A FEW COMMENTS ON MATLIS DUALITY

Year 2014, , 66 - 76, 01.06.2014

Waqas Mahmood

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.

Keywords

Matlis duality, injective hull, local cohomology, flat covers

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: [e-mail]