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

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 , 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.

**[AB58]**Maurice Auslander and David A. Buchsbaum,*Codimension and multiplicity*, Ann. of Math. (2)**68**(1958), 625–657. MR**0099978****[Abe94]**I. M. Aberbach,*Finite phantom projective dimension*, Amer. J. Math.**116**(1994), no. 2, 447–477. MR**1269611**, 10.2307/2374936**[Abe03]**-, personal communication, 2003.**[AHH93]**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**, 10.1515/crll.1993.434.67**[Aus61]**M. Auslander,*Modules over unramified regular local rings*, Illinois J. Math.**5**(1961), 631–647. MR**0179211****[BH97]**Winfried Bruns and Jürgen Herzog,*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956****[Bur72]**Lindsay Burch,*Codimension and analytic spread*, Proc. Cambridge Philos. Soc.**72**(1972), 369–373. MR**0304377****[Eps06]**Neil M. Epstein,*Phantom depth and flat base change*, Proc. Amer. Math. Soc.**134**(2006), no. 2, 313–321. MR**2175997**, 10.1090/S0002-9939-05-08223-7**[Eps05]**Neil M. Epstein,*A tight closure analogue of analytic spread*, Math. Proc. Cambridge Philos. Soc.**139**(2005), no. 2, 371–383. MR**2168094**, 10.1017/S0305004105008546**[Hei93]**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**, 10.1090/S0273-0979-1993-00410-5**[HH90]**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**, 10.1090/S0894-0347-1990-1017784-6**[HH93]**Melvin Hochster and Craig Huneke,*Phantom homology*, Mem. Amer. Math. Soc.**103**(1993), no. 490, vi+91. MR**1144758**, 10.1090/memo/0490**[HH94]**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**, 10.1090/S0002-9947-1994-1273534-X**[HH00]**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**, 10.1307/mmj/1030132721**[HJW01]**Craig Huneke, David A. Jorgensen, and Roger Wiegand,*Vanishing theorems for complete intersections*, J. Algebra**238**(2001), no. 2, 684–702. MR**1823780**, 10.1006/jabr.2000.8603**[Hoc75]**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****[Hun96]**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****[HW94]**Craig Huneke and Roger Wiegand,*Tensor products of modules and the rigidity of 𝑇𝑜𝑟*, Math. Ann.**299**(1994), no. 3, 449–476. MR**1282227**, 10.1007/BF01459794**[HW97]**Craig Huneke and Roger Wiegand,*Tensor products of modules, rigidity and local cohomology*, Math. Scand.**81**(1997), no. 2, 161–183. MR**1612887****[Jor99]**David A. Jorgensen,*Complexity and Tor on a complete intersection*, J. Algebra**211**(1999), no. 2, 578–598. MR**1666660**, 10.1006/jabr.1998.7743**[Lic66]**Stephen Lichtenbaum,*On the vanishing of 𝑇𝑜𝑟 in regular local rings*, Illinois J. Math.**10**(1966), 220–226. MR**0188249****[Mat86]**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****[Mur63]**M. Pavaman Murthy,*Modules over regular local rings*, Illinois J. Math.**7**(1963), 558–565. MR**0156883****[SS04]**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**, 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

Email:
epstein@math.ukans.edu, neilme@umich.edu

DOI:
https://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.