Semi-log canonical vs \(F\)-pure singularities
Let \(R\) be a reduced ring of characteristic \(p>0\) and let \(R^N\) denote its normalization. It is known that each \(R\)-linear map \(\phi:F^e_*R\to R\) admits a unique extension \(\widetilde{\phi}:F^e_*R^N\to R^N\). The paper under review investigates an interesting question:NEWLINENEWLINEQuestion. Let \(R\), \(R^N\), \(\phi\) and \(\widetilde{\phi}\) be as above. Assume that \(\widetilde{\phi}\) is surjective, is \(\phi\) also surjective?NEWLINENEWLINEIt has been known that, without any further assumptions on \(R\), this question has a negative answer. The paper under review discovers a class of rings, called rings with \textit{hereditary surjective trace}, where this question has a positive answer. A ring with \textit{hereditary surjective trace} is defined recursively as follows (note that this definition is characteristic-free).NEWLINENEWLINEDefinition. A reduced ring \(A\) is said to have \textit{hereditary surjective trace} if there exist minimal associated prim ideals of the conductor ideal \(\mathfrak{c}\), \(\mathfrak{p}\in \mathrm{Spec}(A)\) and \(\mathfrak{q}\in \mathrm{Spec}(A^N)\) such that \(A\cap \mathfrak{q}=\mathfrak{p}\) satisfyingNEWLINENEWLINE (1). \(Tr:(A^N/\mathfrak{q})^N\to (A/\mathfrak{p})^N\) is surjective;NEWLINENEWLINE (2). \(A/\mathfrak{p}\) has hereditary surjective trace.NEWLINENEWLINEThe main theorem of the paper under review is:NEWLINENEWLINEMain Theorem. Let \(R\) be a reduced \(F\)-finite ring of characteristic \(p>0\) that has hereditary surjective trace. Assume that \(\phi:F^e_*R\to R\) is an \(R\)-linear map and that its extension \(\widetilde{\phi}:F^e_*R^N\to R^N\) is surjective. Then \(\phi\) is also surjective.NEWLINENEWLINEOne of the corollaries of the main theorem is an analogue of a well-known criterion for semi-log canonicity:NEWLINENEWLINECorollary. Suppose that \(X=\text{Spec}(R)\) is an affine \(F\)-finite scheme satisfying hereditary surjective trace and which is also \(S_2\), \(G_1\) and \(\mathbb{Z}_{(p)}\)-Gorenstein. Set \(X^N\) to be the normalization of \(X\) and \(B\) to be the divisor on \(X^N\) corresponding to the conductor ideal. Then \(X\) is \(F\)-pure if and only \((X^N,B)\) is \(F\)-pure.NEWLINENEWLINEA result on inversion of adjunction for schemes with hereditary surjective trace is also obtained.
scientific article; zbMATH DE number 6037571
scientific article