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ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES

Yıl 2019, , 55 - 63, 08.01.2019
https://doi.org/10.24330/ieja.504110

Öz

This paper is concerned to relationship between the sets of
associated primes of the $d$-local cohomology modules and the
ordinary local cohomology
 modules.  Let $R$ be a commutative Noetherian local ring, $M$ an
  $R$-module and $d, t$ two integers. We prove that
 ${\rm Ass}(H^t_d(M))=\bigcup_{I\in \Phi} {\rm Ass}(H^t_I(M))$ whenever $H^i_d(M)=0$ for all
 $i< t$ and $\Phi=\{I: I  \text{ is an ideal of}\  $R$
 \text{ with} \dim R/I\leq d \}$. We give some information about
 the non-vanishing of the $d$-local cohomology modules. To be more precise, we prove that
$H^i_d(R)\neq 0$ if and only if $i=n-d$ whenever  $R$ is a
Gorenstein ring of dimension $n$. This result leads to an example which shows that ${\rm Ass}(H^{n-d}_d(R))$
is not necessarily a finite set.

Kaynakça

  • J. Azami, R. Naghipour and B. Vakili, Finiteness properties of local cohomology modules for a-minimax modules, Proc. Amer. Math. Soc., 137(2) (2009), 439- 448.
  • C. Banica and M. Stoia, Singular sets of a module and local cohomology, Boll. Un. Mat. Ital. B, 16 (1976), 923-934.
  • M. H. Bijan-Zadeh, Torsion theories and local cohomology over commutative Noetherian rings, J. London Math. Soc., 19(3) (1979), 402-410.
  • K. B. Lorestani, P. Sahandi and S. Yassemi, Artinian local cohomology modules, Canad. Math. Bull., 50(4) (2007), 598-602.
  • M. P. Brodmann and A. L. Faghani, A niteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128(10) (2000), 2851-2853.
  • M. P. Brodmann and R. Y. Sharp, Local Cohomology: an Algebraic Intro- duction with Geometric Applications, Cambridge Studies in Advanced Math- ematics, 60, Cambridge University Press, Cambridge, 1998.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Ad- vanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
  • M. T. Dibaei and S. Yassemi, Associated primes and co niteness of local co- homology modules, Manuscripta Math., 117(2) (2005), 199-205.
  • K. Divaani-Aazar and A. Ma , Associated primes of local cohomology modules, Proc. Amer. Math. Soc., 133(3) (2005), 655-660.
  • C. Huneke, Problems on local cohomology. Free Resolution in Commutative Algebra and Algebraic Geometry. (Sundance, UT, 1990), Res. Notes Math., 2, Jones and Bartlett, Boston, MA, (1992), 93-108.
  • M. Katzman, An example of an in nite set of associated primes of local coho- mology module, J. Algebra, 252(1) (2002), 161-166.
  • H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1986.
  • R. Naghipour, Integral closures, local cohomology and ideal topologies, Rocky Mountain J. Math., 37(3) (2007), 905-916.
  • J. J. Rotman, Introduction to Homological Algebra, Pure and Applied Mathe- matics, 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.
  • R. Y. Sharp, Steps in Commutative Algebra, Second edition, London Math- ematical Society Student Texts, 51, Cambridge University Press, Cambridge, 2000.
  • A. K. Singh, p-Torsion elements in local cohomology modules, Math. Res. Lett., 7(2-3) (2000), 165-176.
  • J. Z. Xu, Minimal injective and at resolutions of modules over Gorenstein rings, J. Algebra, 175(2) (1995), 451-477.
  • N. Zamani, M. H. Bijan-Zadeh and M. S. Sayedsadeghi, d-Transform functor and some niteness and isomorphism results, Vietnam. J. Math., 42(2) (2014), 179-186.
  • N. Zamani, M. H. Bijan-Zadeh and M. S. Sayedsadeghi, Cohomology with support of dimension  d, J. Algebra Appl., 15(3) (2016), 1650042 (10 pp).
Yıl 2019, , 55 - 63, 08.01.2019
https://doi.org/10.24330/ieja.504110

