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A chain morphism for Adams operations on rational algebraic K-theory

Published online by Cambridge University Press:  13 November 2009

Elisenda Feliu
Elisenda Feliu
Affiliation:
Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain, [email protected]
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Abstract

For any regular noetherian scheme X and every k ≥ 1, we define a chain morphism ψk between two chain complexes whose homology with rational coefficients is isomorphic to the algebraic K-groups of X tensored by ℚ. It is shown that the morphisms ψk induce in homology the Adams operations defined by Gillet and Soulé or the ones defined by Grayson.

Keywords

Adams operationshigher algebraic K-theorychain complex of cubestransgression bundle
Type
Research Article
Information
Journal of K-Theory , Volume 5 , Issue 2 , April 2010 , pp. 349 - 402
Copyright
Copyright © ISOPP 2009

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References

1. Théorie des topos et cohomologie étale des schémas. Tome 3, Springer-Verlag, Berlin, 1973, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964.(SGA 4), Artin, Dirigé par M., Grothendieck, A. et Verdier, J. L.. Avec la collaboration de P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics 305.Google Scholar
2.Gil, J. I. Burgos and Wang, S., Higher Bott-Chern forms and Beilinson's regulator, Invent. Math. 132 (1998), no. 2, 261–305.Google Scholar
3.Deligne, P., Le déterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 93–177.CrossRefGoogle Scholar
4.Feliu, E., On uniqueness of characteristic classes, Preprint arxiv:math/0606244v2.Google Scholar
5.Feliu, E., Adams operations on higher arithmetic K-theory, To appear in Publications of the Research Institute for Mathematical Sciences, 2009.CrossRefGoogle Scholar
6.Gillet, H. and Soulé, C., Filtrations on higher algebraic K-theory, Algebraic K-theory (Seattle, WA, 1997), Proc. Sympos. Pure Math. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 89–148.Google Scholar
7.Grayson, D. R., Adams operations on higher K-theory, K-Theory 6 (1992), no. 2, 97– 111.CrossRefGoogle Scholar
8.Lecomte, F., Opérations d'Adams en K-théorie algébrique, K-Theory 13 (1998), no. 2, 179–207.CrossRefGoogle Scholar
9.McCarthy, R., A chain complex for the spectrum homology of the algebraic K-theory of an exact category, Algebraic K-theory (Toronto, ON, 1996), Fields Inst. Commun. 16, Amer. Math. Soc., Providence, RI, 1997, pp. 199–220.Google Scholar
10.Quillen, D., Higher algebraic K-theory. I, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math. 341.Google Scholar
11.Schechtman, V. V., On the delooping of Chern character and Adams operations, K-theory, arithmetic and geometry (Moscow, 1984–1986., Lecture Notes in Math. 1289, Springer, Berlin, 1987, pp. 265–319.Google Scholar
12.Soulé, C., Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics 33, Cambridge University Press, Cambridge, 1992, With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer.CrossRefGoogle Scholar
13.Takeda, Y., Higher arithmetic K-theory, Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, 599–681.CrossRefGoogle Scholar