A characterization of closure operations that induce big Cohen-Macaulay modules

Author:
Geoffrey D. Dietz

Journal:
Proc. Amer. Math. Soc. **138** (2010), 3849-3862

MSC (2000):
Primary 13C14; Secondary 13A35

Published electronically:
May 24, 2010

MathSciNet review:
2679608

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Abstract | References | Similar Articles | Additional Information

Abstract: The intent of this paper is to present a set of axioms that are sufficient for a closure operation to generate a balanced big Cohen-Macaulay module over a complete local domain . Conversely, we show that if such a exists over , then there exists a closure operation that satisfies the given axioms.

**[D]**Geoffrey D. Dietz,*Big Cohen-Macaulay algebras and seeds*, Trans. Amer. Math. Soc.**359**(2007), no. 12, 5959–5989. MR**2336312**, 10.1090/S0002-9947-07-04252-3**[Ho1]**Melvin Hochster,*Topics in the homological theory of modules over commutative rings*, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Nebraska, Lincoln, Neb., June 24–28, 1974; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24. MR**0371879****[Ho2]**Melvin Hochster,*Solid closure*, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 103–172. MR**1266182**, 10.1090/conm/159/01508**[Ho3]**M. HOCHSTER,*Foundations of tight closure theory*, lecture notes available at`http://www.math.lsa.umich.edu/hochster/mse.html`.**[HH1]**Melvin Hochster and Craig Huneke,*Tight closure, invariant theory, and the Briançon-Skoda theorem*, J. Amer. Math. Soc.**3**(1990), no. 1, 31–116. MR**1017784**, 10.1090/S0894-0347-1990-1017784-6**[HH2]**Melvin Hochster and Craig Huneke,*Infinite integral extensions and big Cohen-Macaulay algebras*, Ann. of Math. (2)**135**(1992), no. 1, 53–89. MR**1147957**, 10.2307/2946563**[HH3]**Melvin Hochster and Craig Huneke,*Tight closure of parameter ideals and splitting in module-finite extensions*, J. Algebraic Geom.**3**(1994), no. 4, 599–670. MR**1297848****[HH4]**M. HOCHSTER and C. HUNEKE,*Tight Closure in Equal Characteristic Zero*, preprint available at`http://www.math.lsa.umich.edu/ hochster/msr.html`.**[Sm]**K. E. Smith,*Tight closure of parameter ideals*, Invent. Math.**115**(1994), no. 1, 41–60. MR**1248078**, 10.1007/BF01231753

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Additional Information

**Geoffrey D. Dietz**

Affiliation:
Department of Mathematics, Gannon University, Erie, Pennsylvania 16541

Email:
gdietz@member.ams.org

DOI:
http://dx.doi.org/10.1090/S0002-9939-2010-10417-3

Received by editor(s):
October 28, 2009

Received by editor(s) in revised form:
January 30, 2010

Published electronically:
May 24, 2010

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.