheme-theoretic) preimage of \(Z\) in \(\widetilde{X}\) is divisor and \(\widetilde{X}\to X\) is an isomorphism over the maximal open subscheme \(U\subset X\) on which \(Z\) is already a divisor. Corollary 3. For every CM-quasi-excellent Noetherian scheme \(S\) and every finite type, separated \(S\)-scheme \(X\) that is Cohen-Macaulay, there is an open \(S\)-immersion \(X\hookrightarrow \overline{X}\) into a proper \(S\)-scheme \(\overline{X}\) that is Cohen-Macaulay such that \(\overline{X}\setminus X\) is a (possibly nonreduced) divisor in \(X\).