A Deligne-Riemann-Roch isomorphism

A Deligne-Riemann-Roch isomorphism
[Un isomorphisme de Deligne-Riemann-Roch]

Eriksson, Dennis

Directeur de thèse : Rossler, Damian ; Harari, David

Membres du jury : Burgos-Gil, José Ignacio ; Harari, David ; Kahn, Bruno ; Rossler, Damian ; Soulé, Christophe

Thèses d'Orsay, no. 752 (2008) , 162 p.

Export
Comment citer

@phdthesis{BJHTUP11_2008__0752__P0_0,
     author = {Eriksson, Dennis},
     title = {A {Deligne-Riemann-Roch} isomorphism},
     series = {Th\`eses d'Orsay},
     year = {2008},
     publisher = {Universite Paris-Sud Facult\'e des Sciences d'Orsay},
     number = {752},
     language = {en},
     url = {https://www.numdam.org/item/BJHTUP11_2008__0752__P0_0/}
}

TY  - BOOK
AU  - Eriksson, Dennis
TI  - A Deligne-Riemann-Roch isomorphism
T3  - Thèses d'Orsay
PY  - 2008
IS  - 752
PB  - Universite Paris-Sud Faculté des Sciences d'Orsay
UR  - https://www.numdam.org/item/BJHTUP11_2008__0752__P0_0/
LA  - en
ID  - BJHTUP11_2008__0752__P0_0
ER  -

%0 Book
%A Eriksson, Dennis
%T A Deligne-Riemann-Roch isomorphism
%S Thèses d'Orsay
%D 2008
%N 752
%I Universite Paris-Sud Faculté des Sciences d'Orsay
%U https://www.numdam.org/item/BJHTUP11_2008__0752__P0_0/
%G en
%F BJHTUP11_2008__0752__P0_0

Eriksson, Dennis. A Deligne-Riemann-Roch isomorphism. Thèses d'Orsay, no. 752 (2008), 162 p. https://www.numdam.org/item/BJHTUP11_2008__0752__P0_0/

Sommaire

Part I	p. 1
1. Brauer-Manin obstruction for zero-cycles on curves	p. 3
1.1 Brauer-Manin obstruction	p. 3
1.2 A Short Proof of S. Saito’s Theorem	p. 7
1.3 Brauer-Manin obstruction and Generic Periods	p. 8
1.4 Alternative Description of the Period	p. 10
1.5 Appendix - Suslin homology, $h_{0}$	p. 12
Part II	p. 17
2. Some preliminaries	p. 19
2.1 The virtual category	p. 19
2.2 Algebraic definition	p. 20
2.3 Additional descriptions	p. 22
3. Virtual categories associated to algebraic stacks	p. 33
3.1 Various categories	p. 33
3.2 A splitting principle	p. 39
3.3 Adams and $λ$ -operations on the virtual category	p. 40
3.4 Deformation to the normal cone	p. 43
4. Rigidity and operations on virtual categories	p. 47
5. A functorial excess formula	p. 61
5.1 A rough excess-isomorphism	p. 62
5.2 Excess for projective bundle-morphisms, uniqueness	p. 67
5.3 Excess for closed immersions, uniqueness	p. 68
5.4 Excess for closed immersions, rougher excess and existence	p. 71
5.5 General excess isomorphism	p. 80
6. Applications to functoriality	p. 85
6.1 Explicit construction of characteristic classes	p. 85
6.2 An explicit functorial Lefschetz formula for cyclic diagonal actions	p. 88
6.3 An Adams-Deligne-Riemann-Roch formula	p. 98
6.4 Application to Adams-Riemann-Roch transformations	p. 106
6.5 Mumford’s isomorphism and comparison with Deligne’s isomorphism	p. 108
6.6 A Deligne-Riemann-Roch formula for the Determinant of the cohomology	p. 114
6.7 A conjecture of Köck for the determinant of the cohomology	p. 115
Appendix	p. 119
A. A 1-homotopy theory of schemes	p. 121
B. Localization of Picard categories and the case of quotients of split reductive groups	p. 131
C. Algebraic stacks	p. 133

Bibliographie
Cité par

[1] B. J. Birch, H. P. F. Swinnerton-Dyer, The Hasse problem for rational surfaces. J. reine angew. Math 274/275 (1975), 164-174

[2] J.-L. Colliot-Thélène, Conjectures de type local-global sur l'image des groupes de Chow dans la cohomologie étale, Algebraic K-theory (Seattle, WA, 1997), 1-12, Proc. Sympos. Pure Math., 67.

