Abstract
We prove a projection formula, expressing a relative Buchsbaum–Rim multiplicity in terms of corresponding ones over a module-finite algebra of pure degree, generalizing an old formula for the ordinary (Samuel) multiplicity. Our proof is simple in spirit: after the multiplicities are expressed as sums of intersection numbers, the desired formula results from two projection formulas, one for cycles and another for Chern classes. Similarly, but without using any projection formula, we prove an expansion formula, generalizing the additivity formula for the ordinary multiplicity, a case of the associativity formula.
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Buchsbaum, D., Rim, D.: A generalized Koszul complex. II. Depth and multiplicity. Trans. Amer. Math. Soc 111, 197–224 (1964). (28 #3076)
Chevalley, C.: Intersections of algebraic and algebroid varieties. Trans. Amer. Math. Soc 57, 1–85 (1945). (7, 26c)
Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry.” Graduate Texts in Mathematics, vol. 150. Springer, New York (1995). (97a:13001)
Eisenbud, D., Huneke, C., Ulrich, B.: What is the Rees algebra of a module? Proc. Amer. Math. Soc 131(3), 701–708 (2003). (2003:13003)
Fulton, W.: “Intersection Theory.” Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2. Springer, Berlin (1984). (85k:14004)
Gaffney, T., Rangachev, A.: Pairs of modules and determinantal isolated singularities. arXiv:1501.00201v2 [math.CV]
Huneke, C., Swanson, I.: “Integral closure of ideals, rings, and modules,” London Mathematical Society Lecture Note Series, 336. Cambridge University Press, Cambridge (2006). (2008m:13013)
Kirby, D., Rees, D.: Multiplicities in graded rings. I. The general theory, Commutative algebra: syzygies, multiplicities, and birational algebra, Contemp. Math. Amer. Math. Soc. Provid. R 159, 209–267 (1994). (95b:13002)
Kleiman, S. L.: The Picard scheme, in “Fundamental Algebraic Geometry,” pp. 235–321, Math. Surveys Monogr., 123, Amer. Math. Soc., Providence (2005)
Kleiman, S., Thorup, A.: A geometric theory of the Buchsbaum–Rim multiplicity. J. Algebra 167(1), 168–231 (1994). (96a:14007)
Kleiman, S., Thorup, A.: Mixed Buchsbaum–Rim multiplicities, Amer. J. Math. 118(3), 529–569 (1996). (98g:14008)
Lech, C.: On the associativity formula for multiplicities, Ark. Mat. 3, 301–314 (1957). (19,11a)
Mumford, D.: “Lectures on Curves on an Algebraic Surface.” Annals of Mathematics Studies, vol. 59. Princeton University Press, Princeton (1966). (35 #187)
Ramanujam, C. P.: On a geometric interpretation of multiplicity, Invent. Math. 22, 63–67 (1973/74). (50 #7141)
Samuel, P.: “Algèbre locale.” Mémor. Sci. Math., vol. 123. Gauthier–Villars, Paris (1953). (14, #1012c)
Samuel, P.: “Méthodes d’algèbre abstraite en géométrie algébrique.” Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 4. Springer, Berlin (1955). (17,300b)
Serre, J.-P.: “Algèbra Locale, Multiplicités,” Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, vol. 11. Springer (1965). (34 #1352)
Zariski, O., Samuel, P.: “Commutative Algebra. Vol. II.” The University Series in Higher Mathematics. D. Van Nostrand Co., Inc, Princeton (1960). (22 #11006)
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To the memory of Alexandru (Bobi) Lascu: a dear friend, tireless colleague, and inspiring coauthor.
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Kleiman, S.L. Two formulas for the BR multiplicity. Ann Univ Ferrara 63, 147–158 (2017). https://doi.org/10.1007/s11565-016-0250-2
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DOI: https://doi.org/10.1007/s11565-016-0250-2