Abstract
For a Noetherian local domain R let R + be the absolute integral closure of R and let R ∞ be the perfect closure of R, when R has prime characteristic. In this paper we investigate the projective dimension of residue rings of certain ideals of R + and R ∞. In particular, we show that any prime ideal of R ∞ has a bounded free resolution of countably generated free R ∞-modules. Also, we show that the analogue of this result is true for the maximal ideals of R +, when R has residue prime characteristic. We compute global dimensions of R + and R ∞ in some cases. Some applications of these results are given.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aberbach I.M., Hochster M.: Finite tor dimension and failure of coherence in absolute integral closures. J. Pure Appl. Algebra 122, 171–184 (1997)
Artin M.: On the joins of Hensel rings. Adv. Math. 7, 282–296 (1971)
Asgharzadeh M., Tousi M.: On the notion of Cohen-Macaulayness for non-Noetherian rings. J. Algebra 322, 2297–2320 (2009)
Asgharzadeh, M., Tousi, M.: Almost Cohen-Macaulay rings (unpublished)
Bruns W., Herzog J.: Cohen-Macaulay Rings, vol. 39, revised edition. Cambridge University Press, Cambridge (1998)
Cahen P.J.: Commutative torsion theory. Trans. AMS 184, 73–85 (1973)
Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press (1956)
Faltings G.: Almost étale extensions. In: Cohomologies p-adiques et applications arithmétiques II. Astérisque 279, 185–270 (2002)
Gabber, O., Ramero, L.: Almost Ring Theory, vol. 1800. Springer LNM (2003)
Glaz, S.: Commutative Coherent Rings, vol. 1371. Springer LNM (1989)
Heitmann R.C.: The direct summand conjecture in dimension three. Ann. Math. 156(2), 695–712 (2002)
Heitmann R.C.: Extended plus closure and colon-capturing. J. Algebra 293, 407–426 (2005)
Hochster M.: Canonical elements in local cohomology modules and the direct summand conjecture. J. Algebra 84, 503–553 (1983)
Hochster, M.: Foundations of tight closure theory. Lecture notes from a course taught on the University of Michigan, Fall 2007
Hochster M., Huneke C.: Infinite integral extensions and big Cohen-Macaulay algebras. Ann. Math. 135(2), 53–89 (1992)
Hochster M., Huneke C.: Tigth closure and elements of small order in integral extensions. J. Pure Appl. Algebra 71, 233–247 (1991)
Huneke C., Lyubeznik G.: Absolute integral closure in positive characteristic. Adv. Math. 210(2), 498–504 (2007)
Huneke, C., Swanson, I.: Integral closure of ideals, rings, and modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)
Krause H.: Cohomological quotients and smashing localizations. Am. J. Math. 127(6), 1191–1246 (2005)
Matsumura, H.: Commutative ring theory. Cambridge Studies in Advanced Math., vol. 8, xiv+320 pp. Cambridge University Press, Cambridge (1986)
Northcott D.G.: On the homology theory of general commutative rings. J. Lond. Math. Soc. 36, 231–240 (1961)
Osofsky, B.L.: Homological dimensions of modules. CBMS 12 (1971)
Osofsky B.L.: The subscript of \({\aleph_n}\), projective dimension, and the vanishing of \({{\operatornamewithlimits{\varprojlim}}^{(n)}}\). Bull. AMS 80(1), 8–26 (1974)
Roberts P., Singh A.K., Srinivas V.: Annihilators of local cohomology in characteristic zero. Ill. J. Math. 51(1), 237–254 (2007)
Rotman J.: An Introduction to Homological Algebra. Academic Press, San Diego (1979)
Sally J.D., Vasconcelos W.V.: Stable rings. J. Pure Appl. Algebra 4, 319–336 (1974)
Schenzel P.: Proregular sequences, local cohomology, and completion. Math. Scand. 92(2), 271–289 (2003)
Smith K.E.: Tight closure of parameter ideals. Invent. Math. 115, 41–60 (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Paul Roberts.
Rights and permissions
About this article
Cite this article
Asgharzadeh, M. Homological properties of the perfect and absolute integral closures of Noetherian domains. Math. Ann. 348, 237–263 (2010). https://doi.org/10.1007/s00208-009-0474-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-009-0474-x