The Grothendieck-Riemann-Roch theorem for group scheme actions

Köck, Bernhard

doi:10.1016/s0012-9593(98)80140-7

Köck, Bernhard

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 31 (1998) no. 3, pp. 415-458.

MR Zbl | 4 citations dans Numdam

DOI : 10.1016/s0012-9593(98)80140-7

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     author = {K\"ock, Bernhard},
     title = {The {Grothendieck-Riemann-Roch} theorem for group scheme actions},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {415--458},
     publisher = {Elsevier},
     volume = {Ser. 4, 31},
     number = {3},
     year = {1998},
     doi = {10.1016/s0012-9593(98)80140-7},
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     zbl = {0951.14029},
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     url = {http://archive.numdam.org/articles/10.1016/s0012-9593(98)80140-7/}
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Köck, Bernhard. The Grothendieck-Riemann-Roch theorem for group scheme actions. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 31 (1998) no. 3, pp. 415-458. doi : 10.1016/s0012-9593(98)80140-7. http://archive.numdam.org/articles/10.1016/s0012-9593(98)80140-7/

Bibliographie
Cité par

[At] M. F. Atiyah, Characters and cohomology of finite groups (Publ. Math. IHES, Vol. 9, 1961, pp. 23-64). | Numdam | MR | Zbl

[AT] M. F. Atiyah and D. O. Tall, Group representations, λ-rings and the J-homomorphism (Topology, Vol. 8, 1969, pp. 253-297). | MR | Zbl

[Ba] H. Bass, Algebraic K-theory (Math. Lecture Note Series, Benjamin, New York, 1968). | MR | Zbl

[SGA6] P. Berthelot, A. Grothendieck and L. Illusie, Théorie des Intersections et Théorème de Riemann-Roch (Lecture Notes in Math., Vol. 225, Springer, New York, 1971). | MR | Zbl

[Bo] R. Boltje, Mackey functors and related structures in representation theory and number theory, Report No. 327, Institut für Mathematik der Universität Augsburg, 1995.

[BV] M. Brion and M. Vergne, An equivariant Riemann-Roch theorem for complete, simplicial toric varieties (J. Reine Angew. Math., Vol. 482, 1997, pp. 67-92). | MR | Zbl

[BC] D. Burns and T. Chinburg, Adams operations and integral Hermitian-Galois representations (Amer. J. Math., Vol. 118, 1996, pp. 925-962). | MR | Zbl

[CNT] Ph. Cassou-Noguès and M. J. Taylor, Opérations d'Adams et Groupe des classes d'Algèbre de groupe (J. Algebra, Vol. 95, 1985, pp. 125-152). | MR | Zbl

[CEPT] T. Chinburg, B. Erez, G. Pappas and M. J. Taylor, Riemann-Roch type theorems for arithmetic schemes with a finite group action (J. Reine Angew. Math., Vol. 489, 1997, pp. 151-187). | MR | Zbl

[CR] C. W. Curtis and I. Reiner, Methods of representation theory with applications to finite groups and orders I (Pure Appl. Math., Wiley, New York, 1981). | MR | Zbl

[De] M. Demazure, Désingularisation des variétés de Schubert généralisées (Ann. Sci. École Norm. Sup., (4), Vol. 7, 1974, pp. 53-88). | EuDML | Numdam | MR | Zbl

[SGA3] M. Demazure and A. Grothendieck, Schémas en Groupes I, II, III (Lecture Notes in Math., Vol. 151-152-153, Springer, New York, 1970). | MR | Zbl

[EG] D. Edidin and W. Graham, Equivariant intersection theory, preprint, alg-geom/9603008. | MR | Zbl

[FH] W. Fulton and J. Harris, Representation theory (Grad. Texts in Math., Vol. 129, Springer, New York, 1991). | MR | Zbl

[FL] W. Fulton and S. Lang, Riemann-Roch algebra (Grundlehren Math. Wiss., Vol. 277, Springer, New York, 1985). | MR | Zbl

[FM] W. Fulton and R. Macpherson, Characteristic classes of direct image bundles for covering maps (Ann. of Math., Vol. 125, 1987, pp. 1-92). | MR | Zbl

