# zbMATH — the first resource for mathematics

The six operations for sheaves on Artin stacks. II: Adic coefficients. (English) Zbl 1191.14003
In this second part of their comprehensive work on functorial constructions for cohomology sheaves on Artin stacks, the authors continue their study of the stacky version of Grothendieck’s six operations (or functors) begun in the foregoing first part [ibid. 107, 109–168 (2008; Zbl 1191.14002)]. The objective of the current paper is to extend the theory developed so far to a corresponding theory for “adic” sheaves. In other words, the ground ring $$\Lambda$$ is now assumed to be a complete discrete valuation ring with residue characteristic $$\ell$$. Furthermore, the derived categories (and subcategories) considered in the first part of the work are now replaced by appropriate triangulated categories $$D_c({\mathcal X},\Lambda)$$ and $$D^{(*)}_c({\mathcal X},\Lambda)$$ of constructible $$\Lambda$$-modules on an Artin stack $$\mathcal X$$. Then, for a morphism $$f :{\mathcal X}\to{\mathcal S}$$ of finite type of stacks locally of finite type over a ground scheme $$S$$ (as in the foregoing first part), a theory of Grothendieck’s six fundamental functors ($$Rf_*$$, $$Rf_!$$, $$Lf^*$$, $$Rf^!$$, etc.) between pairs of these categories is elaborated systematically and in great generality.
Again, the authors are forced to work with unbounded complexes, and this requires a careful study of the unbounded derived category of projective systems of $$\Lambda$$-modules at the beginning. The further work is partly an extension of some previous study of the subject, mainly done by K. A. Behrend, T. Ekedahl, and U. Jannsen.
In the course of the paper, the authors complete the overall picture by establishing a satisfactory duality theory (Chapter 7) for the first time in this context.
As the authors point out, the formalism developed in the current paper will be used to study perverse sheaves on Artin attacks in another subsequent-work (cf.: Y. Laszlo and M. Olsson [Math. Z. 261, No. 4, 737–748 (2009; Zbl 1188.14002)].

##### MSC:
 14A20 Generalizations (algebraic spaces, stacks) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 14F20 Étale and other Grothendieck topologies and (co)homologies 18E30 Derived categories, triangulated categories (MSC2010)
Full Text:
##### References:
 [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. · Zbl 1051.14023 [2] A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, in Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque, vol. 100, pp. 5–171, Soc. Math. France, Paris, 1982. [3] M. Bokstedt and A. Neeman, Homotopy limits in triangulated categories, Compos. Math., 86 (1993), 209–234. · Zbl 0802.18008 [4] P. Deligne, La conjecture de Weil II, Publ. Math., Inst. Hautes Étud. Sci., 52 (1980), 137–252. · Zbl 0456.14014 [5] P. Deligne, Cohomologie étale, in Séminaire de Géométrie Algébrique du Bois-Marie (SGA $$\big(4\frac12\big)$$ ), Lect. Notes Math., vol. 569, Springer, Berlin, 1977. [6] T. Ekedahl, On the multiplicative properties of the de Rham–Witt complex. II, Ark. Mat., 23 (1985), 53–102. · Zbl 0575.14017 [7] T. Ekedahl, On the adic formalism, in The Grothendieck Festschrift, vol. II, Progr. Math., vol. 87, pp. 197–218, Birkhäuser, Boston, MA, 1990. · Zbl 0821.14010 [8] O. Gabber, Notes on some t-structures, in Geometric Aspects of Dwork Theory, vol. II, pp. 711–734, Walter de Gruyter, Berlin, 2004. · Zbl 1074.14018 [9] P.-P. Grivel, Catégories dérivées et foncteurs dérivés, in A. Borel (ed.) Algebraic D-Modules, Perspect. Math., vol. 2, Academic Press, Boston, MA, 1987. [10] M. Artin, A. Grothendieck, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, in Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4), Lect. Notes Math. vols. 269, 270, 305, Springer, Berlin, 1972. [11] A. Grothendieck et al., 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. · Zbl 0356.14004 [12] U. Jannsen, Continuous étale cohomology, Math. Ann., 280 (1988), 207–245. · Zbl 0649.14011 [13] B. Keller, On the cyclic homology of ringed spaces and schemes, Doc. Math., 3 (1998), 177–205. · Zbl 0917.19002 [14] Y. Laszlo and M. Olsson, The six operations for sheaves on Artin stacks I: Finite coefficients, Publ. Math., Inst. Hautes Étud. Sci., (2008). · Zbl 1191.14002 [15] Y. Laszlo and M. Olsson, Perverse sheaves on Artin stacks, Math. Z., to appear. · Zbl 1188.14002 [16] G. Laumon and L. Moret-Bailly, Champs algébriques, Ergeb. Math. Grenzgeb., 3. Folge, vol. 39, Springer, Berlin, 2000. [17] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc., 9 (1996), 205–236. · Zbl 0864.14008 [18] A. Neeman, Triangulated Categories, Ann. Math. Stud., vol. 148, Princeton University Press, Princeton, NJ, 2001. · Zbl 0974.18008 [19] M. Olsson, Sheaves on Artin stacks, J. Reine Angew. Math., 603 (2007), 55–112. · Zbl 1137.14004 [20] J. Riou, Dualité (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/iou/doc/dualite.pdf. [21] N. Spaltenstein, Resolutions of unbounded complexes, Compos. Math., 65 (1988), 121–154. · Zbl 0636.18006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.