Duke Mathematical Journal

Higher algebraic K-theory of group actions with finite stabilizers

Gabriele Vezzosi and Angelo Vistoli

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Abstract

We prove a decomposition theorem for the equivariant $K$-theory of actions of affine group schemes $G$ of finite type over a field on regular separated Noetherian algebraic spaces, under the hypothesis that the actions have finite geometric stabilizers and satisfy a rationality condition together with a technical condition that holds, for example, for $G$ abelian or smooth. We reduce the problem to the case of a ${\rm GL}\sb n$-action and finally to a split torus action.

Article information

Source
Duke Math. J., Volume 113, Number 1 (2002), 1-55.

Dates
First available in Project Euclid: 18 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1087575224

Digital Object Identifier
doi:10.1215/S0012-7094-02-11311-8

Mathematical Reviews number (MathSciNet)
MR1905391

Zentralblatt MATH identifier
1012.19002

Subjects
Primary: 19E08: $K$-theory of schemes [See also 14C35]
Secondary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]

Citation

Vezzosi, Gabriele; Vistoli, Angelo. Higher algebraic K -theory of group actions with finite stabilizers. Duke Math. J. 113 (2002), no. 1, 1--55. doi:10.1215/S0012-7094-02-11311-8. https://projecteuclid.org/euclid.dmj/1087575224


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References

  • M. Artin, A. Grothendieck, and J. L. Verdier, Théorie des topos et cohomologie étale des schémas, Vol. 3, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4), Lecture Notes in Math. 305, Springer, Berlin, 1973.
  • P. Berthelot, A. Grothendieck, and L. Illusie, Théorie des intersections et théorème de Riemann-Roch, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 6), Lecture Notes in Math. 225, Springer, Berlin, 1971.
  • M. Demazure and P. Gabriel, Groupes algébriques, Vol. 1: Géométrie algébrique, généralités, groupes commutatifs, Masson, Paris, 1970.
  • M. Demazure and A. Grothendieck, Schémas en groupes, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 3), Lecture Notes in Math. 151, 152, 153, Springer, Berlin, 1970.,
  • D. Edidin and W. Graham, Riemann-Roch for equivariant Chow groups, Duke Math. J. 102 (2000), 567--594.
  • A. Grothendieck, Revêtements étales et groupe fondamental, Séminaire de Géométrie algébrique du Bois-Marie (SGA 1), Lecture Notes in Math. 224, Springer, Berlin, 1971.
  • A. Grothendieck and J. A. Dieudonné, Eléments de géométrie algébrique, I: Le langage des schémas, Springer, Berlin, 1971.
  • R. Joshua, Higher intersection theory on algebraic stacks, II, preprint, 2000, http://math.ohio-state.edu/~joshua/pub.html
  • G. Laumon and L. Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb. (3) 39, Springer, Berlin, 2000.
  • A. S. Merkurjev, Comparison of the equivariant and the standard $K$-theory of algebraic varieties, St. Petersburg Math. J. 9 (1998), 815--850.
  • D. Quillen, ``Higher algebraic $K$-theory, I'' in Algebraic $K$-Theory, I: Higher $K$-Theories (Seattle, 1972), Lecture Notes in Math. 341, Springer, Berlin, 1973.
  • J.-P. Serre, Linear representations of finite groups, Grad. Texts in Math. 42, Springer, New York, 1977.
  • V. Srinivas, Algebraic $K$-Theory, 2d ed., Progr. Math. 90, Birkhäuser, Boston, 1996.
  • R. W. Thomason, Comparison of equivariant algebraic and topological $K$-theory, Duke Math. J. 53 (1986), 795--825.
  • --. --. --. --., Lefschetz-Riemann-Roch theorem and coherent trace formula, Invent. Math. 85 (1986), 515--543.
  • --. --. --. --., ``Algebraic $K$-theory of group scheme actions'' in Algebraic Topology and Algebraic $K$-Theory (Princeton, 1983), Ann. of Math. Stud. 113, Princeton Univ. Press, Princeton, 1987, 539--563.
  • --. --. --. --., Equivariant algebraic vs. topological $K$-homology Atiyah-Segal-style, Duke Math. J. 56 (1988), 589--636.
  • --. --. --. --., Une formule de Lefschetz en $K$-théorie équivariante algébrique, Duke Math. J. 68 (1992), 447--462.
  • --. --. --. --., Les $K$-groupes d'un schéma éclaté et une formule d'intersection excédentaire, Invent. Math. 112 (1993), 195--215.
  • R. W. Thomason and T. Trobaugh, ``Higher algebraic $K$-theory of schemes and of derived categories'' in The Grothendieck Festschrift, Vol. III, Progr. Math. 88, Birkhäuser, Boston, 1990, 247--435.
  • B. Toen, Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford, $K$-Theory 18 (1999), 33--76.
  • --------, Notes on $G$-theory of Deligne-Mumford stacks, preprint.
  • A. Vistoli, Higher equivariant $K$-theory for finite group actions, Duke Math. J. 63 (1991), 399--419.
  • --. --. --. --., ``Equivariant Grothendieck groups and equivariant Chow groups'' in Classification of Irregular Varieties (Trento, Italy, 1990), ed. E. Ballico, F. Catanese, and C. Ciliberto, Lecture Notes in Math. 1515, Springer, Berlin, 1992, 112--133.