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

MathSciNet review:
1415354

Full-text PDF Free Access

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 the complexity of the tensor product of two finitely generated modules is the sum of the complexities of each if for . 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.

**[Au]**M. Auslander,*Modules over unramified regular local rings*, Illinois J. Math.**5**(1961), 631–647. MR**0179211****[Av]**L. L. Avramov,*Modules of finite virtual projective dimension*, Invent. Math.**96**(1989), no. 1, 71–101. MR**981738**, 10.1007/BF01393971**[E]**David Eisenbud,*Homological algebra on a complete intersection, with an application to group representations*, Trans. Amer. Math. Soc.**260**(1980), no. 1, 35–64. MR**570778**, 10.1090/S0002-9947-1980-0570778-7**[Gu]**Tor H. Gulliksen,*A change of ring theorem with applications to Poincaré series and intersection multiplicity*, Math. Scand.**34**(1974), 167–183. MR**0364232****[HW1]**Craig Huneke and Roger Wiegand,*Tensor products of modules and the rigidity of 𝑇𝑜𝑟*, Math. Ann.**299**(1994), no. 3, 449–476. MR**1282227**, 10.1007/BF01459794**[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]**Stephen Lichtenbaum,*On the vanishing of 𝑇𝑜𝑟 in regular local rings*, Illinois J. Math.**10**(1966), 220–226. MR**0188249****[M]**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**

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:
http://dx.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