LAUSR

AbstractLet (R,????)$(R,\mathfrak {m})$ 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 HIi(M)$\mathrm {H}^{i}_{I}(M)$, i ≥ 0. In particular, we show that if R is catenary, M an equidimensional R-module of dimension d, and x1,x2,…,xt$x_{1},x_{2},\dots ,x_{t}$ is an I-filter regular sequence on M, then (0:HId−j(M〈x1,x2,…,xi−1〉M)xi)$(0:_{\mathrm {H}^{d-j}_{I}(\frac {M}{\langle x_{1},x_{2},\dots ,x_{i-1}\rangle M})} x_{i})$ is I-cofinite for all i=1,2,…,t$i = 1,2,\dots ,t$ and all i ≤ j ≤ t if and only if HId−j(M〈x1,x2,…,xi−1〉M)$\mathrm {H}^{d-j}_{I}(\frac {M}{\langle x_{1},x_{2},\dots ,x_{i-1}\rangle M})$ is I-cofinite for all i=1,2,…,t$i = 1,2,\dots ,t$ 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 HIW(I,M)(M)IHIW(I,M)(M)$\frac {\mathrm {H}^{\mathcal {W}(I,M)}_{I}(M)}{I\mathrm {H}^{\mathcal {W}(I,M)}_{I}(M)}$ has finite support, where W(I,M):=Max{i:HIi(M)is not weakly Laskerian}.$$\mathcal{W}(I,M) := \text{Max} \{i : \mathrm{H}^{i}_{I}(M) \text{~is not weakly Laskerian}\}. $$