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.

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

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%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
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Eriksson, Dennis. A Deligne-Riemann-Roch isomorphism. Thèses d'Orsay, no. 752 (2008), 162 p. http://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

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