What is a system of parameters?
Authors:
Louiza Fouli and Craig Huneke
Journal:
Proc. Amer. Math. Soc. 139 (2011), 2681-2696
MSC (2010):
Primary 13A35, 13C40, 13D45
DOI:
https://doi.org/10.1090/S0002-9939-2011-10790-1
Published electronically:
February 8, 2011
MathSciNet review:
2801607
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper we discuss various refinements and generalizations of a theorem of Sankar Dutta and Paul Roberts. Their theorem gives a criterion for 
elements in a 
-dimensional Noetherian Cohen-Macaulay local ring to be a system of parameters, i.e., to have height 
. We chiefly remove the assumption that the ring be Cohen-Macaulay and discuss similar theorems.
- 1. Nguyen Tu Cuong and Le Thanh Nhan, Pseudo Cohen-Macaulay and pseudo generalized Cohen-Macaulay modules, J. Algebra 267 (2003), no. 1, 156–177. MR 1993472, https://doi.org/10.1016/S0021-8693(03)00225-4
- 2. Sankar P. Dutta and Paul C. Roberts, A characterization of systems of parameters, Proc. Amer. Math. Soc. 124 (1996), no. 3, 671–675. MR 1283547, https://doi.org/10.1090/S0002-9939-96-02955-3
- 3. Shiro Goto and Hideto Sakurai, The equality 𝐼²=𝑄𝐼 in Buchsbaum rings, Rend. Sem. Mat. Univ. Padova 110 (2003), 25–56. MR 2033000
- 4. Raymond C. Heitmann, The direct summand conjecture in dimension three, Ann. of Math. (2) 156 (2002), no. 2, 695–712. MR 1933722, https://doi.org/10.2307/3597204
- 5. 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
- 6. 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, https://doi.org/10.1090/S0894-0347-1990-1017784-6
- 7. Melvin Hochster and Craig Huneke, Indecomposable canonical modules and connectedness, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 197–208. MR 1266184, https://doi.org/10.1090/conm/159/01509
- 8. Thomas Marley, Mark W. Rogers, and Hideto Sakurai, Gorenstein rings and irreducible parameter ideals, Proc. Amer. Math. Soc. 136 (2008), no. 1, 49–53. MR 2350387, https://doi.org/10.1090/S0002-9939-07-08958-7
- 9. Günter Scheja and Uwe Storch, Differentielle Eigenschaften der Lokalisierungen analytischer Algebren, Math. Ann. 197 (1972), 137–170 (German). MR 306172, https://doi.org/10.1007/BF01419591
- 10. Anne-Marie Simon and Jan R. Strooker, Gorenstein rings call the tune, Commutative algebra, Lect. Notes Pure Appl. Math., vol. 244, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 227–238. MR 2184800, https://doi.org/10.1201/9781420028324.ch16
- 11. K. E. Smith, Tight closure of parameter ideals, Invent. Math. 115 (1994), no. 1, 41–60. MR 1248078, https://doi.org/10.1007/BF01231753
- 12. Jan R. Strooker, Homological questions in local algebra, London Mathematical Society Lecture Note Series, vol. 145, Cambridge University Press, Cambridge, 1990. MR 1074178
in Buchsbaum rings, Rend. Sem. Mat. Univ. Padova 110 (2003), 25-56. Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 13A35, 13C40, 13D45
Retrieve articles in all journals with MSC (2010): 13A35, 13C40, 13D45
Additional Information
Louiza Fouli
Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
Email:
lfouli@math.nmsu.edu
Craig Huneke
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email:
huneke@math.ku.edu
DOI:
https://doi.org/10.1090/S0002-9939-2011-10790-1
Keywords:
System of parameters,
local cohomology,
limit closure,
tight closure,
plus closure
Received by editor(s):
March 15, 2010
Received by editor(s) in revised form:
August 2, 2010
Published electronically:
February 8, 2011
Additional Notes:
The first author was partially supported by the NSF-AWM Mentoring Travel Grant, grant DMS-0839954. She thanks the Department of Mathematics at the University of Kansas for its hospitality.
The second author was partially supported by the National Science Foundation, grant DMS-0756853.
Communicated by:
Irena Peeva
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
