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A generalization of the cofiniteness problem in local cohomology modules. (English) Zbl 1096.13522
From the paper: $$R$$ denotes a commutative noetherian ring with non-zero identity and $$M$$ is a finitely generated $$R$$-module.
Definition. Let $$\Phi$$ be a system of ideals of $$R$$. The general local cohomology module $$H^j_\Phi(M)$$ is defined to be $$\Phi$$-cofinite if there exists an ideal $$I\in\Phi$$ such that $$\text{Ext}^i_R(R/I, H^j_\Phi(M))$$ is finitely generated for all $$i$$.
Definition. The sequence $$x_1,\dots, x_n$$ of elements of $$R$$ is called a $$d$$-sequence on $$M$$, for each $$i= 0,1,\dots, n-1$$, the equality $$(x_1,\dots, x_i)M:_M x_{i+1} x+k= (x_1,\dots, x_i)M:_M x_k$$ holds for all $$k\geq 1$$ it is an unconditioned strong $$d$$-sequence on $$M$$ if $$x^{\alpha_1}_1,\dots, x^{\alpha_n}_n$$ is a $$d$$-sequence in any order for all positive integers $$\alpha_1,\dots, \alpha_n$$.
The authors prove that if an ideal $$I$$ of $$R$$ is generated by a unconditioned strong $$d$$-sequence on $$M$$ then the locally cohomology module $$H^i_I(M)$$ is $$I$$-cofinite. Furthermore, for any system of ideals $$\Phi$$ of $$R$$, they study the cofiniteness problem in the context of general local cohomology modules.

##### MSC:
 13D45 Local cohomology and commutative rings 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
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