[3] J.-L. Colliot-Thélène, Un théorème de finitude pour le groupe de Chow des zéro-cycles d'un groupe algébrique linéaire sur un corps p-adique, Inventiones mathematicae 159, 2005, 589-606. | MR | Zbl | DOI

[4] A. Grothendieck, Le groupe de Brauer III, Dix exposés sur la cohomologie des schémas, North-Holland Publis. Co., Amsterdam et Masson et Cie, Paris, 1968. | Zbl

[5] D. Harari, T. Szamuely, Local-global principles for 1-motives Duke Math. J. 143, no 3, 531-557 (2008). | MR | Zbl | DOI

[6] D. Harari, Méthode des fibrations et obstruction de Manin, Duke Math. J. 75, 1994, 221-260. | MR | Zbl | DOI

[7] W. Fulton, Intersection Theory Springer Verlag, 1998. | MR | Zbl | DOI

[8] Yu. I. Manin, Le groupe de Brauer-Grothendieck en géométrie diophantienne, Actes du Congrès International des Mathématiciens (Nice, 1970), 1, 1970, 401-411. | MR | Zbl

[9] C. Mazza, V. Voevodsky, C. Weibel, Lectures on Motivic Cohomology, Clay Monographs in Math 2, 2006, AMS. | MR | Zbl

[10] J. S. Milne, Abelian varieties, in "Arithmetic Geometry", ed. by Cornell and Silvermann, Springer Verlag, 1986. | MR | Zbl | DOI

[11] J. S. Milne, Comparison of the Brauer group with the Tate-Šafarevič group, J. Fac. Sci. Univ. Tokyo Sect. IA, Vol. 28:3, 1982, 735-743. | MR | Zbl

[12] B. Poonen, M. Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. Volume 150, Number 3 (1999), pp. 1109-1149. | MR | Zbl | DOI

[13] N. Ramachandran, Duality of Albanese and Picard 1-motives, K-Theory 22 (2001), no. 3, pp. 271-301. | MR | Zbl

[14] S. Saito, Some observations on motivic cohomology of arithmetic schemes, Invent. Math. 98, 1989, 371-404. | MR | Zbl | DOI

[15] J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. reine angew. Math 327 (1981), 12-80. | MR | Zbl

[16] J.-P. Serre, Corps locaux, Hermann, Paris, 1962

[17] J.-P. Serre, Groupes algébriques et corps de classes, Hermann, Paris, 1959. | MR | Zbl

[18] J.-P. Serre, Morphismes universels et variétés d'Albanese, Variétés de Picard, Seminaire C. Chevalley, E.N.S. Paris, 1958-1959. | Zbl | Numdam

[19] A. N. Skorobogatov, Torsors and rational points, Cambridge University Press, 2001.

[20] M. Spiess, T. Szamuely, On the Albanese map for smooth quasi-projective varieties, Mathematische Annalen 325 (2003), pp. 1-17. | MR | Zbl | DOI

[21] O. Wittenberg, On Albanese torsors and the elementary obstruction, Mathematische Annalen 340 (2008), no. 4, 805-838. | MR | Zbl | DOI

[AD61] D. Puppe A. Dold, Homologie nicht-additiver funktoren. anwendungen, Annwendungen, Ann. Inst. Fourier 11 (1961), 201-312. | MR | Zbl | Numdam | DOI

[AG67] J. Dieudonné A. Grothendieck, Éléments de Géométrie Algébrique iv, Publications Mathématiques de l'IHÉS, 4 (1960) | MR | Numdam

[AG67] J. Dieudonné A. Grothendieck, Éléments de Géométrie Algébrique iv, Publications Mathématiques de l'IHÉS, 8 (1961)