[Gr] D. R. Grayson, Exterior power operations on higher K-theory (K-theory, Vol. 3, 1989, pp. 247-260). | MR | Zbl

[Gro1] A. Grothendieck, Sur quelques propriétés fondamentales en théorie des intersections, Séminaire C. Chevalley, 2e année, Anneaux de Chow et applications, Secr. Math. Paris, 1958. | EuDML | Numdam | MR

[Gro2] A. Grothendieck, Classes de Chern et représentations linéaires des groupes discrets, in J. Giraud et al., Dix exposés sur la cohomologie des schémas (Adv. Stud. Pure Math., North-Holland Publishing Company, Amsterdam, 1968, pp. 215-305). | MR | Zbl

[EGA] A. Grothendieck and J. A. Dieudonné, Eléments de Géométrie Algébrique I (Grundlehren Math. Wiss., Vol. 166, Springer, New York, 1971) ; ...II, III, (Publ. Math. IHES, Vol. 8, 1961, Vol. 11, 1961, Vol. 17, 1963). | Numdam | MR | Zbl

[Ha] R. Hartshorne, Algebraic geometry (Graduate Texts in Math., Vol. 52, Springer, New York, 1977). | MR | Zbl

[Hi] H. L. Hiller, λ-rings and algebraic K-theory (J. Pure Appl. Algebra, Vol. 20, 1981, pp. 241-266). | MR | Zbl

[J] J. C. Jantzen, Representations of algebraic groups (Pure Appl. Math., Vol. 131, Academic Press, Boston, 1987). | MR | Zbl

[Ke] M. Kervaire, Opérations d'Adams en Théorie des représentations linéaires des groupes finis (Enseign. Math., Vol. 22, 1976, pp. 1-28). | MR | Zbl

[Ko0] B. Köck, Das Lefschetz- und Riemann-Roch-Theorem in der höheren äquivarianten K-Theorie Dissertation, Regensburg, 1989. | Zbl

[Ko1] B. Köck, Chow motif and higher Chow theory of G / P (Manuscripta Math., Vol. 70, 1991, pp. 363-372). | EuDML | MR | Zbl

[Ko2] B. Köck, Das Adams-Riemann-Roch-Theorem in der höheren äquivarianten K-Theorie (J. Reine Angew. Math., Vol. 421, 1991, pp. 189-217). | EuDML | MR | Zbl

[Ko3] B. Köck, The Lefschetz theorem in higher equivariant K-theory (Comm. Algebra, Vol. 19, 1991, pp. 3411-3422). | MR | Zbl

[Ko4] B. Köck, Higher K'-groups of integral group rings (K-Theory, Vol. 4, 1991, pp. 177-187). | MR | Zbl

[Ko5] B. Köck, Shuffle products in higher K-theory (J. Pure Appl. Algebra, Vol. 92, 1994, pp. 269-307). | MR | Zbl

[Ko6] B. Köck, The Grothendieck-Riemann-Roch theorem in the higher K-theory of group scheme actions Habilitationsschrift, Karlsruhe, 1995.

[Ko7] B. Köck, On Adams operations on the higher K-theory of group rings, in : G. Banaszak et al. (eds.), Algebraic K-theory (Poznań, 1995) (Contemp. Math., Vol. 199, Amer. Math. Soc., Providence, 1996, pp. 139-150). | MR | Zbl

[Ko8] B. Köck, Adams operations for projective modules over group rings (Math. Proc. Cambridge Philos. Soc., Vol. 122, 1997, pp. 55-71). | MR | Zbl

[Ko9] B. Köck, Operations on locally free classgroups, preprint, Karlsruhe, 1997. | MR | Zbl

[Ko10] B. Köck, Riemann-Roch for tensor power, preprint, Karlsruhe, 1998. | MR | Zbl

[KK1] B. Kostant and S. Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G (Adv. Math., Vol. 62, 1986, pp. 187-237). | MR | Zbl

[KK2] B. Kostant and S. Kumar, T-equivariant K-theory of generalized flag varieties (J. Differential Geom., Vol. 32, 1990, pp. 549-603). | MR | Zbl

