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Intersection multiplicities in commutative algebra. (English) Zbl 0596.13012
The author proves the vanishing of intersection multiplicity over a complete intersection or a local ring whose singular locus is of dimension not greater than one under the assumption that modules are of finite projective dimension, that is: Let A be a complete intersection or a local ring with dim Sing(A)$$\leq 1$$, and let M and N be finitely generated A-modules of finite projective dimension such that $$M\otimes_ AN$$ is of finite length. If dim M$$+\dim N<\dim A$$, then $$\sum_{i\geq 0}(-1)^ i length(Tor^ A_ i(M,N))=0\quad [see$$ the author, ”Local Chern characters and intersection multiplicity”, and Bull. Am. Math. Soc., New Ser. 13, 127-130 (1985; Zbl 0585.13004)]. The author’s method largely depends on the theory of local Chern characters which was mostly developed in the book ”Intersection theory” by W. Fulton (1984; Zbl 0541.14005). The author establishes some further properties of local Chern characters and proves the vanishing stated above. In this paper the author explains how the theory of local Chern characters works in the theory of intersection multiplicity and in demonstrating his result of its vanishing.
Reviewer: Y.Aoyama
##### MSC:
 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 14M10 Complete intersections 13H15 Multiplicity theory and related topics