On finite type epimorphisms of rings

On flat epimorphisms of rings and pointwise localizations

In this paper all rings are commutative. We prove some new results on flat epimorphisms of rings and pointwise localizations. Especially among them, it is proved that a ring $R$ is an absolutely flat (von-Neumann regular) ring if and only if it is isomorphic to the pointwise localization $R^{(-1)}R$, or equivalently, each $R-$algebra is $R-$flat. For a given minimal prime ideal $\mathfrak{p}$ of a ring $R$, the surjectivity of the canonical map $R\rightarrow R_{\mathfrak{p}}$ is characterized. Finally, we give a new proof to the fact that in a flat epimorphism of rings, the contraction-extensi…

Tilting modules arising from ring epimorphisms

Given a ring R, we investigate tilting modules of the form S \oplus S/R for some injective ring epimorphism R \to S. In particular, we are interested in tilting modules arising from Schofield's universal localization. For some rings, in this way one obtains a classification of all tilting modules.

Flat ring epimorphisms and universal localisations of commutative rings

We study different types of localisations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localisation in the sense of Cohn and Schofield; and (b) when such universal localisations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialisation closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localisations. Moreove…

Flat commutative ring epimorphisms of almost Krull dimension zero

We consider flat epimorphisms of commutative rings $R\to U$ such that, for every ideal $I\subset R$ for which $IU=U$, the quotient ring $R/I$ is semilocal of Krull dimension zero. Under these assumptions, we show that the projective dimension of the $R$-module $U$ does not exceed $1$. We also describe the Geigle-Lenzing perpendicular subcategory $U^{\perp_{0,1}}$ in $R\mathsf{-Mod}$. Assuming additionally that the ring $U$ and all the rings $R/I$ are perfect, we show that all flat $R$-modules are $U$-strongly flat. Thus we obtain a generalization of some results of the paper arXiv:1801.04820, …

局部上同调模的满射，一个一般化的Peskine-Szpiro定理，以及应用于增厚子方案上同调消失的应用

Epimorphisms of local cohomology modules, a general Peskine-Szpiro theorem, and an application to sheaf cohomology vanishing for thickenings

本文研究局部上同调模的满射性质，推广Peskine-Szpiro定理，并证明了增厚子方案上同调的Kodaira型消失定理。

Rees Algebras of Diagonal Ideals

There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the diagonal ideal, the kernel of the multiplication map. We prove that the diagonal ideal is of linear type and recover the defining ideal of the Rees algebra in some special cases. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety.