Generalized intersection multiplicities of modules

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 $\dim R/I + \dim R/J < \dim R$.