The six operations for sheaves on Artin stacks I: Finite coefficients
Laszlo, Yves ; Olsson, Martin
Publications Mathématiques de l'IHÉS, Tome 108 (2008), p. 109-168 / Harvested from Numdam

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},
     volume = {108},
     year = {2008},
     pages = {109-168},
     doi = {10.1007/s10240-008-0011-6},
     mrnumber = {2434692},
     zbl = {1191.14002},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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, Tome 108 (2008) pp. 109-168. doi : 10.1007/s10240-008-0011-6. http://gdmltest.u-ga.fr/item/PMIHES_2008__107__109_0/

[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 1963494 | Zbl 1051.14023

[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 751966 | Zbl 0536.14011

[3] Bokstedt, M., Neeman, A. (1993) Homotopy limits in triangulated categories. Compos. Math. 86: pp. 209-234 | Numdam | MR 1214458 | Zbl 0802.18008

[4] Deligne, P. (1977) Cohomologie étale | MR 463174 | Zbl 0345.00010

[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 0122.16102

[6] Freitag, E., Kiehl, R. (1988) Étale cohomology and the Weil conjecture. Springer, Berlin | MR 926276

[7] Frenkel, E., Gaitsgory, D., Vilonen, K. (2002) On the geometric Langland's conjecture. J. Amer. Math. Soc. 15: pp. 367-417 | MR 1887638 | Zbl 1071.11039

[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 1971516 | Zbl 1059.14026

[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 2017446 | Zbl 1039.14001

[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 1039.14001

[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 463174 | Zbl 0345.00011

[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 2182076 | Zbl 1118.18001

[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 2434693 | Zbl 1191.14003

[18] Y. Laszlo and M. Olsson, Perverse sheaves on Artin stacks, Math. Z., to appear. | MR 2480756

[19] Laumon, G. (2003) Transformation de Fourier homogène. Bull. Soc. Math. Fr. 131: pp. 527-551 | Numdam | MR 2044494 | Zbl 1088.11044

[20] Laumon, G., Moret-Bailly, L. (2000) Champs algébriques. Springer, Berlin | MR 1771927 | Zbl 0945.14005

[21] G. Laumon and B. C. Ngo, Le lemme fondamental pour les groupes unitaires, preprint (2004), http://arxiv.org/abs/math/0404454. | MR 2434884 | Zbl 1179.22019

[22] Neeman, A. (1996) The Grothendieck duality theorem via Bousfield's techniques and Brown representability. J. Amer. Math. Soc. 9: pp. 205-236 | MR 1308405 | Zbl 0864.14008

[23] Olsson, M. (2007) Sheaves on Artin stacks. J. Reine Angew. Math. 603: pp. 55-112 | MR 2312554 | Zbl 1137.14004

[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 1948842 | Zbl 1033.18007

[26] Serre, J.-P. (1994) Cohomologie galoisienne. Springer, Berlin | MR 1324577 | Zbl 0812.12002

[27] Spaltenstein, N. (1988) Resolutions of unbounded complexes. Compos. Math. 65: pp. 121-154 | Numdam | MR 932640 | Zbl 0636.18006