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Cominimaxness of local cohomology modules. (English) Zbl 07088770
Summary: Let $$R$$ be a commutative Noetherian ring, $$I$$ an ideal of $$R$$. Let $$t\in\mathbb{N}_0$$ be an integer and $$M$$ an $$R$$-module such that $$\mathrm{Ext}^i_R(R/I,M)$$ is minimax for all $$i\leq t+1$$. We prove that if $$H^i_I(M)$$ is $$\mathrm{FD}_{\leq 1}$$ (or weakly Laskerian) for all $$i<t$$, then the $$R$$-modules $$H^i_I(M)$$ are $$I$$-cominimax for all $$i<t$$ and $$\mathrm{Ext}^i_R(R/I,H^t_I(M))$$ is minimax for $$i=0,1$$. Let $$N$$ be a finitely generated $$R$$-module. We prove that $$\mathrm{Ext}^j_R(N,H^i_I(M))$$ and $$\mathrm{Tor}^R_j(N,H^i_I(M))$$ are $$I$$-cominimax for all $$i$$ and $$j$$ whenever $$M$$ is minimax and $$H^i_I(M)$$ is $$\mathrm{FD}_{\leq 1}$$ (or weakly Laskerian) for all $$i$$.

##### 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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