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Attached primes of local cohomology modules of complexes. (English) Zbl 1521.13027

In the paper under review, the author describes the attached primes of top local cohomology modules in derived categories. More precisely, let \((R, \mathfrak{m})\) be a local ring, \(\mathcal{S}\) a specialization closed subset and \(X\not\simeq 0\) an \(R\)-complex with finitely generated and bounded homology and finite dimension. Assume that \(H^d_{\mathcal{S}}(X)\neq 0\). Then \[\mathrm{Att}_RH^d_{\mathcal{S}}(X)=\{\mathfrak{p}\in \mathrm{Supp}_R(X)|cd(\mathcal{S}, R/\mathfrak{p})-inf X_{\mathfrak{p}}=d\}.\]
In addition, the author gives a generalization of the Lichtenbaum-Hartshorne vanishing theorem for complexes of \(R\)-modules.

MSC:

13D45 Local cohomology and commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13D09 Derived categories and commutative rings
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References:

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