Complexity of tensor products of modules and a theorem of Huneke-Wiegand
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- by Claudia Miller PDF
- Proc. Amer. Math. Soc. 126 (1998), 53-60 Request permission
Abstract:
This paper concerns the notion of complexity, a measure of the growth of the Betti numbers of a module. We show that over a complete intersection $R$ the complexity of the tensor product $M\otimes _{R} N$ of two finitely generated modules is the sum of the complexities of each if $\operatorname {Tor}_{i}^{R}(M,N)=0$ for $i\geq 1$. One of the applications is simplification of the proofs of central results in a paper of C. Huneke and R. Wiegand on the tensor product of modules and the rigidity of Tor.References
- M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631โ647. MR 179211, DOI 10.1215/ijm/1255631585
- L. L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), no.ย 1, 71โ101. MR 981738, DOI 10.1007/BF01393971
- David Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no.ย 1, 35โ64. MR 570778, DOI 10.1090/S0002-9947-1980-0570778-7
- Tor H. Gulliksen, A change of ring theorem with applications to Poincarรฉ series and intersection multiplicity, Math. Scand. 34 (1974), 167โ183. MR 364232, DOI 10.7146/math.scand.a-11518
- 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
- C. Huneke and R. Wiegand, Tensor products of modules, rigidity, and local cohomology, submitted.
- D. Jorgensen, Complexity and Tor on a complete intersection, submitted.
- Stephen Lichtenbaum, On the vanishing of $\textrm {Tor}$ in regular local rings, Illinois J. Math. 10 (1966), 220โ226. MR 188249
- Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
Additional Information
- Claudia Miller
- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
- Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- Email: cmiller@math.uiuc.edu
- Received by editor(s): March 25, 1996
- Received by editor(s) in revised form: July 5, 1996
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 53-60
- MSC (1991): Primary 13C14, 13C40, 13D05, 13D40, 13H10
- DOI: https://doi.org/10.1090/S0002-9939-98-04017-9
- MathSciNet review: 1415354