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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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