Cofiniteness of Local Cohomology Modules over Homomorphic Image of Cohen-Macaulay Rings

Abstract

Let $(R,\mathfrak {m})$ be a Noetherian local ring, M a non-zero finitely generated R-module, and let I be an ideal of R. In this paper, we establish some new properties of local cohomology modules $\mathrm {H}^{i}_{I}(M)$, i ≥ 0. In particular, we show that if R is catenary, M an equidimensional R-module of dimension d, and $x_{1},x_{2},\dots ,x_{t}$ is an I-filter regular sequence on M, then $(0:_{\mathrm {H}^{d-j}_{I}(\frac {M}{\langle x_{1},x_{2},\dots ,x_{i-1}\rangle M})} x_{i})$ is I-cofinite for all $i = 1,2,\dots ,t$ and all i ≤ j ≤ t if and only if $\mathrm {H}^{d-j}_{I}(\frac {M}{\langle x_{1},x_{2},\dots ,x_{i-1}\rangle M})$ is I-cofinite for all $i = 1,2,\dots ,t$ and all i ≤ j ≤ t. Also we study the cofiniteness of local cohomology modules over homomorphic image of Cohen-Macaulay rings and we show that $\frac {\mathrm {H}^{\mathcal {W}(I,M)}_{I}(M)}{I\mathrm {H}^{\mathcal {W}(I,M)}_{I}(M)}$ has finite support, where

$$\mathcal{W}(I,M) := \text{Max} \{i : \mathrm{H}^{i}_{I}(M) \text{~is not weakly Laskerian}\}. $$