Roberts, Paul The vanishing of intersection multiplicities of perfect complexes. (English) Zbl 0585.13004 Bull. Am. Math. Soc., New Ser. 13, 127-130 (1985). 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 Cited in 13 ReviewsCited in 44 Documents MSC: 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 Keywords:perfect complexes; Noetherian local ring; intersection multiplicity; Krull dimensions; regular local rings; complete intersections; isolated singularities Citations:Zbl 0587.13007 PDFBibTeX XMLCite \textit{P. Roberts}, Bull. Am. Math. Soc., New Ser. 13, 127--130 (1985; Zbl 0585.13004) Full Text: DOI References: [1] Paul Baum, William Fulton, and Robert MacPherson, Riemann-Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 101 – 145. · Zbl 0332.14003 [2] Sankar P. Dutta, M. Hochster, and J. E. McLaughlin, Modules of finite projective dimension with negative intersection multiplicities, Invent. Math. 79 (1985), no. 2, 253 – 291. · Zbl 0588.13020 · doi:10.1007/BF01388973 [3] Hans-Bjørn Foxby, The MacRae invariant, Commutative algebra: Durham 1981 (Durham, 1981) London Math. Soc. Lecture Note Ser., vol. 72, Cambridge Univ. Press, Cambridge, 1982, pp. 121 – 128. · Zbl 0544.13012 [4] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005 [5] Henri Gillet and Christophe Soulé, \?-théorie et nullité des multiplicités d’intersection, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 3, 71 – 74 (French, with English summary). · Zbl 0587.13007 [6] Christian Peskine and Lucien Szpiro, Syzygies et multiplicités, C. R. Acad. Sci. Paris Sér. A 278 (1974), 1421 – 1424 (French). · Zbl 0281.13004 [7] Jean-Pierre Serre, Algèbre locale. Multiplicités, Cours au Collège de France, 1957 – 1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, vol. 11, Springer-Verlag, Berlin-New York, 1965 (French). · Zbl 0142.28603 [8] L. Szpiro, Sur la théorie des complexes parfaits, Commutative algebra: Durham 1981 (Durham, 1981) London Math. Soc. Lecture Note Ser., vol. 72, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 83 – 90. · Zbl 0554.13005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.