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The six operations for sheaves on Artin stacks. I: Finite coefficients. (English) Zbl 1191.14002

The paper under review is the first one in a series of articles devoted to fundamental constructions and techniques in the theory of constructible sheaves on Artin stacks. The authors’ motivation for developing such an extended general framework is based on the fact that it has already turned out to provide the necessary tools needed in various recent papers on the geometric Langlands program mainly by G. Laumon, E. Frenkel, D. Gaitsgory, K. Vilonen, and others. Also, it is hoped that this framework will shed further light on various Lefschetz-type trace formulas for stacks (or even n-stacks) of rather general kind. As for the basic set-up, let \(\Lambda\) be a Gorenstein local ring of dimension zero and characteristic \(\ell\). Fix an affine excellent scheme \(S\) of finite dimension such that \(\ell\) is invertible on \(S\). Then, for an algebraic stack \(\mathfrak X\) locally of finite type over \(S\), consider the usual derived categories \(D^*({\mathfrak X})\), where \(*\in\{+,-,b,\emptyset,[a,b]\}\), and therein the full subcategory \(D^*_c({\mathfrak X})\) of complexes of \(\Lambda\)-modules on the “lisse-étale site” of \(\mathfrak X\) with constructible cohomology sheaves, respectively. Consider also the variant subcategories \(D^{(*)}({\mathfrak X})\) of \(D^*({\mathfrak X})\) consisting of complexes \(K\) such that for any quasi-compact open \({\mathcal U}\subset{\mathfrak X}\), the restriction \(K|_{\mathcal U}\) is in \(D^*_c({\mathcal U})\).
The aim of the present paper is to develop a systematic and rigorous theory of six particular functors between some of these subcategories, which are defined with respect to a finite type morphism \(f :{\mathfrak X} \to{\mathfrak Y}\) of stacks locally of finite type over \(S\). These functors are to be generalizations of A. Grothendieck’s classical functors \(Rf_*, Rf_!, Lf^*, Rf^!\), etc. in the theory of schemes, satisfying all the usual adjointness properties known from there.
The construction of these six Grothendieck operations on lisse-étale constructible sheaves on Artin stacks (that are locally of finite type over special excellent schemes of finite Krull dimension) is carried out in full detail, with a tremendous amount of subtle technicalities and refined categorical tools. The outcome is a powerful general formalism which also leads to generalizations of the classical base change theorems and the Künneth formula to such Artin stacks, including new results about cohomological descent even for unbounded constructible complexes.
For the second part, cf. ibid. 107, 169–210 (2008; Zbl 1191.14003).

MSC:

14A20 Generalizations (algebraic spaces, stacks)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

Citations:

Zbl 1191.14003
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Full Text: DOI arXiv

References:

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