Resultants and the algebraicity of the join pairing on Chow varieties

Plümer, Judith

doi:10.1090/S0002-9947-97-01888-6

Resultants and the algebraicity of the join pairing on Chow varieties
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by Judith Plümer PDF

Trans. Amer. Math. Soc. 349 (1997), 2187-2209 Request permission

Abstract:

The Chow/Van der Waerden approach to algebraic cycles via resultants is used to give a purely algebraic proof for the algebraicity of the complex suspension. The algebraicity of the join pairing on Chow varieties then follows. The approach implies a more algebraic proof of Lawson’s complex suspension theorem in characteristic 0. The continuity of the action of the linear isometries operad on the group completion of the stable Chow variety is a consequence.

References

Daniel Barlet, Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie, Fonctions de plusieurs variables complexes, II (Sém. François Norguet, 1974–1975) Lecture Notes in Math., Vol. 482, Springer, Berlin, 1975, pp. 1–158 (French). MR 0399503
D. Barlet, Preprint (1993).
J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 0420609, DOI 10.1007/BFb0068547
Charles P. Boyer, H. Blaine Lawson Jr., Paulo Lima-Filho, Benjamin M. Mann, and Marie-Louise Michelsohn, Algebraic cycles and infinite loop spaces, Invent. Math. 113 (1993), no. 2, 373–388. MR 1228130, DOI 10.1007/BF01244311
W.–L. Chow and B. L. van der Waerden, ZAG 9 Über zugeordnete Formen und algebraische Systeme von algebraischen Mannigfaltigkeiten. Math. Annalen 113 (1937), 692 - 704.
J. Dalbec, The Join of Two Varieties. Preprint.
Albrecht Dold and René Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. (2) 67 (1958), 239–281 (German). MR 97062, DOI 10.2307/1970005
Eric M. Friedlander, Algebraic cycles, Chow varieties, and Lawson homology, Compositio Math. 77 (1991), no. 1, 55–93. MR 1091892
I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
Phillip Griffiths, Topics in algebraic and analytic geometry, Mathematical Notes, No. 13, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Written and revised by John Adams. MR 0355119
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Ernst Kunz, Introduction to commutative algebra and algebraic geometry, Birkhäuser Boston, Inc., Boston, MA, 1985. Translated from the German by Michael Ackerman; With a preface by David Mumford. MR 789602
H. Blaine Lawson Jr., Algebraic cycles and homotopy theory, Ann. of Math. (2) 129 (1989), no. 2, 253–291. MR 986794, DOI 10.2307/1971448
H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Algebraic cycles, Bott periodicity, and the Chern characteristic map, The mathematical heritage of Hermann Weyl (Durham, NC, 1987) Proc. Sympos. Pure Math., vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 241–263. MR 974339, DOI 10.1090/pspum/048/974339
Paulo Lima-Filho, Completions and fibrations for topological monoids, Trans. Amer. Math. Soc. 340 (1993), no. 1, 127–147. MR 1134758, DOI 10.1090/S0002-9947-1993-1134758-4
J. Peter May, $E_{\infty }$ ring spaces and $E_{\infty }$ ring spectra, Lecture Notes in Mathematics, Vol. 577, Springer-Verlag, Berlin-New York, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave. MR 0494077
M. C. McCord, Classifying spaces and infinite symmetric products, Trans. Amer. Math. Soc. 146 (1969), 273–298. MR 251719, DOI 10.1090/S0002-9947-1969-0251719-4
David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
J. Plümer, Stabilisierte Chow–Varietäten und die Chernklassenabbildung. Diplomarbeit, Universität Osnabrück (1993).
I. R. Shafarevich, Basic algebraic geometry, Die Grundlehren der mathematischen Wissenschaften, Band 213, Springer-Verlag, New York-Heidelberg, 1974. Translated from the Russian by K. A. Hirsch. MR 0366917, DOI 10.1007/978-3-642-96200-4
B. L. van der Waerden, Algebra. Teil II, Heidelberger Taschenbücher, Band 23, Springer-Verlag, Berlin-New York, 1967 (German). Unter Benutzung von Vorlesungen von E. Artin und E. Noether; Fünfte Auflage. MR 0233647, DOI 10.1007/978-3-642-96045-1
B. L. van der Waerden, ZAG 6 Algebraische Korrespondenzen und rationale Abbildungen. Math. Annalen 110 (1935), 134 - 160.