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Extension functors of cominimax modules. (English) Zbl 1360.13044
Let $$R$$ be a commutative Noetherian ring with identity and $$I$$ an ideal of $$R$$. Recall that an $$R$$-module $$X$$ is called minimax if it admits a finitely generated submodule $$Y$$ such that the $$R$$-module $$X/Y$$ is Artinian. An $$R$$-module $$M$$ is called $$I$$-cominimax if $$\text{Supp}_RM\subseteq \text{V}(I)$$ and $$\text{Ext}_R^i(R/I,M)$$ is minimax for all $$i\geq 0$$. Finally, an $$R$$-module $$M$$ is called $$I$$-weakly cofinite if $$\text{Supp}_RM\subseteq \text{V}(I)$$ and for each $$i\geq 0$$, any quotient module of $$\text{Ext}_R^i(R/I,M)$$ has finitely many associated primes.
Let $$M$$ be an $$I$$-cominimax $$R$$-module and $$N$$ a finitely generated $$R$$-module. The authors prove that if either $$\dim_RM\leq 1$$, or $$\dim_RN\leq 2$$, then $$\text{Ext}_R^i(N,M)$$ is $$I$$-cominimax for all $$i\geq 0$$. As an application, they show that if $$R$$ is local and $$M$$ is a minimax $$R$$-module such that $$\dim_R(\text{H}_I^i(M))\leq 2$$ for all $$i\geq 0$$ , then $$\text{Ext}_R^j(N,\text{H}_I^i(M))$$ is $$I$$-weakly cofinite for all finitely generated $$R$$-modules $$N$$, and all $$i,j\geq 0$$.

##### 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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