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A FEW COMMENTS ON MATLIS DUALITY

Year 2014, Volume: 15 Issue: 15, 66 - 76, 01.06.2014
https://doi.org/10.24330/ieja.266238

Abstract

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.

References

  • 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
Year 2014, Volume: 15 Issue: 15, 66 - 76, 01.06.2014
https://doi.org/10.24330/ieja.266238

Abstract

References

  • 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
There are 16 citations in total.

Details

Other ID JA67UZ27CM
Journal Section Articles
Authors

Waqas Mahmood This is me

Publication Date June 1, 2014
Published in Issue Year 2014 Volume: 15 Issue: 15

Cite

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. June 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, no. 15 (June 2014): 66-76. https://doi.org/10.24330/ieja.266238.
EndNote Mahmood W (June 1, 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, vol. 15, no. 15, pp. 66–76, 2014, doi: 10.24330/ieja.266238.
ISNAD Mahmood, Waqas. “A FEW COMMENTS ON MATLIS DUALITY”. International Electronic Journal of Algebra 15/15 (June 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, vol. 15, no. 15, 2014, pp. 66-76, doi:10.24330/ieja.266238.
Vancouver Mahmood W. A FEW COMMENTS ON MATLIS DUALITY. IEJA. 2014;15(15):66-7.