arXiv subjects

Linquan Ma

Publications and source records attributed to Linquan Ma.

58 records · Page 4Linked to original sources

arXiv2015

F-injectivity and Buchsbaum singularities

Let (R,m) be a local ring that contains a field. We show that, when R has equal characteristic p>0 and when H_m^i(R) has finite length for all i<dimR, then R is F-injective if and only if every ideal generated by a system of parameters is Frobenius closed. As a corollary, we show that such an R is in fact a Buchsbaum ring. This answers positively a question of S. Takagi that F-injective singularities with isolated non-Cohen-Macaulay locus are Buchsbaum. We also study the characteristic 0 analogue of this question and we show that Du Bois singularities with isolated non-Cohen-Macaulay locus are Buchsbaum in the graded case.

math.AC↗

arXiv2013

A sufficient condition for F-purity

It is well known that nice conditions on the canonical module of a local ring have a strong impact in the study of strong F-regularity and F-purity. In this note, we prove that if (R,m) is an equidimensional and S_2 local ring that admits a canonical ideal I such that R/I is F-pure, then R is F-pure. We also provide examples to show that not all Cohen-Macaulay F-pure local rings satisfy this property.

math.AC↗

arXiv2013

Finiteness properties of local cohomology for F-pure local rings

In this paper, we show that for an $F$-pure local ring $(R,\m)$, all local cohomology modules $H_{\m}^i(R)$ have finitely many Frobenius compatible submodules. This answers positively an open question raised by F.Enescu and M.Hochster. We also prove that if $(R,\m)$ is excellent and is $F$-pure on the punctured spectrum, then all local cohomology modules have finite length in the category of $R$-modules with Frobenius action. Finally, we show that the property that all $H_{\m}^(R)$ have finitely many Frobenius compatible submodules passes to localizations.

math.AC↗

arXiv2013

The category of F-modules has finite global dimension

Let R be a regular ring of characteristic p. Hochster showed that the category of Lyubeznik's F-modules has enough injectives, so that every F-module has an injective resolution in this category. We show that under mild conditions on R, for example when R is essentially of finite type over an F-finite regular local ring, the category of F-modules has finite global dimension d+1 where d=dimR. We also give examples to show that for F-finite F-modules, $\Ext_{F_R}^1(M,N)$ need not be finite.

math.AC↗