[Kr] Ch. Kratzer, λ-Structure en K-théorie algébrique (Comment. Math. Helv., Vol. 55, 1980, pp. 233-254). | EuDML | MR | Zbl

[La] A. Lascoux, Anneau de Grothendieck de la variété de drapeaux, in The Grothendieck Festschrift, vol. III (Progr. Math., Vol. 88, Birkhäuser, Boston, 1990, pp. 1-34). | MR | Zbl

[Le] H. W. Lenstra, Grothendieck groups of abelian group rings (J. Pure Appl. Algebra, Vol. 20, 1981, pp. 173-193). | MR | Zbl

[Man] Y. I. Manin, Lectures on the K-functor in algebraic geometry (Russian Math. Surveys, Vol. 24, No. 5, 1969, pp. 1-89). | MR | Zbl

[Mo] R. Morelli, The K theory of a toric variety (Adv. Math., Vol. 100, 1993, pp. 154-182). | MR | Zbl

[Mum] D. Mumford, Geometric invariant theory (Ergeb. Math. Grenzgeb., Vol. 34, Springer, New York, 1965). | MR | Zbl

[Q] D. Quillen, Higher algebraic K-theory : I, in H. Bass (ed.), Algebraic K-Theory I (Seattle, 1972) (Lecture Notes in Math., Vol. 341, Springer, New York, 1973, pp. 85-147). | MR | Zbl

[Sn] V. P. Snaith, Invariants of representations, in J. F. Jardine and V. P. Snaith (eds.), Algebraic K-theory : Connections with geometry and topology (Lake Louise, 1987) (NATO Adv. Sci. Inst. Ser. C : Math. Phys. Sci., Vol. 279, Kluwer Acad. Publ., Dordrecht, 1989, pp. 445-508). | MR | Zbl

[So] C. Soulé, Opérations en K-Théorie algébrique (Canad. J. Math., Vol. 37, No. 3, 1985, pp. 488-550). | MR | Zbl

[Su] H. Sumihiro, Equivariant completion II (J. Math. Kyoto Univ., Vol. 15, 1975, pp. 573-605). | MR | Zbl

[Ta] G. Tamme, The theorem of Riemann-Roch, in M. Rapoport, N. Schappacher and P. Schneider (eds.), Beilinson's conjectures on special values of L-functions (Perspect. in Math., Vol. 4, Academic Press, Boston, 1988, pp. 103-168). | MR | Zbl

[Tho] C. B. Thomas, Characteristic classes and the cohomology of finite groups (Cambridge Stud. Adv. Math., Vol. 9, Cambridge Univ. Press, Cambridge, 1986). | MR | Zbl

[Th1] R. W. Thomason, Absolute cohomological purity (Bull. Soc. Math. France, Vol. 112, 1984, pp. 397-406). | EuDML | Numdam | MR | Zbl

[Th2] R. W. Thomason, Lefschetz-Riemann-Roch theorem and coherent trace formula (Invent. Math., Vol. 85, 1986, pp. 515-543). | EuDML | MR | Zbl

[Th3] R. W. Thomason, Algebraic K-theory of group scheme actions, in W. Browder (ed.), Algebraic topology and algebraic K-theory (Princeton, 1983) (Ann. Math. Stud., Vol. 113, Univ. Press, Princeton, 1987, pp. 539-563). | MR | Zbl

[Th4] R. W. Thomason, Une formule de Lefschetz en K-théorie équivariante algébrique (Duke Math. J., Vol. 68, 1992, pp. 447-462). | MR | Zbl

[Th5] R. W. Thomason, Les K-groupes d'un schéma éclaté et une formule d'intersection excédentaire (Invent., Math., Vol. 112, 1993, pp. 195-215). | EuDML | MR | Zbl

[We] C. A. Weibel, Homotopy algebraic K-theory, in M. R. Stein and R. K. Dennis (eds.), Algebraic K-theory and algebraic number theory (Honolulu, 1987) (Contemp. Math., Vol. 83, Amer. Math. Soc., Providence, 1989, pp. 461-488). | MR | Zbl

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