The finiteness dimension of local cohomology modules and its dual notion |
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Authors: | Mohsen Asgharzadeh Kamran Divaani-Aazar Massoud Tousi |
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Institution: | a Department of Mathematics, Shahid Beheshti University, Tehran, Iran b Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran c Department of Mathematics, Az-Zahra University, Vanak, Post Code 19834, Tehran, Iran |
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Abstract: | Let a be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. We explore the behavior of the two notions fa(M), the finiteness dimension of M with respect to a, and, its dual notion qa(M), the Artinianness dimension of M with respect to a. When (R,m) is local and r?fa(M) is less than , the m-finiteness dimension of M relative to a, we prove that is not Artinian, and so the filter depth of a on M does not exceed fa(M). Also, we show that if M has finite dimension and is Artinian for all i>t, where t is a given positive integer, then is Artinian. This immediately implies that if q?qa(M)>0, then is not finitely generated, and so fa(M)≤qa(M). |
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Keywords: | 13D45 13Exx |
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