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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Generalized intersection multiplicities of modules


Author: Sankar P. Dutta
Journal: Trans. Amer. Math. Soc. 276 (1983), 657-669
MSC: Primary 13H15; Secondary 13D99, 13H10
DOI: https://doi.org/10.1090/S0002-9947-1983-0688968-4
MathSciNet review: 688968
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Abstract: In this paper we study intersection multiplicities of modules as defined by Serre and prove that over regular local rings of $ \dim \leqslant 5$, given two modules $ M,N$ with $ l(M\otimes_{R}N) < \infty $ and $ \dim\;M + \dim \;N < \dim \;R,\chi (M,N) = \sum\nolimits_{i = 0}^{\dim\; R}( - 1)^i l(\operatorname{Tor}_i^R(M,N)) = 0 $. We also study multiplicity in a more general set up. Finally we extend Serre's result from pairs of modules to pairs of finite free complexes whose homologies are killed by $ {I^n},{J^n}$, respectively, for some $ n > 0$, with $ \dim \,R/I + \dim \,R/J < \dim \,R$.


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  • [A-B] M. Auslander and D. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625-657. MR 0099978 (20:6414)
  • [C-E] H. Cartan and S. Eilenberg, Homological algebra, Chap. XVI, Princeton Univ. Press, Princeton, N. J., 1956. MR 0077480 (17:1040e)
  • [D1] S. P. Dutta, Weak linking and intersection multiplicity, J. Pure Appl. Algebra (to appear). MR 687746 (84g:13036)
  • [D2] -, Frobenius and multiplicities, J. Algebra (to appear). MR 725094 (85f:13022)
  • [F] H. B. Foxby, Intersection multiplicities of modules (preprint).
  • [F-F-I] R. M. Fossum, H. B. Foxby and B. Iverson, A characteristic class in algebraic $ k$-theory (preprint).
  • [H] M. Hochster, Conference on Commutative Algebra, Lecture Notes in Math., vol. 311, Springer-Verlag, Berlin and New York, 1973, pp. 120-152. MR 0340251 (49:5006)
  • [L] S. Lichtenbaum, On the vanishing of $ \operatorname{Tor}$ in regular local rings, Illinois J. Math. 10 (1966), 220-226. MR 0188249 (32:5688)
  • [M] M. P. Mallaivain-Brameret, Une remarque sur les anneaux locaux réguliers, Sém. Dubreil-Pisot (Algèbre et Théorie des Nombres), 1970/71, n$ ^\circ$13.
  • [P-S] C. Peskine and L. Szpiro, Syzygies et multiplicité, C. R. Acad. Sci. Paris Ser. A-B 278 (1974), 1421-1424. MR 0349659 (50:2152)
  • [S] J. P. Serre, Algèbre locale. Multiplicité, Lecture Notes in Math., vol. 11, Springer-Verlag, Berlin and New York, 1975.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0688968-4
Article copyright: © Copyright 1983 American Mathematical Society

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