Sheaves on Artin stacks.

*(English)*Zbl 1137.14004The main goal of the paper under review is to develop a theory of quasi-coherent and constructible sheaves on algebraic stacks. This is done by modifying a previous construction of Laumon and Moret-Bailly (which wrongly assumed that morphisms of algebraic stacks induce morphisms of lisse-étale topoi).

In the first part of the paper (Sections 2 to 5) is recalled some aspects of the theory of cohomological descent and basic definitions of the lisse-étale site, cartesian sheaves over a sheaf of algebras and verify some basic properties of such sheaves It is also shown the connection between the derived category of cartesian sheaves over some sheaf of rings with various derived categories of sheaves on the simplicial space obtained from a covering of the algebraic stack by an algebraic space. In the second part the author specializes the previous discussion to quasi-coherent sehaves by showing that the triangulated category \(D^+_{\text{ qcoh}}(\mathcal{X})\) of bounded below complexes of \({\mathcal O}_{\mathcal{X}_{\text{lis-et}}}\)-modules with quasi-coherent cohomology sheaves satisfies all the basic properties from the theory for schemes. Then he focuses in defining the cotangent complex of a morphism of algebraic stacks and how the previous methods of the first part can be used to develop a theory of constructible sheaves on an algebraic stalk. Finally a comparison between the theory of quasi-coherent sheaves in the lisse-étale topology and the theory of quasi-coherent sheaves in the big flat and big étale topologies is discussed. As an application of these results, it is proved finiteness of coherent cohomology for proper Artin stacks. The paper finishes with stack-theoretic versions of Grothendieck’s fundamental theorem for proper morphisms, Grothendieck’s existence theorem and Zariski’s connectedness theorem.

In the first part of the paper (Sections 2 to 5) is recalled some aspects of the theory of cohomological descent and basic definitions of the lisse-étale site, cartesian sheaves over a sheaf of algebras and verify some basic properties of such sheaves It is also shown the connection between the derived category of cartesian sheaves over some sheaf of rings with various derived categories of sheaves on the simplicial space obtained from a covering of the algebraic stack by an algebraic space. In the second part the author specializes the previous discussion to quasi-coherent sehaves by showing that the triangulated category \(D^+_{\text{ qcoh}}(\mathcal{X})\) of bounded below complexes of \({\mathcal O}_{\mathcal{X}_{\text{lis-et}}}\)-modules with quasi-coherent cohomology sheaves satisfies all the basic properties from the theory for schemes. Then he focuses in defining the cotangent complex of a morphism of algebraic stacks and how the previous methods of the first part can be used to develop a theory of constructible sheaves on an algebraic stalk. Finally a comparison between the theory of quasi-coherent sheaves in the lisse-étale topology and the theory of quasi-coherent sheaves in the big flat and big étale topologies is discussed. As an application of these results, it is proved finiteness of coherent cohomology for proper Artin stacks. The paper finishes with stack-theoretic versions of Grothendieck’s fundamental theorem for proper morphisms, Grothendieck’s existence theorem and Zariski’s connectedness theorem.

Reviewer: Sergio Estrada (Murcia)

##### MSC:

14A20 | Generalizations (algebraic spaces, stacks) |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

18E15 | Grothendieck categories (MSC2010) |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

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DOI

##### References:

[1] | Mem. Amer. Math. Soc. pp 774– (2003) |

[2] | DOI: 10.1007/BF02685881 · Zbl 0237.14003 · doi:10.1007/BF02685881 |

[3] | J. Alg. Geom. 12 pp 357– (2003) |

[4] | Comp. Math. 65 pp 121– (1988) |

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