• 제목, 요약, 키워드: cohomological dimension

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ON THE COHOMOLOGICAL DIMENSION OF FINITELY GENERATED MODULES

  • Bahmanpour, Kamal;Samani, Masoud Seidali
    • 대한수학회보
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    • v.55 no.1
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    • pp.311-317
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    • 2018
  • Let (R, m) be a commutative Noetherian local ring and I be an ideal of R. In this paper it is shown that if cd(I, R) = t > 0 and the R-module $Hom_R(R/I,H^t_I(R))$ is finitely generated, then $$t={\sup}\{{\dim}{\widehat{\hat{R}_p}}/Q:p{\in}V(I{\hat{R}}),\;Q{\in}mAss{_{\widehat{\hat{R}_p}}}{\widehat{\hat{R}_p}}\;and\;p{\widehat{\hat{R}_p}}=Rad(I{\wideha{\hat{R}_p}}=Q)\}$$. Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R{\subset}R[X]$ will be included.

A NOTE ON COHOMOLOGICAL DIMENSION OVER COHEN-MACAULAY RINGS

  • Bagheriyeh, Iraj;Bahmanpour, Kamal;Ghasemi, Ghader
    • 대한수학회보
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    • v.57 no.2
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    • pp.275-280
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    • 2020
  • Let (R, m) be a Noetherian local Cohen-Macaulay ring and I be a proper ideal of R. Assume that βR(I, R) denotes the constant value of depthR(R/In) for n ≫ 0. In this paper we introduce the new notion γR(I, R) and then we prove the following inequalities: βR(I, R) ≤ γR(I, R) ≤ dim R - cd(I, R) ≤ dim R/I. Also, some applications of these inequalities will be included.

EXACTNESS OF IDEAL TRANSFORMS AND ANNIHILATORS OF TOP LOCAL COHOMOLOGY MODULES

  • BAHMANPOUR, KAMAL
    • 대한수학회지
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    • v.52 no.6
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    • pp.1253-1270
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    • 2015
  • Let (R, m) be a commutative Noetherian local domain, M a non-zero finitely generated R-module of dimension n > 0 and I be an ideal of R. In this paper it is shown that if $x_1,{\ldots },x_t$ ($1{\leq}t{\leq}n$) be a sub-set of a system of parameters for M, then the R-module $H^t_{(x_1,{\ldots },x_t)}$(R) is faithful, i.e., Ann $H^t_{(x_1,{\ldots },x_t)}$(R) = 0. Also, it is shown that, if $H^i_I$ (R) = 0 for all i > dim R - dim R/I, then the R-module $H^{dimR-dimR/I}_I(R)$ is faithful. These results provide some partially affirmative answers to the Lynch's conjecture in [10]. Moreover, for an ideal I of an arbitrary Noetherian ring R, we calculate the annihilator of the top local cohomology module $H^1_I(M)$, when $H^i_I(M)=0$ for all integers i > 1. Also, for such ideals we show that the finitely generated R-algebra $D_I(R)$ is a flat R-algebra.

ON THE LOCAL COHOMOLOGY OF MINIMAX MODULES

  • Mafi, Amir
    • 대한수학회보
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    • v.48 no.6
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    • pp.1125-1128
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    • 2011
  • Let R be a commutative Noetherian ring, a an ideal of R, and M a minimax R-module. We prove that the local cohomology modules $H^j_a(M)$ are a-cominimax; that is, $Ext^i_R$(R/a, $H^j_a(M)$) is minimax for all i and j in the following cases: (a) dim R/a = 1; (b) cd(a) = 1, where cd is the cohomological dimension of a in R; (c) dim $R{\leq}2$. In these cases we also prove that the Bass numbers and the Betti numbers of $H^j_a(M)$ are finite.