Phantom depth and stable phantom exactness

Author:
Neil M. Epstein

Journal:
Trans. Amer. Math. Soc. **359** (2007), 4829-4864

MSC (2000):
Primary 13A35; Secondary 13C15, 13D25

Published electronically:
May 11, 2007

MathSciNet review:
2320653

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Abstract | References | Similar Articles | Additional Information

Abstract: Phantom depth, phantom nonzerodivisors, and phantom exact sequences are analogues of the non-``phantom'' notions which have been useful in tackling the (very difficult) localization problem in tight closure theory. In the present paper, these notions are developed further and partially reworked. For instance, although no analogue of a long exact sequence arises from a short stably phantom exact sequence of complexes, we provide a method for recovering the kind of information obtainable from such a long sequence. Also, we give alternate characterizations of the notion of phantom depth, including one based on Koszul homology, which we use to show that with very mild conditions on a finitely generated module , any two maximal phantom -regular sequences in an ideal have the same length. In order to do so, we prove a ``Nakayama lemma for tight closure'', which is of independent interest. We strengthen the connection of phantom depth with minheight, we explore several analogues of ``associated prime'' in tight closure theory, and we discuss a connection with the problem of when tight closure commutes with localization.

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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.ukans.edu, neilme@umich.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-07-04118-9

Keywords:
Tight closure,
phantom depth,
phantom homology,
rigidity

Received by editor(s):
September 3, 2004

Received by editor(s) in revised form:
May 4, 2005

Published electronically:
May 11, 2007

Additional Notes:
The author was partially supported by the National Science Foundation.

Article copyright:
© Copyright 2007
American Mathematical Society

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