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Attached primes and Artinian modules. (English) Zbl 1404.13017
Let \((R,\mathfrak{m})\) be a commutative Noetherian local ring with identity and \(I\) an ideal of \(R\).
Assume that \(A\) is a non-finitely generated Artinian \(R\)-module and let \(\mathcal{S}\) denote the set of all prime ideals \(\mathfrak{p}\in \text{Spec} R\) such that \(A\) is not \(\mathfrak{p}\)-cofinite and \(\dim R/\mathfrak{p}=1\). The first main result of this paper asserts that \[ \mathrm{Rad}(\text{Ann}_RA)=\bigcap_{\mathfrak{p}\in \mathcal{S}} \mathfrak{p}, \] provided \(R\) is \(\mathfrak{m}\)-adically complete. Recall that an \(R\)-module \(X\) is called \(I\)-cofinite if \(\text{Supp}_RX\subseteq \text{V}(I)\) and the \(R\)-module \(\text{Ext}_R^i(R/I,X)\) is finitely generated for all \(i\in \mathbb{N}_0\).
Let \(N\) be a \(d\)-dimensional finitely generated \(R\)-module. The second main result of the paper deals with the attached prime ideals of the local cohomology modules \[ \text{H}_{\mathfrak m}^i(N):=\varinjlim_n \;\text{Ext}^i_R(R/\mathfrak{m}^n,N);\;\;i\in \mathbb{N}_0. \] Suppose that \(R\) is a homomorphic image of a Gorenstein local ring. For every integer \(t=0, \dots, d\), the authors show that \[ \{\mathfrak{p}\in \text{Att}_R(\text{H}_{\mathfrak m}^t(N)) \mid \dim R/\mathfrak {p}=t\}= \{\mathfrak {p}\in \text{Ass}_RN\mid \dim R/\mathfrak {p}=t\}. \] We remind that for an Artinian \(R\)-module \(A\), the set of attached prime ideals of \(A\) is defined by \(\text{Att}_RA:=\{\mathfrak p\in \text{Spec}R \mid \mathfrak p=\text{Ann}_RL \text{ for some quotient } L \text{ of } A \}.\)
MSC:
13D45 Local cohomology and commutative rings
13E05 Commutative Noetherian rings and modules
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Full Text: Euclid