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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)
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