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Artinianness of local cohomology modules of ZD-modules. (English) Zbl 1090.13012

Let \(R\) denote a noetherian ring. An \(R\)-module \(M\) is called a ZD-module whenever for any submodule \(N\) of \(M\) the set of zero divisors of \(M/N\) is a union of finitely many prime ideals of \(\text{Ass}_R M/N.\) The authors’ intention is to extend the classical finiteness results for the local cohomology of finitely generated \(R\)-modules to ZD-modules. In particular the following is shown:
Let \(\mathfrak a\) denote an ideal of \(R\) and let \(M\) be a ZD-module. Suppose that the \(\mathfrak a\)-relative Goldie dimension of any quotient is finite. Then:
(a) \(H^i_{\mathfrak a}(M)\) is Artinian for all \(i \in \mathbb Z,\) provided \(\dim R/\mathfrak a = 0.\)
(b) \(H^d_{\mathfrak a}(M)\) is Artinian, whenever \(d = \dim M\) is finite.

MSC:

13D45 Local cohomology and commutative rings
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
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References:

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