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K-théorie et nullité des multiplicités d’intersection. (K-theory and vanishing of intersection multiplicities). (French) Zbl 0587.13007
In this paper a conjecture of J.-P. Serre is proved. Let R be a regular ring and let $$(x_ 1,...,x_ n)$$ be a regular sequence in R such that $$A=R/(x_ 1,...,x_ n)$$ is a local noetherian ring. Let $$M_ 1$$ and $$M_ 2$$ be two A-modules of finite type such that $$Supp(M_ 1)\cap Supp(M_ 2)$$ is the maximal ideal of A. Define the intersection multiplicity of $$M_ 1$$ and $$M_ 2$$ to be $$\chi (M_ 1,M_ 2)=\sum_{j\geq 0}(-1)^ j$$ $$length(Tor^ A_ j(M_ 1,M_ 2)).$$
Theorem: If $$\dim (Supp(M_ 1))+\dim (Supp(M_ 2))<\dim (Spec A)$$, then $$\chi (M_ 1,M_ 2)=0.$$
The proof uses $$\lambda$$-structures on algebraic K-theory and $$\gamma$$- filtrations. The result was proved independently and by different techniques by P. Roberts [Bull. Am. Math.-Soc., New Ser. 13, 127- 130 (1985; Zbl 0585.13004)].
Reviewer: S.Geller

MSC:
 13D15 Grothendieck groups, $$K$$-theory and commutative rings 13C15 Dimension theory, depth, related commutative rings (catenary, etc.) 13H05 Regular local rings 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry 18F25 Algebraic $$K$$-theory and $$L$$-theory (category-theoretic aspects)