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Some results on the local cohomology of minimax modules. (English) Zbl 1340.13009
Let $$R$$ be a commutative Noetherian ring with identity. Let $$I$$ be an ideal of $$R$$ and $$X$$ an $$R$$-module. This paper studies the finiteness properties of the local cohomology modules $H_I^i(X):=\underset {n} {\varinjlim}\text{Ext}^i_R\left(\frac {R}{I^n},X\right);\;\;i\in \mathbb {N}_0.$ Recall that $$X$$ is said to be minimax if it has a finitely generated submodule $$Y$$ such that $$X/Y$$ is an Artinian $$R$$-module. Also, the $$R$$-module $$X$$ is said to be weakly Laskerian if each quotient module of $$X$$ has finitely many associated prime ideals.
Let $$M$$ be a minimax $$R$$-module such that $$\dim_R \text{Supp}_R(H^i_I(M))\leq 1$$ for all integers $$i$$. The authors show that $$\text{Ext}_R^j(R/I, H^i_I(M))$$ is minimax for all integers $$i$$ and $$j$$.
Assume that $$R$$ is local and $$t$$ is a non-negative integer. Let $$N$$ be a weakly Laskerian $$R$$-module such that $$\dim_R \text{Supp}_R(H^i_I(N)) \leq 2$$ for all $$i<t$$. The authors prove that $$\text{Ext}_R^j(R/I,H^i_I(N))$$ and $$\text{Hom}_R(R/I,H^t_I(N))$$ are weakly Laskerian for all $$i<t$$ and all $$j\geq 0$$.

##### MSC:
 13D45 Local cohomology and commutative rings 13E10 Commutative Artinian rings and modules, finite-dimensional algebras 13C05 Structure, classification theorems for modules and ideals in commutative rings
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