Local cohomology: Associated primes, artinianness and asymptotic behaviour

TL;DR

This paper investigates the finiteness, associated primes, artinianness, and asymptotic properties of local cohomology modules over noetherian rings, focusing on the natural maps between Ext and local cohomology and their kernels and cokernels.

Contribution

It provides new insights into the finiteness and asymptotic behavior of kernels and cokernels of natural maps involving local cohomology modules, extending understanding of their structure.

Findings

Finiteness properties of kernels and cokernels of the natural map are established.

Results on associated primes and artinianness of local cohomology modules are derived.

Asymptotic behavior of these modules is analyzed in the graded case.

Abstract

Let $R$ be a noetherian ring, $\fa$ an ideal of $R$, $M$ an $R$--module and $n$ a non-negative integer. In this paper we first will study the finiteness properties of the kernel and the cokernel of the natural map $f:\Ext^n_{R}(R/\fa,M)\lo \Hom_{R}(R/\fa,\lc^{n}_{\fa}(M))$. Then we will get some corollaries about the associated primes and artinianness of local cohomology modules. Finally we will study the asymptotic behaviour of the kernel and the cokernel of this natural map in the graded case.