A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-... more A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category. We also associate, in a 2-functorial way, to a 1-morphism of geometric stacks f : X → Y, an adjunction f * ⊣ f * for the corresponding categories of quasi-coherent sheaves that agrees with the classical one defined for schemes. This construction is described both geometrically in terms of the small flat site and algebraically in terms of comodules over the Hopf algebroid. CONTENTS 25 6. Properties and functoriality of quasi-coherent sheaves 33 7. Describing functoriality via comodules 40 8. Deligne-Mumford stacks and functoriality for the étale topology 45 References 49
A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-... more A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category. We also associate, in a 2-functorial way, to a 1-morphism of geometric stacks f : X → Y, an adjunction f * ⊣ f * for the corresponding categories of quasi-coherent sheaves that agrees with the classical one defined for schemes. This construction is described both geometrically in terms of the small flat site and algebraically in terms of comodules over the Hopf algebroid. CONTENTS 25 6. Properties and functoriality of quasi-coherent sheaves 33 7. Describing functoriality via comodules 40 8. Deligne-Mumford stacks and functoriality for the étale topology 45 References 49
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Papers by Alonso Tarrío