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Minimaxness and cofiniteness properties of local cohomology modules. (English) Zbl 1273.13025
Let \(R\) be a Noetherian commutative ring with nonzero identity. Let \(I\) be an ideal of \(R\) and \(M\) a finitely generated \(R\)-module. This interesting paper investigates the finiteness properties of the local cohomology modules \(H_I^i(M)\).
Recall that the finiteness dimension of \(M\) with respect to \(I\) is defined by \[ f_I(M):= \inf \{i\in \mathbb{N}_0|H_I^i(M) \text{ is not finitely generated} \}. \;\;(*) \] It is easy to verify that \(f_I(M)=\inf \{f_{IR_{\mathfrak p}}(M_{\mathfrak p})|{\mathfrak p} \in \mathrm{Supp}_R(M/IM)\}.\) This leads the authors to define nth finiteness dimension of \(M\) with respect to \(I\) by \[ f^n_I(M):=\inf \{f_{IR_{\mathfrak p}}(M_{\mathfrak p})|{\mathfrak p}\in \mathrm{Supp}_R(M/IM) \text{ and } \dim R/{\mathfrak p}\geq n\}, \] for any nonnegative integer \(n.\) So, \(f^0_I(M)\) is nothing else but \(f_I(M)\).
The authors establish a description similar to \((*)\) for each of \(f^1_I(M)\) and \(f^2_I(M)\). To illustrate their related results, we first recall the notions of minimax modules and weakly Laskerian modules. An \(R\)-module \(X\) is said to be minimax if it has a finitely generated submodule \(Y\) such that the quotient module \(X/Y\) is Artinian. Also, an \(R\)-module \(X\) is called weakly Laskerian if each quotient module of \(X\) has finitely many associated prime ideals. They authors show that \(f^1_I(M)=\inf \{i\in \mathbb{N}_0|H_I^i(M) \text{ is not minimax} \}\) and in the case \(R\) is similocal, they prove that \(f^2_I(M)=\inf \{i\in \mathbb{N}_0|H_I^i(M) \text{ is not weakly Laskerian} \}\).

MSC:
13D45 Local cohomology and commutative rings
14B15 Local cohomology and algebraic geometry
13E05 Commutative Noetherian rings and modules
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