[AG67] J. Dieudonné A. Grothendieck, Éléments de Géométrie Algébrique iv, Publications Mathématiques de l'IHÉS, 11 (1961)

[AG67] J. Dieudonné A. Grothendieck, Éléments de Géométrie Algébrique iv, Publications Mathématiques de l'IHÉS, 17 (1963)

[AG67] J. Dieudonné A. Grothendieck, Éléments de Géométrie Algébrique iv, Publications Mathématiques de l'IHÉS, 20 (1964) | MR | Numdam | Zbl

[AG67] J. Dieudonné A. Grothendieck, Éléments de Géométrie Algébrique iv, Publications Mathématiques de l'IHÉS, 24 (1965) | Zbl | Numdam

[AG67] J. Dieudonné A. Grothendieck, Éléments de Géométrie Algébrique iv, Publications Mathématiques de l'IHÉS, 28 (1966) | Zbl | Numdam

[AG67] J. Dieudonné A. Grothendieck, Éléments de Géométrie Algébrique iv, Publications Mathématiques de l'IHÉS, 32 (1967). | Zbl | Numdam

[AG70a] M. Demazure et al A. Grothendieck, Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 1. Propriétés Génénerales des Schemas en Groupes., Lecture notes in mathematics, vol. 151, Springer Verlag, 1970.

[AG70b] M. Demazure et al A. Grothendieck, Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 2 - Groupes de type multiplicatif, et structure des schemas en groupes generaux., Lecture notes in mathematics, vol. 152, Springer Verlag, 1970.

[AG70c] M. Demazure et al A. Grothendieck, Séminaire de Géométrie Algébrique du Bois Marie - 1962-64 - Schémas en groupes - (SGA 3) - vol. 3. Structure des Schémas en Groupes Réductifs, Lecture notes in mathematics, vol. 153, Springer Verlag, 1970. | Zbl

[AK04] A. Vistoli A. Kresch, On coverings of Deligne-Mumford stacks and surjectivity of the Brauer map, Bull. London Math. Soc. 36 (2004), no. 2, 188-192. | MR | Zbl | DOI

[Ara74a] S. Arakelov, Intersection theory of divisors on an arithmetic surface, Izv. Akad. Nauk. 86 (1974), 1167-1180. | MR | Zbl

[Ara74b] S. Arakelov, Theory of intersections on the arithmetic surfaces, Proc. ICM Vancouver, 1974, pp. 405-408. | MR | Zbl

[Blo87] S. Bloch, Cycles on arithmetic schemes and Euler characteristics of curves, Proceedings of Symposia in Pure Mathematics, Volume 46, 1987, pp. 421-450. | MR | Zbl | DOI

[Bor74] A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. Ec. Norm. Sup. 7 (1974), 235-272. | MR | Zbl | Numdam | DOI

[Bre] M. Breuning, Determinant functors on triangulated categories, http://front.math.ucdavis.edu/math.CT/0610435.

[Del74] P. Deligne, Théorie de Hodge : III, Publications Mathématiques de l'lHES 44 (1974), 5-77. | MR | Zbl | Numdam | DOI

[Del77] P. Deligne, Séminaire de Géométrie Algébrique du Bois Marie - Co- homologie étale - (SGA $4 \frac{1}{2}$ ), Lecture notes in mathematics, vol. 569, Springer Verlag, 1977.

[Del87] P. Deligne, Le déterminant de la cohomologie, Contemp. Mathematics 67 (1987), 93-177.

[DM82] J. Fogarty et al D. Mumford, Geometric invariant theory, 2nd enlarged edition ed., Springer Verlag, 1982. | MR | Zbl

[Elk89] R. Elkik, Fibres d'intersection et integrales de classes de Chern, Ann. scient. Ec. Norm. Sup. 22 (1989), 195-226. | MR | Zbl | Numdam | DOI

[Fal84] G. Faltings, Calculus on arithmetic surfaces, Ann. of Math. 119 (1984), no. 2, 387-424. | MR | Zbl | DOI

[FK76] D. Mumford F. Knudsen, The projectivity of the moduli space of stable curves i: Preliminaries on "det" and "div", Math. Scand. 39 (1976), no. 1, 19-55. | MR | Zbl

[Fra] J. Franke, Riemann-Roch in Functorial Form, unpublished manuscript.

