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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


Generalized intersection multiplicities of modules

Author: Sankar P. Dutta
Journal: Trans. Amer. Math. Soc. 276 (1983), 657-669
MSC: Primary 13H15; Secondary 13D99, 13H10
MathSciNet review: 688968
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Abstract: In this paper we study intersection multiplicities of modules as defined by Serre and prove that over regular local rings of $ \dim \leqslant 5$, given two modules $ M,N$ with $ l(M\otimes_{R}N) < \infty $ and $ \dim\;M + \dim \;N < \dim \;R,\chi (M,N) = \sum\nolimits_{i = 0}^{\dim\; R}( - 1)^i l(\operatorname{Tor}_i^R(M,N)) = 0 $. We also study multiplicity in a more general set up. Finally we extend Serre's result from pairs of modules to pairs of finite free complexes whose homologies are killed by $ {I^n},{J^n}$, respectively, for some $ n > 0$, with $ \dim \,R/I + \dim \,R/J < \dim \,R$.

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PII: S 0002-9947(1983)0688968-4
Article copyright: © Copyright 1983 American Mathematical Society

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