Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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: 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 $ M$, any two maximal phantom $ M$-regular sequences in an ideal $ I$ 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

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.