On the second rigidity theorem of Huneke and Wiegand
HTML articles powered by AMS MathViewer
- by Olgur Celikbas and Ryo Takahashi PDF
- Proc. Amer. Math. Soc. 147 (2019), 2733-2739 Request permission
Abstract:
In 2007 Huneke and Wiegand announced in an erratum that one of the conclusions of their depth formula theorem is flawed due to an incorrect convention for the depth of the zero module. Since then, the deleted claim has remained unresolved. In this paper we give examples to prove that the deleted claim is false, in general. Moreover, we point out several places in the literature which relied upon this deleted claim or the initial argument from 2007.References
- Tokuji Araya and Yuji Yoshino, Remarks on a depth formula, a grade inequality and a conjecture of Auslander, Comm. Algebra 26 (1998), no.ย 11, 3793โ3806. MR 1647079, DOI 10.1080/00927879808826375
- M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631โ647. MR 179211
- Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
- Majid Rahro Zargar, Olgur Celikbas, Mohsen Gheibi, and Arash Sadeghi, Homological dimensions of rigid modules, Kyoto J. Math. 58 (2018), no.ย 3, 639โ669. MR 3843394, DOI 10.1215/21562261-2017-0033
- Olgur Celikbas and Greg Piepmeyer, Syzygies and tensor product of modules, Math. Z. 276 (2014), no.ย 1-2, 457โ468. MR 3150213, DOI 10.1007/s00209-013-1208-z
- Olgur Celikbas, Arash Sadeghi, and Ryo Takahashi, Bounds on depth of tensor products of modules, J. Pure Appl. Algebra 219 (2015), no.ย 5, 1670โ1684. MR 3299700, DOI 10.1016/j.jpaa.2014.07.005
- Olgur Celikbas, Srikanth B. Iyengar, Greg Piepmeyer, and Roger Wiegand, Criteria for vanishing of $\textrm {Tor}$ over complete intersections, Pacific J. Math. 276 (2015), no.ย 1, 93โ115. MR 3366029, DOI 10.2140/pjm.2015.276.93
- Hailong Dao, Some homological properties of modules over a complete intersection, with applications, Commutative algebra, Springer, New York, 2013, pp.ย 335โ371. MR 3051378, DOI 10.1007/978-1-4614-5292-8_{1}0
- S. P. Dutta, On modules of finite projective dimension, Nagoya Math. J. 219 (2015), 87โ111. MR 3413574, DOI 10.1215/00277630-3140702
- E. Graham Evans and Phillip Griffith, Syzygies, London Mathematical Society Lecture Note Series, vol. 106, Cambridge University Press, Cambridge, 1985. MR 811636, DOI 10.1017/CBO9781107325661
- Craig Huneke, David A. Jorgensen, and Roger Wiegand, Vanishing theorems for complete intersections, J. Algebra 238 (2001), no.ย 2, 684โ702. MR 1823780, DOI 10.1006/jabr.2000.8603
- Craig Huneke and Roger Wiegand, Tensor products of modules and the rigidity of $\textrm {Tor}$, Math. Ann. 299 (1994), no.ย 3, 449โ476. MR 1282227, DOI 10.1007/BF01459794
- Craig Huneke and Roger Wiegand, Tensor products of modules, rigidity and local cohomology, Math. Scand. 81 (1997), no.ย 2, 161โ183. MR 1612887, DOI 10.7146/math.scand.a-12871
- Wolmer V. Vasconcelos, Reflexive modules over Gorenstein rings, Proc. Amer. Math. Soc. 19 (1968), 1349โ1355. MR 237480, DOI 10.1090/S0002-9939-1968-0237480-2
- Craig Huneke and Roger Wiegand, Correction to: โTensor products of modules and the rigidity of Torโ [Math. Ann. 299 (1994), no. 3, 449โ476; MR1282227], Math. Ann. 338 (2007), no.ย 2, 291โ293. MR 2302064, DOI 10.1007/s00208-006-0076-9
- Graham J. Leuschke and Roger Wiegand, Cohen-Macaulay representations, Mathematical Surveys and Monographs, vol. 181, American Mathematical Society, Providence, RI, 2012. MR 2919145, DOI 10.1090/surv/181
Additional Information
- Olgur Celikbas
- Affiliation: Department of Mathematics, West Virginia University, Morgantown, West Virginia 26506-6310
- MR Author ID: 942955
- ORCID: 0000-0002-5306-7589
- Email: olgur.celikbas@math.wvu.edu
- Ryo Takahashi
- Affiliation: Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan; Department of Mathematics, University of Kansas, Snow Hall, 1460 Jayhawk Blvd., Lawrence, Kansas 66045-7523
- MR Author ID: 674867
- Email: takahashi@math.nagoya-u.ac.jp
- Received by editor(s): May 26, 2018
- Published electronically: April 9, 2019
- Additional Notes: The second author was partially supported by JSPS Grant-in-Aid for Scientific Research 16K05098 and 16KK0099.
- Communicated by: Claudia Polini
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2733-2739
- MSC (2010): Primary 13D07; Secondary 13C13, 13C14, 13H10
- DOI: https://doi.org/10.1090/proc/14564
- MathSciNet review: 3973877
Dedicated: Dedicated to the memory of Ragnar-Olaf Buchweitz