Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Complexity of tensor products of modules
and a theorem of Huneke-Wiegand


Author: Claudia Miller
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [Au] M. Auslander, Modules over unramified regular local rings, Ill. J. Math. 5 (1961), 631-647. MR 31:3460
  • [Av] L. Avramov, Modules of finite virtual projective dimension, Invent. Math. 96 (1989), 71-101. MR 90g:13027
  • [E] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35-64. MR 82d:13013
  • [Gu] T. Gulliksen, A change of rings theorem, with applications to Poincaré series and intersection multiplicity, Math. Scand. 34 (1974), 167-183. MR 51:487
  • [HW1] C. Huneke and R. Wiegand, Tensor products of modules and the rigidity of Tor, Math. Ann. 299 (1994), 449-476. MR 95m:13008
  • [HW2] C. Huneke and R. Wiegand, Tensor products of modules, rigidity, and local cohomology, submitted.
  • [J] D. Jorgensen, Complexity and Tor on a complete intersection, submitted.
  • [L] S. Lichtenbaum, On the vanishing of Tor in regular local rings, Ill. J. Math. 10 (1966), 220-226. MR 32:5688
  • [M] H. Matsumura, Commutative Ring Theory, in Cambridge Studies in Advanced Mathematics, no. 8, Cambridge Univ. Press, Cambridge, 1989. MR 90i:13001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 13C14, 13C40, 13D05, 13D40, 13H10

Retrieve articles in all journals with MSC (1991): 13C14, 13C40, 13D05, 13D40, 13H10


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

DOI: https://doi.org/10.1090/S0002-9939-98-04017-9
Keywords: Complexity, complete intersection, hypersurface, rigidity, tensor product
Received by editor(s): March 25, 1996
Received by editor(s) in revised form: July 5, 1996
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society