Öz

Kaynakça

  • J. Azami, R. Naghipour and B. Vakili, Finiteness properties of local cohomology modules for a-minimax modules, Proc. Amer. Math. Soc., 137(2) (2009), 439- 448.
  • C. Banica and M. Stoia, Singular sets of a module and local cohomology, Boll. Un. Mat. Ital. B, 16 (1976), 923-934.
  • M. H. Bijan-Zadeh, Torsion theories and local cohomology over commutative Noetherian rings, J. London Math. Soc., 19(3) (1979), 402-410.
  • K. B. Lorestani, P. Sahandi and S. Yassemi, Artinian local cohomology modules, Canad. Math. Bull., 50(4) (2007), 598-602.
  • M. P. Brodmann and A. L. Faghani, A niteness result for associated primes of local cohomology modules, Proc. Amer. Math. Soc., 128(10) (2000), 2851-2853.
  • M. P. Brodmann and R. Y. Sharp, Local Cohomology: an Algebraic Intro- duction with Geometric Applications, Cambridge Studies in Advanced Math- ematics, 60, Cambridge University Press, Cambridge, 1998.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Ad- vanced Mathematics, 39, Cambridge University Press, Cambridge, 1993.
  • M. T. Dibaei and S. Yassemi, Associated primes and co niteness of local co- homology modules, Manuscripta Math., 117(2) (2005), 199-205.
  • K. Divaani-Aazar and A. Ma , Associated primes of local cohomology modules, Proc. Amer. Math. Soc., 133(3) (2005), 655-660.
  • C. Huneke, Problems on local cohomology. Free Resolution in Commutative Algebra and Algebraic Geometry. (Sundance, UT, 1990), Res. Notes Math., 2, Jones and Bartlett, Boston, MA, (1992), 93-108.
  • M. Katzman, An example of an in nite set of associated primes of local coho- mology module, J. Algebra, 252(1) (2002), 161-166.
  • H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, Cambridge, 1986.
  • R. Naghipour, Integral closures, local cohomology and ideal topologies, Rocky Mountain J. Math., 37(3) (2007), 905-916.
  • J. J. Rotman, Introduction to Homological Algebra, Pure and Applied Mathe- matics, 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.
  • R. Y. Sharp, Steps in Commutative Algebra, Second edition, London Math- ematical Society Student Texts, 51, Cambridge University Press, Cambridge, 2000.
  • A. K. Singh, p-Torsion elements in local cohomology modules, Math. Res. Lett., 7(2-3) (2000), 165-176.
  • J. Z. Xu, Minimal injective and at resolutions of modules over Gorenstein rings, J. Algebra, 175(2) (1995), 451-477.
  • N. Zamani, M. H. Bijan-Zadeh and M. S. Sayedsadeghi, d-Transform functor and some niteness and isomorphism results, Vietnam. J. Math., 42(2) (2014), 179-186.
  • N. Zamani, M. H. Bijan-Zadeh and M. S. Sayedsadeghi, Cohomology with support of dimension  d, J. Algebra Appl., 15(3) (2016), 1650042 (10 pp).
Toplam 19 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Makaleler
Yazarlar

Z. Rahimi-molaei Bu kişi benim

Sh. Payrovi Bu kişi benim

S. Babaei Bu kişi benim

Yayımlanma Tarihi 8 Ocak 2019
Yayımlandığı Sayı Yıl 2019

Kaynak Göster

APA Rahimi-molaei, Z., Payrovi, S., & Babaei, S. (2019). ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES. International Electronic Journal of Algebra, 25(25), 55-63. https://doi.org/10.24330/ieja.504110
AMA Rahimi-molaei Z, Payrovi S, Babaei S. ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES. IEJA. Ocak 2019;25(25):55-63. doi:10.24330/ieja.504110
Chicago Rahimi-molaei, Z., Sh. Payrovi, ve S. Babaei. “ON THE ASSOCIATED PRIMES OF THE D-LOCAL COHOMOLOGY MODULES”. International Electronic Journal of Algebra 25, sy. 25 (Ocak 2019): 55-63. https://doi.org/10.24330/ieja.504110.
EndNote Rahimi-molaei Z, Payrovi S, Babaei S (01 Ocak 2019) ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES. International Electronic Journal of Algebra 25 25 55–63.
IEEE Z. Rahimi-molaei, S. Payrovi, ve S. Babaei, “ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES”, IEJA, c. 25, sy. 25, ss. 55–63, 2019, doi: 10.24330/ieja.504110.
ISNAD Rahimi-molaei, Z. vd. “ON THE ASSOCIATED PRIMES OF THE D-LOCAL COHOMOLOGY MODULES”. International Electronic Journal of Algebra 25/25 (Ocak 2019), 55-63. https://doi.org/10.24330/ieja.504110.
JAMA Rahimi-molaei Z, Payrovi S, Babaei S. ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES. IEJA. 2019;25:55–63.
MLA Rahimi-molaei, Z. vd. “ON THE ASSOCIATED PRIMES OF THE D-LOCAL COHOMOLOGY MODULES”. International Electronic Journal of Algebra, c. 25, sy. 25, 2019, ss. 55-63, doi:10.24330/ieja.504110.
Vancouver Rahimi-molaei Z, Payrovi S, Babaei S. ON THE ASSOCIATED PRIMES OF THE d-LOCAL COHOMOLOGY MODULES. IEJA. 2019;25(25):55-63.