Phantom depth and stable phantom exactness
HTML articles powered by AMS MathViewer
- by Neil M. Epstein PDF
- Trans. Amer. Math. Soc. 359 (2007), 4829-4864 Request permission
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.References
- Maurice Auslander and David A. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625–657. MR 99978, DOI 10.2307/1970159
- I. M. Aberbach, Finite phantom projective dimension, Amer. J. Math. 116 (1994), no. 2, 447–477. MR 1269611, DOI 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 10.1515/crll.1993.434.67
- M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647. MR 179211, DOI 10.1215/ijm/1255631585
- 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 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 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 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 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 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 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 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 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 10.1006/jabr.2000.8603
- Melvin Hochster, Topics in the homological theory of modules over commutative rings, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24, 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. MR 0371879, DOI 10.1090/cbms/024
- 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, DOI 10.1016/0167-4889(95)00136-0
- Craig Huneke and Roger Wiegand, Tensor products of modules and the rigidity of $\textrm {Tor}$, Math. Ann. 299 (1994), no. 3, 449–476. MR 1282227, DOI 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 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 10.1006/jabr.1998.7743
- Stephen Lichtenbaum, On the vanishing of $\textrm {Tor}$ 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 10.1155/S1073792804133424
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
- 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.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2320653