Local cohomology and pure morphisms

Strong $F$-regularity and generating morphisms of local cohomology modules

We establish a criterion for the strong $F$-regularity of a (non-Gorenstein) Cohen-Macaulay reduced complete local ring of dimension at least $2$, containing a perfect field of prime characteristic $p$. We also describe an explicit generating morphism (in the sense of Lyubeznik) for the top local cohomology module with support in certain ideals arising from an $n\times (n-1)$ matrix $X$ of indeterminates. For $p\geq 5$, these results led us to derive a simple, new proof of the well-known fact that the generic determinantal ring defined by the maximal minors of $X$ is strongly $F$-regular.

Powers of ideals and the cohomology of stalks and fibers of morphisms

We first provide here a very short proof of a refinement of a theorem of Kodiyalam and Cutkosky, Herzog and Trung on the regularity of powers of ideals. This result implies a conjecture of Hà and generalizes a result of Eisenbud and Harris concerning the case of ideals primary for the graded maximal ideal in a standard graded algebra over a field. It also implies a new result on the regularities of powers of ideal sheaves. We then compare the cohomology of the stalks and the cohomology of the fibers of a projective morphism to the effect of comparing the maximum over fibers and over stalks of …

Comparison of relative cohomology theories with respect to semidualizing modules

We compare and contrast various relative cohomology theories that arise from resolutions involving semidualizing modules. We prove a general balance result for relative cohomology over a Cohen-Macaulay ring with a dualizing module, and we demonstrate the failure of the naive version of balance one might expect for these functors. We prove that the natural comparison morphisms between relative cohomology modules are isomorphisms in several cases, and we provide a Yoneda-type description of the first relative Ext functor. Finally, we show by example that each distinct relative cohomology constru…

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.

等维映射到分裂环是纯的

Equidimensional morphisms onto splinters are pure

论文证明了Noetherian环是分裂环当且仅当每个等维满射映射的像环映射是纯的，揭示了大量自动纯的环映射。同时，证明了局部Noetherian方案局部为分裂环当且仅当每个局部等维映射是强纯的。

纯态射下伴随理想的行为研究

On the behavior of adjoint ideals under pure morphisms

该研究通过超积刻画伴随理想层，分析其在纯态射下的行为，证明了满足特定条件时簇对的plt型性质可沿纯态射逆向传递，正面解答了Z. Zhuang提出的相关问题。