Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-17T20:05:25.005Z Has data issue: false hasContentIssue false
  • English
  • Français

Balanced big Cohen-Macaulay modules and ring extensions

Published online by Cambridge University Press:  14 November 2011

Liam O'Carroll
Liam O'Carroll
Affiliation:
Department of Mathematics, University of Edinburgh, Mayfield Road, Edinburgh EH93JZ
Get access Rights & Permissions [Opens in a new window]

Extract

Let A and B be commutative Noetherian local rings such that B contains A and B is flat and integral over A. It is shown that if M is a balanced big Cohen-Macaulay A-module (that is, every system of parameters for A is an M-sequence), then M⊗AB is a balanced big Cohen-Macaulay B-module. An example of a ring A is given such that, if B is the completion of A, then the analogous result is false in this case. This answers a question posed by Riley in the negative.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 99 , Issue 1-2 , 1984 , pp. 171 - 172
Copyright
Copyright © Royal Society of Edinburgh 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Ferrand, D. and Raynaud, M.. Fibres formelles d'un anneau local noetherien. Ann. Sci. Ecole Norm. Sup. 3 (1970), 295–311.CrossRefGoogle Scholar
2Hochster, M.. Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors. Proceedings of the conference on Commutative Algebra, Queen's University, Kingston, Ontario, 1975 (Queen's University papers in Pure and Applied Mathematics No. 24, 1975, pp. 106–195).Google Scholar
3Nagata, M.. Local rings (New York: Wiley, 1962).Google Scholar
4O'Carroll, L.. On the generalized fractions of Sharp and Zakeri. J. London Math. Soc. 28 (1983), 417–427.CrossRefGoogle Scholar
5Ogoma, T.. Fiber product of Noetherian rings and its applications. Math. Proc. Cambridge Philos. Soc., to appear.Google Scholar
7Sharp, R. Y.. Cohen-Macaulay properties for balanced big Cohen-Macaulay modules. Math. Proc. Cambridge Philos. Soc. 90 (1981), 229–238.CrossRefGoogle Scholar
8Sharp, R. Y.. A Cousin complex characterisation of balanced big Cohen-Macaulay modules. Quart. J. Math. Oxford 33 (1982), 471–485.CrossRefGoogle Scholar