[Fra90] J. Franke, Chow categories, Compositio Mathematica 76 (1990), no. 1-2, 101-162.

[Fra91] J. Franke, Chern Functors, Arithmetic algebraic geometry, Texel, Progress in Math. 89, (1989) (van der Geer et al., ed.), Birkhäuser, 1991, pp. 75-152. | MR | Zbl | DOI

[Ful98] W. Fulton, Intersection theory, 2nd ed., Springer Verlag, 1998. | MR | Zbl

[GL00] L. Moret-Bailly G. Laumon, Champs algebriques, Springer Verlag, 2000.

[Gra92] D. Grayson, Adams operations on higher K-theory, K-theory 6 (1992), 97-111. | MR | Zbl

[Gra95] D. Grayson, Weight filtrations via commuting automorphisms, K-theory 9 (1995), 139-172. | MR | Zbl

[GV02] A. Vistoli G. Vezzosi, Higher Algebaric K-theory of group actions with finite stabilizers, Duke Math. J. 113 (2002), no. 1, 1-55 | MR | Zbl

[HG84] C. Soulé H. Gillet, Intersections sur les variétés d'Arakelov, Comptes Rendus Mathématique. Académie des Sciences 299 (1984), no. 12, 563-566. | MR | Zbl

[HG87] C. Soulé H. Gillet, Intersection theory using Adams operations, Inventiones Mathematicae 90 (1987), no. 2, 243-277.

[Hol] S. Hollander, A homotopy theory for stacks, http://arxiv.org/abs/math/0110247v2, To appear in Israel Journal of Math. | MR | Zbl

[KK01] D. Roessler K. Köhler, A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof, Inventiones Mathematicae 145 (2001), no. 2, 333-396. | MR | Zbl | DOI

[Knu] F. Knudsen, The cohomological determinant, Unpublished manuscript.

[Knu83] F. Knudsen, The projectivity of the moduli space of stable curves II: the stacks $M_{g, n}$ , Math. Scand. 52 (1983), 161-199. | MR | Zbl | DOI

[Köc98] B. Köck, The Grothendieck-Riemann-Roch theorem for group scheme actions, Annales Scientifiques de l'Ecole Normale Superieure 31 (1998), no. 3, 415-458. | MR | Zbl | Numdam | DOI

[Lev99] M. Levine, K-theory and motivic cohomology of schemes, unpublished manuscript, 1999.

[Lev05] M. Levine, The homotopy coniveau tower, preprint, 2005.

[Lic68] S. Lichtenbaum, Curves over discrete valuation rings, Amer. J. Math. 90 (1968), 380-405. | MR | Zbl | DOI

[MB89] L. Moret-Bailly, La formule de Noether pour les surfaces arithmetiques, Inventiones Mathematicae 98 (1989), 491-498. | MR | Zbl | DOI

[Mor99] F. Morel, Théorie homotopique des schémas, Astérisque, vol. 256, Société Mathématique de France, 1999.

[Mum77] D. Mumford, Stability of projective varieties, Einseign. Math. 23 (1977), 39-100. | MR | Zbl

[Ols05] M. Olsson, On proper coverings of Artin stacks, Advances in Mathematics 198 (2005), 93-106. | MR | Zbl | DOI

[P71] L. Illusie P, Berthelot; A. Grothendieck, Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch (SGA 6), Lecture notes in mathematics, vol. 225, Springer Verlag, 1971. | Zbl

[PB75] R. Macpherson P. Baum, W. Fulton, Riemann-Roch for singular varieties., Publications Mathématiques de l'IHÉS 45 (1975), 101-145. | MR | Zbl | Numdam | DOI

[PD69] D. Mumford P. Deligne, The irreducibility of the space of curves of a given genus, Publications Mathématiques de l'IHÉS 36 (1969), 75-109. | MR | Zbl | Numdam | DOI

[PGG99] J. F. Jardine P. G. Goerss, Simplicial Homotopy Theory, Birkhäuser Verlag, 1999. | MR | Zbl

[Qui73] D. Quillen, Higher algebraic K-theory, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash.,1972) (Lecture Notes in Mathematics 341), Springer Verlag,. 1973, pp. 85-147.

