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Limit Koszul cohomology and torsion theory. (Russian. English summary) Zbl 0977.13007

Let \(A\) be a commutative ring with the unit element. A category \(K\) is considered, whose set of objects \(\Omega\) consists of some finite sequences \(x=(x_1, \dots, x_n)\) of elements of \(A\) (i.e., \(\Omega \subseteq L(A)\), where \(L(A)\) is the set of all finite sequences of elements of \(A\)) and the set of morphisms \(H(x,y)\) for two elements \(x=(x_1, \dots, x_n)\) and \(y=(y_1, \dots, y_m)\) consists of some \(m\times n\) matrices \(H\) such that \(Hx^t=y^t\) (where \((\cdot)^t\) means matrix transposition), the following conditions being fulfilled:
(1) \(x\in\Omega \& y\in\Omega \Rightarrow \exists z(z\in\Omega \& H(x,z) \neq \emptyset \& H(y,z)\neq \emptyset))\) (2) \(x\in\Omega \& \alpha= (\alpha_1, \dots, \alpha_n)\) (where \(\alpha_1\) are positive integers) \(\Rightarrow x^\alpha= (x_1^{a_1}, \dots, x_n^{a_n}) \in\Omega \&\) \(\text{diag} (x_1^{a_1-1}, \dots, x_n^{a_n-1}) \in H(x,x^\alpha)\), (where diag means the diagonal matrix).
One has here a generalization of notions of triangular sets [R. Y. Sharp and H. Zakerhi, Mathematika 29, 32-41 (1982; Zbl 0497.13006)] and rectangular sets [E. S. Golod, Vestn. Mosk. Univ. Ser. I 7-13 (1986; Zbl 0614.13007)]. (The author calls \(K\) a filter set and a category of filter sets simultaneously.) For \(A\)-modules \(M,N\) and \(x\in \Omega\), denote by \(K_\bullet (x,M)\) the corresponding Koszul complex, \(K^\bullet (x,M,N) = \text{Hom}_A (K_\bullet (x,M),N)\) and \(H^i(x,M,N)\) the corresponding cohomology modules.
The limit cohomology functors \(H^i(\Omega,M,N)\) are introduced and studied. With their help, a generalization of Golod’s result on local cohomology functors (loc. cit.) is obtained, as well as a new characterization of the Herzog-Bijan-Zadeh generalized local cohomology functors [M. Bijan-Zadeh, Glasg. Math. J. 21, 173-181 (1980; Zbl 0438.13009); J. London Math. Soc. II. Ser. 19, 402-410 (1979; Zbl 0404.13010)], connected, in its turn, with the torsion theory.

MSC:

13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13D30 Torsion theory for commutative rings
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
13D45 Local cohomology and commutative rings

Keywords:

Koszul complex
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