2013/04/30 by Leovigildo Alonso, Leovigildo Alonso Tarrío, Ana Jeremías López +5
Mathematics · #Adjunction #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number #Algebraic structures and combinatorial models #Coherent sheaf #Derived category #Formalism (music) #Grothendieck group #Homotopy and Cohomology in Algebraic Topology #Stack (abstract data type) #math.AG #msc:14A20 #msc:14F05 #msc:18F20
paper · pdf · doi:10.1016/j.exmath.2014.12.007
published as Expo. Math., 33 (2015) pp. 452-501 · 51 pages. V2: new introduction, discussion of qc-sheaves on a quotient stack and several further improvements
arxiv created 2014/11/21 · openalex publication_date 2014/12/30 · openalex created_date 2016/06/24 · arxiv updated 2017/04/27 · openalex updated_date 2026/08/05
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 associate, in a 2-functorial way, to a 1-morphism of geometric stacks f, 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 the Hopf algebroid.