[Rio06] J. Riou, Opérations sur la k-théorie algébrique et régulateurs viala théorie homotopique des schémas, Ph.D. thesis, l'Universite Paris 7 - Denis Diderot, http://www.math.upsud.fr/riou/these/these.pdf, 2006.

[Riv72] N. Saavedra Rivano, Catégories tannakiennes, LNM 265, Springer Verlag, 1972. | MR | Zbl

[RT90] T. Trobaugh R.W. Thomason, Higher algebraic K-theory of schemes and of derived categories, Grothendieck Festschrifft III, Birkhäuser, 1990, pp. 247- 435. | MR

[Sai88a] T. Saito, Conductor, Discriminant, and the Noether formula of arithmetic surfaces, Duke Math. Journal 57 (1988), no. 1, 151-173. | MR | Zbl | DOI

[Sai88b] T. Saito, Self-intersection 0-cycles and coherent sheaves on arithmetic schemes, Duke Mathematical Journal (1988), 555-578. | MR | Zbl

[Seg68] G. Segal, The representation ring of a compact Lie group, Publications Mathématiques de l'IHÉS 34 (1968), 113-128. | MR | Zbl | Numdam | DOI

[Sou85] C. Soulé, Opérations en k-théorie algébrique, Canad. J. Math. 37 (1985), no. 3, 488-550. | MR | Zbl | DOI

[Sou92] C. Soulé, Lectures on Arakelov geometry, Cambridge studies in advanced mathematics 33, 1992. | MR | Zbl

[Tho85] R.W. Thomason, Algebraic K-theory and étale cohomology, Ann. Sci. Ec. Norm. Sup. 18 (1985), 437-552. | MR | Zbl | Numdam | DOI

[Tho86] R.W. Thomason, Lefschetz-Riemann-Roch theorem and coherent trace formula, Inventiones Mathematicae 85 (1986), no. 3, 515-543.

[Tho87a] R.W. Thomason, Algebraic K-theory of group scheme actions, Algebraic topology and algebraic K-theory, Ann. of Math. Stud., 113 ((Princeton, N.J.,1983)), Princeton Univ. Press, 1987, pp. 539-563. | MR | Zbl

[Tho87b] R.W. Thomason, Equivariant resolution, linearization and Hilbert's fourteenth problem over arbitrary base-schemes, Adv. Math. 65 (1987), 16-34. | MR | Zbl | DOI

[Tho92] R.W. Thomason, Une formule de Lefschetez en K-théorie équivariante algébrique, Duke Math. J. 68 (1992), no. 3, 447-462.

[Toe99a] B. Toen, $K$ -theory and cohomology of algebraic stacks: Riemann-Roch theorems, $𝒟$ -modules and GAGA theorems, Ph.D. thesis, Université Paul Sabatier de Toulouse, , 1999 | arXiv

[Toe99b] B. Toen, Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford, K-theory (1999), 33-76. | MR | Zbl

[Tot04] B. Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004), 1-22.

[VV99] F. Morel V. Voevodsky, $A^{1}$ -homotopy theory of schemes, Publications Mathématiques de l'IHÉS 90 (1999), 45-143.

[Wal85] F. Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology (Lecture Notes in Math. 1126), . 1985, pp. 318-419.

[WD89] J. Smith. W. Dwyer, D. Kan, Homotopy commutative diagrams and their realizations, J. Pure Appl. Algebra 57 (1989), 5-24. | MR | Zbl | DOI

[Wei89] C. Weibel, Homotopy algebraic K-theory, AMS Contemp. Math. 83 (1989), 461-488. | MR | Zbl | DOI

[WF83] H. Gillet W. Fulton, Riemann-Roch for general algebraic varieties, Bulletin de la Société Mathématique de France 111 (1983), 287-300. | MR | Zbl | Numdam

[WF85] S. Lang W. Fulton, Riemann-Roch algebra, Springer Verlag, 1985. | MR | Zbl