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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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