Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Big Cohen-Macaulay algebras and seeds

Author: Geoffrey D. Dietz
Journal: Trans. Amer. Math. Soc. 359 (2007), 5959-5989
MSC (2000): Primary 13C14, 13A35; Secondary 13H10, 13B99
Published electronically: June 26, 2007
MathSciNet review: 2336312
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this article, we delve into the properties possessed by algebras, which we have termed seeds, that map to big Cohen-Macaulay algebras. We will show that over a complete local domain of positive characteristic any two big Cohen-Macaulay algebras map to a common big Cohen-Macaulay algebra. We will also strengthen Hochster and Huneke's ``weakly functorial" existence result for big Cohen-Macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic. We also show that every seed over a positive characteristic ring $ (R,m)$ maps to a balanced big Cohen-Macaulay $ R$-algebra that is an absolutely integrally closed, $ m$-adically separated, quasilocal domain.

References [Enhancements On Off] (What's this?)

  • [AFH] L. AVRAMOV, H.-B. FOXBY, and B. HERZOG, Structure of local homomorphisms, J. Algebra 164 (1994), no. 1, 124-145.MR 1268330 (95f:13029)
  • [BS] J. BARTIJN and J.R. STROOKER, Modifications Monomiales, Séminaire d'Algèbre Dubreil-Malliavin, Paris 1982, Lecture Notes in Mathematics, 1029, Springer-Verlag, Berlin, 1983, 192-217.MR 0732476 (85j:13035)
  • [BH] W. BRUNS and J. HERZOG, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993. MR 1251956 (95h:13020)
  • [D] G. DIETZ, Closure Operations in Positive Characteristic and Big Cohen-Macaulay Algebras, Thesis, Univ. of Michigan, 2005.
  • [E] D. EISENBUD, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.MR 1322960 (97a:13001)
  • [Heit] R. HEITMANN, The direct summand conjecture in dimension three, Ann. of Math. (2) 156 (2002), no. 2, 695-712.MR 1933722 (2003m:13008)
  • [Ho1] M. HOCHSTER, Topics in the homological theory of modules over commutative rings, CBMS Regional Conference Series 24, Amer. Math. Soc., 1975.MR 0371879 (51:8096)
  • [Ho2] M. HOCHSTER, Solid closure, Contemp. Math. 159 (1994), 103-172.MR 1266182 (95a:13011)
  • [Ho3] M. HOCHSTER, Big Cohen-Macaulay algebras in dimension three via Heitmann's theorem, J. Algebra 254 (2002), no. 2, 395-408.MR 1933876 (2004c:13011)
  • [HH1] M. HOCHSTER and C. HUNEKE, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), 31-116.MR 1017784 (91g:13010)
  • [HH2] M. HOCHSTER and C. HUNEKE, Infinite integral extensions and big Cohen-Macaulay algebras, Annals of Math. 135 (1992), 53-89.MR 1147957 (92m:13023)
  • [HH3] M. HOCHSTER and C. HUNEKE, Tight closure of parameter ideals and splitting in module-finite extensions, J. Algebraic Geom. 3 (1994), no. 4, 599-670. MR 1297848 (95k:13002)
  • [HH4] M. HOCHSTER and C. HUNEKE, $ F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1-62. MR 1273534 (95d:13007)
  • [HH5] M. HOCHSTER and C. HUNEKE, Applications of the existence of big Cohen-Macaulay algebras, Adv. Math. 113 (1995), no. 1, 45-117. MR 1332808 (96d:13014)
  • [HJ] K. HRBACEK and T. JECH, Introduction to Set Theory, Second edition. Monographs and Textbooks in Pure and Applied Mathematics, 85, Marcel Dekker, Inc., New York, 1984. MR 0758796 (85m:04001)
  • [Mat] H. MATSUMURA, Commutative Algebra, Benjamin-Cummings, New York, 1980.MR 0575344 (82i:13003)
  • [N1] M. NAGATA, Lectures on the Fourteenth Problem of Hilbert, Tata Institute of Fundamental Research, Bombay, 1965. MR 0215828 (35:6663)
  • [N2] M. NAGATA, Local Rings, Interscience, New York, 1972.MR 0460307 (57:301)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13C14, 13A35, 13H10, 13B99

Retrieve articles in all journals with MSC (2000): 13C14, 13A35, 13H10, 13B99

Additional Information

Geoffrey D. Dietz
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019-0315
Address at time of publication: Department of Mathematics, Gannon University, Erie, Pennsylvania 16541

Keywords: Big Cohen-Macaulay algebras, tight closure
Received by editor(s): August 22, 2005
Published electronically: June 26, 2007
Additional Notes: The author was supported in part by a VIGRE grant from the National Science Foundation.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society