On the second rigidity theorem of Huneke and Wiegand

Celikbas, Olgur; Takahashi, Ryo

doi:10.1090/proc/14564

On the second rigidity theorem of Huneke and Wiegand
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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. 11, 3793–3806. MR 1647079, DOI 10.1080/00927879808826375
M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647. 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. 3, 639–669. MR 3843394, DOI 10.1215/21562261-2017-0033
Olgur Celikbas and Greg Piepmeyer, Syzygies and tensor product of modules, Math. Z. 276 (2014), no. 1-2, 457–468. 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. 5, 1670–1684. 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. 1, 93–115. 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. 335–371. 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), 87–111. 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. 2, 684–702. 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. 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
Wolmer V. Vasconcelos, Reflexive modules over Gorenstein rings, Proc. Amer. Math. Soc. 19 (1968), 1349–1355. MR 237480, DOI 10.1090/S0002-9939-1968-0237480-2
Craig Huneke and Roger Wiegand, Correction to: “Tensor products of modules and the rigidity of Tor” [Math. Ann. 299 (1994), no. 3, 449–476; MR1282227], Math. Ann. 338 (2007), no. 2, 291–293. 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