BibTex RIS Kaynak Göster

A FEW COMMENTS ON MATLIS DUALITY

Yıl 2014, Cilt: 15 Sayı: 15, 66 - 76, 01.06.2014
https://doi.org/10.24330/ieja.266238

Öz

For a Noetherian local ring (R, m) with p ∈ Spec(R), we denote the
R-injective hull of R/p by ER(R/p). We show that it has an Rˆp
-module structure, and there is an isomorphism ER(R/p) ∼= ERˆp (Rˆp/pRˆp
), where Rˆp stands for the p-adic completion of R. Moreover, for a complete Cohen-Macaulay ring
R, the module D(ER(R/p)) is isomorphic to Rˆp provided that dim(R/p) = 1,
where D(·) denotes the Matlis dual functor HomR(·, ER(R/m)). Here, Rˆp
denotes the completion of Rp with respect to the maximal ideal pRp. These
results extend those of Matlis (see [11]) shown in the case of the maximal ideal
m.

Kaynakça

  • M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, University of Oxford, 1969.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Univ. Press, 39, M. Brodmann and R. Sharp, Local Cohomology, An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics No. 60. Cambridge University Press, 1998.
  • E. E. Enochs, Injective and Flat Covers, Envelopes and Resolvents, Israel J. Math., 39 (1981), 189-209.
  • E. E. Enochs, Flat Covers and Flat Cotorsion Modules, Proc. Amer. Math. Soc., 92 (1984), 179-184.
  • E. E. Enochs and O.M.G. Jenda, Relative Homological Algebra(de Gruyter Expositions in Mathematics, 30), Walter de Gruyter, Berlin, 2000.
  • R. Fossum, H.-B. Foxby, B. Griffith and I. Reiten, Minimal Injective Reso- lutions with Applications to Dualizig Modules and Gorenstein Modules, Publ. Math. Inst. Hautes Etudues Sci., 45 (1976), 193-215.
  • A. Grothendieck, Local Cohomology(Notes by R. Hartshorne), Lecture Notes in Math. vol.41, Springer, 1967.
  • C. Huneke, Lectures on Local Cohomology (with an Appendix by Amelia Tay- lor), Contemp. Math., 436 (2007), 51-100.
  • M. Hellus, Local Cohomology and Matils Duality, arXiv:math/0703124v1.
  • E. Matlis, Injective Modules Over Noetherian Rings, Pacific J. Math., 8 (1958), 528.
  • H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1986.
  • E. Miller, S. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, A.K. Singh and U. Walther, Twenty Four Hours of Local Cohomology (Graduate Studies in Mathematics), American Mathematical Society, Vol. 87, 2007.
  • W. Mahmood, On Cohomologically Complete Intersections in Cohen-Macaulay Rings, submitted. P. Schenzel, On Birational Macaulayfications and Cohen-Macaulay Canonical Modules, J. Algebra, 275 (2004), 751-770.
  • P. Schenzel, On Formal Local Cohomology and Connectedness, J. Algebra, (2) (2007), 894-923.
  • P. Schenzel, A Note on the Matlis Dual of a Certain Injective Hull, arXiv:1306.3311v1.
  • C. Weibel, An Introduction to Homological Algebra, Cambridge Univ. Press, Waqas Mahmood Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan e-mail: waqassms@gmail.com
Yıl 2014, Cilt: 15 Sayı: 15, 66 - 76, 01.06.2014
https://doi.org/10.24330/ieja.266238

