The six operations for sheaves on Artin stacks I: Finite coefficients
Publications Mathématiques de l'IHÉS, Volume 107 (2008), pp. 109-168.

In this paper we develop a theory of Grothendieck's six operations of lisse-étale constructible sheaves on Artin stacks locally of finite type over certain excellent schemes of finite Krull dimension. We also give generalizations of the classical base change theorems and Kunneth formula to stacks, and prove new results about cohomological descent for unbounded complexes.

@article{PMIHES_2008__107__109_0,
     author = {Laszlo, Yves and Olsson, Martin},
     title = {The six operations for sheaves on {Artin} stacks {I:} {Finite} coefficients},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {109--168},
     publisher = {Institut des Hautes \'Etudes Scientifiques},
     volume = {107},
     year = {2008},
     doi = {10.1007/s10240-008-0011-6},
     mrnumber = {2434692},
     zbl = {1191.14002},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-008-0011-6/}
}
TY  - JOUR
AU  - Laszlo, Yves
AU  - Olsson, Martin
TI  - The six operations for sheaves on Artin stacks I: Finite coefficients
JO  - Publications Mathématiques de l'IHÉS
PY  - 2008
SP  - 109
EP  - 168
VL  - 107
PB  - Institut des Hautes Études Scientifiques
UR  - http://archive.numdam.org/articles/10.1007/s10240-008-0011-6/
DO  - 10.1007/s10240-008-0011-6
LA  - en
ID  - PMIHES_2008__107__109_0
ER  - 
%0 Journal Article
%A Laszlo, Yves
%A Olsson, Martin
%T The six operations for sheaves on Artin stacks I: Finite coefficients
%J Publications Mathématiques de l'IHÉS
%D 2008
%P 109-168
%V 107
%I Institut des Hautes Études Scientifiques
%U http://archive.numdam.org/articles/10.1007/s10240-008-0011-6/
%R 10.1007/s10240-008-0011-6
%G en
%F PMIHES_2008__107__109_0
Laszlo, Yves; Olsson, Martin. The six operations for sheaves on Artin stacks I: Finite coefficients. Publications Mathématiques de l'IHÉS, Volume 107 (2008), pp. 109-168. doi : 10.1007/s10240-008-0011-6. http://archive.numdam.org/articles/10.1007/s10240-008-0011-6/

[1] K. A. Behrend, Derived l-Adic Categories for Algebraic Stacks, Mem. Amer. Math. Soc., vol. 163, no. 774, Amer. Math. Soc., Providence, RI, 2003. | MR | Zbl

[2] Beĭlinson, A.A., Bernstein, J., Deligne, P. (1982) Faisceaux pervers. Analysis and Topology on Singular Spaces, I (Luminy, 1981). Soc. Math. France, Paris | MR | Zbl

[3] Bokstedt, M., Neeman, A. (1993) Homotopy limits in triangulated categories. Compos. Math. 86: pp. 209-234 | Numdam | MR | Zbl

[4] Deligne, P. (1977) Cohomologie étale | MR | Zbl

[5] J. Dieudonné and A. Grothendieck, Éléments de géométrie algébrique, Publ. Math., Inst. Hautes Étud. Sci., 4, 8, 11, 17, 20, 24, 28, 32 (1961-1967). | Numdam | Zbl

[6] Freitag, E., Kiehl, R. (1988) Étale cohomology and the Weil conjecture. Springer, Berlin | MR

[7] Frenkel, E., Gaitsgory, D., Vilonen, K. (2002) On the geometric Langland's conjecture. J. Amer. Math. Soc. 15: pp. 367-417 | MR | Zbl

[8] K. Fujiwara, A proof of the absolute purity conjecture (after Gabber), in S. Usui, M. Green, L. Illusie, K. Kato, E. Looijenga, S. Mukai, S. Saito (eds.), Algebraic Geometry 2000, Azumino, Adv. Stud. Pure Math., vol. 36, pp. 153-183, Math. Soc. Japan, Tokyo, 2002. | MR | Zbl

[9] O. Gabber, A finiteness theorem for non abelian H1 of excellent schemes, in Conférence en l'honneur de L. Illusie, Orsay (2005), http://www.math.polytechnique.fr/~laszlo/gdtgabber/non-abelien.pdf.

