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The vanishing of intersection multiplicities of perfect complexes. (English) Zbl 0585.13004
Let R be a commutative Noetherian local ring and M, N finitely generated R-modules of finite projective dimension such that \(M\otimes_ RN\) is a module of finite length. then the intersection multiplicity \(\chi (M,N)=\sum_{i\geq 0}(-1)^ ilength(Tor_ i(M,N))\) is well defined. When R is a regular local ring, the Krull dimensions of M and N are known to satisfy dim M\(+\dim N\leq \dim R\). The author proves that the following statement holds for arbitrary regular local rings (as conjectured by Serre) and for complete intersections and isolated singularities as well: if dim M\(+\dim N<\dim R\), then \(\chi (M,N)=0.\)
[See also H. Gillet and C. Soulé, C. R. Acad. Sci., Sér. I 300, 71-74 (1985; Zbl 0587.13007).]
Reviewer: T.W.Hungerford

13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
13H05 Regular local rings
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
13D25 Complexes (MSC2000)
13E05 Commutative Noetherian rings and modules
14M10 Complete intersections
13C12 Torsion modules and ideals in commutative rings
Full Text: DOI
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