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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)].

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: DOI arXiv
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