A Splitting Theorem for Local Cohomology and its Applications

TL;DR

This paper proves a splitting theorem for local cohomology modules over Noetherian rings, providing new insights and simplified proofs for existing results on the structure and finiteness properties of local cohomology.

Contribution

It introduces a splitting theorem for local cohomology modules under certain finiteness conditions, generalizing previous results with more straightforward proofs.

Findings

Established a splitting isomorphism for local cohomology modules with respect to filter regular elements.

Generalized finiteness results for associated primes of local cohomology.

Provided simplified proofs of known theorems in local cohomology theory.

Abstract

Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. We show in this paper that, for an integer $t$, if the local cohomology module $H^{i}_\mathfrak{a}(M)$ with respect to an ideal $\frak a$ is finitely generated for all $i<t$, then $$H^{i}_\mathfrak{a}(M/xM)\cong H^{i}_\mathfrak{a}(M)\oplus H^{i+1}_\mathfrak{a}(M)$ for all $\frak a$-filter regular elements $x$ containing in a enough large power of $\frak a$ and all $i<t-1$. As consequences we obtain generalizations, by very short proofs, of the main results of M. Brodmann and A.L. Faghani (A finiteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128(2000), 2851-2853) and of H.L. Truong and the first author (Asymptotic behavior of parameter ideals in generalized Cohen-Macaulay module, J. Algebra, 320(2008),158-168).