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Vanishing of the top local cohomology modules over Noetherian rings

Authors:

Kamran Divaani-Aazar

Affiliation:

(1) Department of Mathematics, Az-Zahra University, Vanak, Post Code 19834, Tehran, Iran

Abstract:

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let $$
mathfrak{a}
$$

be an ideal of R and $$
mathfrak{M}
$$

denote the intersection of all prime ideals $mathfrak{p} in Supp_R H_mathfrak{a}^d (M)$ . It is shown that

$H_mathfrak{a}^d (M) simeq H_mathfrak{M}^d (M)/sumlimits_{n in mathbb{N}} {langle mathfrak{M}rangle } (0:_{H_mathfrak{M}^d (M)} mathfrak{a}^n ),$

where for an Artinian R-module A we put $langle mathfrak{M}rangle A = cap _{n in mathbb{N}} mathfrak{M}^n$ A. As a consequence, it is proved that for all ideals $$
mathfrak{a}
$$

of R, there are only finitely many non-isomorphic top local cohomology modules $H_mathfrak{a}^d (M)$ having the same support. In addition, we establish an analogue of the Lichtenbaum-Hartshorne vanishing theorem over rings that need not be local.

Keywords:

Artinian modules attached prime ideals cohomological dimension formally isolated local cohomology secondary representations

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