[10] O. Gabber, Finiteness theorems for étale cohomology of excellent schemes, in Conference in Honor of P. Deligne on the Occasion of his 61st Birthday, IAS, Princeton (2005), http://www.math.polytechnique.fr/~laszlo/gdtgabber/abelien.pdf.

[11] P.-P. Grivel, Catégories dérivées et foncteurs dérivés, in A. Borel (ed.), Algebraic D-Modules, Perspect. Math., vol. 2, Academic Press, Boston, MA, 1987.

[12] Grothendieck, A. (2003) Revétements étales et groupe fondamental. Séminaire de Géométrie Algébrique du Bois-Marie (SGA 1). Société Mathématique de France, Paris | MR | Zbl

[13] Artin, M., Grothendieck, A., Verdier, J.-L. (1972) Revétements étales et groupe fondamental. Séminaire de Géométrie Algébrique du Bois-Marie (SGA 1). Société Mathématique de France, Paris | Zbl

[14] A. Grothendieck, Cohomologie l-adique et fonctions L, in L. Illusie (ed.), Séminaire de Géometrie Algébrique du Bois-Marie (SGA 5), Lect. Notes Math., vol. 589, Springer, Berlin, 1977. | MR | Zbl

[15] L. Illusie, Y. Laszlo, and F. Orgogozo, Théorème de finitude en cohomologie étale, d'après Gabber, in preparation, École Polytechnique (2007), http://www.math.polytechnique.fr/~laszlo/gdtgabber/gdtgabber.html.

[16] Kashiwara, M., Schapira, P. (2006) Categories and Sheaves. Springer, Berlin | MR | Zbl

[17] Y. Laszlo and M. Olsson, The six operations for sheaves on Artin stacks II: Adic Coefficients, Publ. Math., Inst. Hautes Étud. Sci., (2008). | Numdam | MR | Zbl

[18] Y. Laszlo and M. Olsson, Perverse sheaves on Artin stacks, Math. Z., to appear. | MR

[19] Laumon, G. (2003) Transformation de Fourier homogène. Bull. Soc. Math. Fr. 131: pp. 527-551 | Numdam | MR | Zbl

[20] Laumon, G., Moret-Bailly, L. (2000) Champs algébriques. Springer, Berlin | MR | Zbl

[21] G. Laumon and B. C. Ngo, Le lemme fondamental pour les groupes unitaires, preprint (2004), http://arxiv.org/abs/math/0404454. | MR | Zbl

[22] Neeman, A. (1996) The Grothendieck duality theorem via Bousfield's techniques and Brown representability. J. Amer. Math. Soc. 9: pp. 205-236 | MR | Zbl

[23] Olsson, M. (2007) Sheaves on Artin stacks. J. Reine Angew. Math. 603: pp. 55-112 | MR | Zbl

[24] J. Riou, Pureté (d'après Ofer Gabber), in Théorèmes de finitude en cohomogie étale d'après Ofer Gabber, in preparation, preprint (2007), http://www.math.u-psud.fr/~riou/doc/gysin.pdf.

[25] Serpét, C. (2003) Resolution of unbounded complexes in Grothendieck categories. J. Pure Appl. Algebra 177: pp. 103-112 | MR | Zbl

[26] Serre, J.-P. (1994) Cohomologie galoisienne. Springer, Berlin | MR | Zbl

[27] Spaltenstein, N. (1988) Resolutions of unbounded complexes. Compos. Math. 65: pp. 121-154 | Numdam | MR | Zbl

Cited by Sources: