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
DOI:
https://doi.org/10.1090/S0002-9947-07-04118-9
Published electronically:
May 11, 2007
MathSciNet review:
2320653
Full-text PDF Free Access
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 $M$, 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.
- Maurice Auslander and David A. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625–657. MR 99978, DOI https://doi.org/10.2307/1970159
- I. M. Aberbach, Finite phantom projective dimension, Amer. J. Math. 116 (1994), no. 2, 447–477. MR 1269611, DOI https://doi.org/10.2307/2374936
- ---, personal communication, 2003.
- Ian M. Aberbach, Melvin Hochster, and Craig Huneke, Localization of tight closure and modules of finite phantom projective dimension, J. Reine Angew. Math. 434 (1993), 67–114. MR 1195691, DOI https://doi.org/10.1515/crll.1993.434.67
- M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647. MR 179211
- Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
- Lindsay Burch, Codimension and analytic spread, Proc. Cambridge Philos. Soc. 72 (1972), 369–373. MR 304377, DOI https://doi.org/10.1017/s0305004100047198
- Neil M. Epstein, Phantom depth and flat base change, Proc. Amer. Math. Soc. 134 (2006), no. 2, 313–321. MR 2175997, DOI https://doi.org/10.1090/S0002-9939-05-08223-7
- Neil M. Epstein, A tight closure analogue of analytic spread, Math. Proc. Cambridge Philos. Soc. 139 (2005), no. 2, 371–383. MR 2168094, DOI https://doi.org/10.1017/S0305004105008546
- Raymond C. Heitmann, A counterexample to the rigidity conjecture for rings, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 94–97. MR 1197425, DOI https://doi.org/10.1090/S0273-0979-1993-00410-5
- 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, DOI https://doi.org/10.1090/S0894-0347-1990-1017784-6
- Melvin Hochster and Craig Huneke, Phantom homology, Mem. Amer. Math. Soc. 103 (1993), no. 490, vi+91. MR 1144758, DOI https://doi.org/10.1090/memo/0490
- Melvin Hochster and Craig Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1–62. MR 1273534, DOI https://doi.org/10.1090/S0002-9947-1994-1273534-X
- Melvin Hochster and Craig Huneke, Localization and test exponents for tight closure, Michigan Math. J. 48 (2000), 305–329. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786493, DOI https://doi.org/10.1307/mmj/1030132721
- Craig Huneke, David A. Jorgensen, and Roger Wiegand, Vanishing theorems for complete intersections, J. Algebra 238 (2001), no. 2, 684–702. MR 1823780, DOI https://doi.org/10.1006/jabr.2000.8603
- 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
- Craig Huneke, Tight closure and its applications, CBMS Regional Conference Series in Mathematics, vol. 88, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With an appendix by Melvin Hochster. MR 1377268
- Craig Huneke and Roger Wiegand, Tensor products of modules and the rigidity of ${\rm Tor}$, Math. Ann. 299 (1994), no. 3, 449–476. MR 1282227, DOI https://doi.org/10.1007/BF01459794
- Craig Huneke and Roger Wiegand, Tensor products of modules, rigidity and local cohomology, Math. Scand. 81 (1997), no. 2, 161–183. MR 1612887, DOI https://doi.org/10.7146/math.scand.a-12871
- David A. Jorgensen, Complexity and Tor on a complete intersection, J. Algebra 211 (1999), no. 2, 578–598. MR 1666660, DOI https://doi.org/10.1006/jabr.1998.7743
- Stephen Lichtenbaum, On the vanishing of ${\rm Tor}$ in regular local rings, Illinois J. Math. 10 (1966), 220–226. MR 188249
- Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
- M. Pavaman Murthy, Modules over regular local rings, Illinois J. Math. 7 (1963), 558–565. MR 156883
- Anurag K. Singh and Irena Swanson, Associated primes of local cohomology modules and of Frobenius powers, Int. Math. Res. Not. 33 (2004), 1703–1733. MR 2058025, DOI https://doi.org/10.1155/S1073792804133424
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13A35, 13C15, 13D25
Retrieve articles in all journals with MSC (2000): 13A35, 13C15, 13D25
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
MR Author ID:
768826
Email:
epstein@math.ukans.edu, neilme@umich.edu
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.