Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T17:35:16.294Z Has data issue: false hasContentIssue false
  • English
  • Français

Torsion in tensor powers of modules

Part of: Theory of modules and ideals Commutative algebra: Homological methods

Published online by Cambridge University Press:  11 January 2016

Olgur Celikbas ,
Srikanth B. Iyengar ,
Greg Piepmeyer  and
Roger Wiegand
Olgur Celikbas
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA
Srikanth B. Iyengar
Affiliation:
Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130, USA
Greg Piepmeyer
Affiliation:
Columbia Basin College Pasco, Washington 99301, USA
Roger Wiegand
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Tensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules N with the property that M ⊗R N has nonzero torsion unless M is very special. An important example of such a module N is the Frobenius power peR over a complete intersection domain R of characteristic p > 0.

MSC classification

Secondary: 13C12: Torsion modules and ideals 13D07: Homological functors on modules (Tor, Ext, etc.)
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 219 , September 2015 , pp. 113 - 125
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

[1] Auslander, M., Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631–647. MR 0179211.CrossRefGoogle Scholar
[2] Avramov, L. L., Hochster, M., Iyengar, S. B., and Yao, Y., Homological invariants of modules over contracting endomorphisms, Math. Ann. 353 (2012), 275–291. MR 2915536. DOI 10.1007/s00208-011-0682-z.Google Scholar
[3] Avramov, L. L. and Miller, C., Frobenius powers of complete intersections, Math. Res. Lett. 8 (2001), 225–232. MR 1825272. DOI 10.4310/MRL.2001.v8.n2.a10.CrossRefGoogle Scholar
[4] Bruns, W. and Herzog, J., Cohen-Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993. MR 1251956.Google Scholar
[5] Herzog, J., Ringe der Charakteristik p und Frobeniusfunktoren, Math. Z. 140 (1974), 67–78. MR 0352081.CrossRefGoogle Scholar
[6] Huneke, C., Jorgensen, D. A., and Wiegand, R., Vanishing theorems for complete intersections, J. Algebra 238 (2001), 684–702. MR 1823780. DOI 10.1006/jabr.2000.8603.CrossRefGoogle Scholar
[7] Huneke, C. and Wiegand, R., Tensor products of modules, rigidity and local cohomology, Math. Scand. 81 (1997), 161–183. MR 1612887.Google Scholar
[8] Kunz, E., Characterization of regular local rings for characteristic p, Amer. J. Math. 91 (1969), 772–784. MR 0252389.CrossRefGoogle Scholar
[9] Leuschke, G. J. and Wiegand, R., Cohen-Macaulay Representations, Math. Surveys Monogr. 181, Amer. Math. Soc., Providence, 2012. MR 2919145. DOI 10.1090/surv/181.Google Scholar
[10] Lichtenbaum, S., On the vanishing of TOR in regular local rings, Illinois J. Math. 10 (1966), 220–226. MR 0188249.CrossRefGoogle Scholar
[11] Mac Lane, S., Homology, reprint of the 1975 original, Springer, Berlin, 1995. MR 1344215.Google Scholar
[12] Peskine, C. and Szpiro, L., Dimension projective finie et cohomologie locale, applications à la démonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck, Publ. Math. Inst. Hautes Études Sci. 42 (1973), 47–119. MR 0374130. CrossRefGoogle Scholar
[13] Speyer, C., Torsion-free tensor powers, preprint, http://mathoverflow.net/ questions/73120/torsion-freetensor-powers (accessed 11 June 2015).Google Scholar