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On the associated primes of local cohomology modules. (English) Zbl 0940.13013
The main theorem of this paper is: Let \(I\) be an ideal of a Noetherian ring \(A\), and let \(M\) be a finitely generated \(A\)-module. For a positive integer \(n\), the following conditions are equivalent:
(i) \(H^i_I (M)\) is finitely generated for \(i=1, \dots,n-1;\)
(ii) there exists an unconditioned strong \(d\)-sequence of length \(n\) on \(M\) which is an \(I\)-filter regular sequence on \(M\). \((x_1, \dots, x_n\) is an unconditioned strong \(d\)-sequence on \(M\) if \(x_1^{i_1}, \dots, x_n^{i_n}\) is a \(d\)-sequence in any order for all positive integers \(i_j\).)
An application to associated primes of \(H^i_I(M)\) is given.

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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References:
[1] DOI: 10.1007/BF01226087 · Zbl 0368.14004
[2] DOI: 10.1007/BF01450555 · Zbl 0451.13008
[3] Goto S., The theory of unconditioned strong d-sequences and modules of finite local cohomology
[4] Grothendieck A., Lecture Notes in Math 41 (1967)
[5] DOI: 10.1016/0001-8708(82)90045-7 · Zbl 0505.13004
[6] Huneke C., Free Resolutions in commutative Algebra and algebric Geometry pp 93– (1992)
[7] DOI: 10.2307/2154297 · Zbl 0785.13005
[8] DOI: 10.1080/00927879808826293 · Zbl 0909.13007
[9] Khashyarmanesh K., Nagoya Math. J 151 pp 37– (1998)
[10] Matsumura H., Commutative ring theory (1986) · Zbl 0603.13001
[11] Nagel U., Commutative algebra: Syzygies, multiplicities, and birational algebra pp 307– (1994)
[12] DOI: 10.1112/jlms/s2-28.3.417 · Zbl 0497.13007
[13] Raghavan K., Ph.D. Thesis (1991)
[14] DOI: 10.1017/S0305004100071164 · Zbl 0786.13005
[15] DOI: 10.1002/mana.19780850106 · Zbl 0398.13014
[16] Styückrad J., Buchsbaum rings and applications (1986)
[17] DOI: 10.1017/S0305004100060308 · Zbl 0509.13024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.