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Phantom depth and flat base change


Author: Neil M. Epstein
Journal: Proc. Amer. Math. Soc. 134 (2006), 313-321
MSC (2000): Primary 13A35; Secondary 13B40, 13C15, 13H10
DOI: https://doi.org/10.1090/S0002-9939-05-08223-7
Published electronically: September 21, 2005
MathSciNet review: 2175997
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Abstract: We prove that if $f: (R,\mathfrak{m}) \rightarrow (S,\mathfrak{n})$ is a flat local homomorphism, $S/\mathfrak{m} S$ is Cohen-Macaulay and $F$-injective, and $R$and $S$ share a weak test element, then a tight closure analogue of the (standard) formula for depth and regular sequences across flat base change holds. As a corollary, it follows that phantom depth commutes with completion for excellent local rings. We give examples to show that the analogue does not hold for surjective base change.


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Additional Information

Neil M. Epstein
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: epstein@math.ku.edu, neilme@umich.edu

DOI: https://doi.org/10.1090/S0002-9939-05-08223-7
Keywords: Tight closure, phantom depth, base change
Received by editor(s): May 17, 2004
Published electronically: September 21, 2005
Additional Notes: The author was partially supported by the National Science Foundation.
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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