2007/01/27 by Martin Olsson · 135 citations
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Morphism #Functor #Mathematics #Pullback #Stack (abstract data type) #Pure mathematics #Coherent sheaf #Mistake #Social connectedness #Algebra over a field #Computer science #Programming language
paper · doi:10.1515/crelle.2007.012
published in Journal für die reine und angewandte Mathematik (Crelles Journal) 2007(603) (De Gruyter)
openalex publication_date 2007/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We develop a theory of quasi-coherent and constructible sheaves on algebraic stacks correcting a mistake in the recent book of Laumon and Moret-Bailly. We study basic cohomological properties of such sheaves, and prove stack-theoretic versions of Grothendieck's Fundamental Theorem for proper morphisms, Grothendieck's Existence Theorem, Zariski's Connectedness Theorem, as well as finiteness theorems for proper push forwards of coherent and constructible sheaves. We also explain how to define a derived pullback functor which enables one to carry through the construction of a cotangent complex for a morphism of algebraic stacks due to Laumon and Moret-Bailly.