Öz

Kaynakça

  • M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, University of Oxford, 1969.
  • W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Univ. Press, 39, M. Brodmann and R. Sharp, Local Cohomology, An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics No. 60. Cambridge University Press, 1998.
  • E. E. Enochs, Injective and Flat Covers, Envelopes and Resolvents, Israel J. Math., 39 (1981), 189-209.
  • E. E. Enochs, Flat Covers and Flat Cotorsion Modules, Proc. Amer. Math. Soc., 92 (1984), 179-184.
  • E. E. Enochs and O.M.G. Jenda, Relative Homological Algebra(de Gruyter Expositions in Mathematics, 30), Walter de Gruyter, Berlin, 2000.
  • R. Fossum, H.-B. Foxby, B. Griffith and I. Reiten, Minimal Injective Reso- lutions with Applications to Dualizig Modules and Gorenstein Modules, Publ. Math. Inst. Hautes Etudues Sci., 45 (1976), 193-215.
  • A. Grothendieck, Local Cohomology(Notes by R. Hartshorne), Lecture Notes in Math. vol.41, Springer, 1967.
  • C. Huneke, Lectures on Local Cohomology (with an Appendix by Amelia Tay- lor), Contemp. Math., 436 (2007), 51-100.
  • M. Hellus, Local Cohomology and Matils Duality, arXiv:math/0703124v1.
  • E. Matlis, Injective Modules Over Noetherian Rings, Pacific J. Math., 8 (1958), 528.
  • H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1986.
  • E. Miller, S. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, A.K. Singh and U. Walther, Twenty Four Hours of Local Cohomology (Graduate Studies in Mathematics), American Mathematical Society, Vol. 87, 2007.
  • W. Mahmood, On Cohomologically Complete Intersections in Cohen-Macaulay Rings, submitted. P. Schenzel, On Birational Macaulayfications and Cohen-Macaulay Canonical Modules, J. Algebra, 275 (2004), 751-770.
  • P. Schenzel, On Formal Local Cohomology and Connectedness, J. Algebra, (2) (2007), 894-923.
  • P. Schenzel, A Note on the Matlis Dual of a Certain Injective Hull, arXiv:1306.3311v1.
  • C. Weibel, An Introduction to Homological Algebra, Cambridge Univ. Press, Waqas Mahmood Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan e-mail: waqassms@gmail.com
Toplam 16 adet kaynakça vardır.

Ayrıntılar

Diğer ID JA67UZ27CM
Bölüm Makaleler
Yazarlar

Waqas Mahmood Bu kişi benim

Yayımlanma Tarihi 1 Haziran 2014
Yayımlandığı Sayı Yıl 2014 Cilt: 15 Sayı: 15

Kaynak Göster

APA Mahmood, W. (2014). A FEW COMMENTS ON MATLIS DUALITY. International Electronic Journal of Algebra, 15(15), 66-76. https://doi.org/10.24330/ieja.266238
AMA Mahmood W. A FEW COMMENTS ON MATLIS DUALITY. IEJA. Haziran 2014;15(15):66-76. doi:10.24330/ieja.266238
Chicago Mahmood, Waqas. “A FEW COMMENTS ON MATLIS DUALITY”. International Electronic Journal of Algebra 15, sy. 15 (Haziran 2014): 66-76. https://doi.org/10.24330/ieja.266238.
EndNote Mahmood W (01 Haziran 2014) A FEW COMMENTS ON MATLIS DUALITY. International Electronic Journal of Algebra 15 15 66–76.
IEEE W. Mahmood, “A FEW COMMENTS ON MATLIS DUALITY”, IEJA, c. 15, sy. 15, ss. 66–76, 2014, doi: 10.24330/ieja.266238.
ISNAD Mahmood, Waqas. “A FEW COMMENTS ON MATLIS DUALITY”. International Electronic Journal of Algebra 15/15 (Haziran 2014), 66-76. https://doi.org/10.24330/ieja.266238.
JAMA Mahmood W. A FEW COMMENTS ON MATLIS DUALITY. IEJA. 2014;15:66–76.
MLA Mahmood, Waqas. “A FEW COMMENTS ON MATLIS DUALITY”. International Electronic Journal of Algebra, c. 15, sy. 15, 2014, ss. 66-76, doi:10.24330/ieja.266238.
Vancouver Mahmood W. A FEW COMMENTS ON MATLIS DUALITY. IEJA. 2014;15(15):66-7.