2014/02/13 by Daniel Halpern-Leistner, Halpern-Leistner, Daniel, Anatoly Preygel +1 · 4 citations
Computer Science · Mathematics · #14A20 #14F05 #18-XX #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1402.3204
openalex publication_date 2014/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
One fundamental consequence of a scheme X being proper is that the functor classifying maps from X to any other suitably nice scheme or algebraic stack is representable by an algebraic stack. This result has been generalized by replacing X with a proper algebraic stack. We show, however, that it also holds when X is replaced by many examples of algebraic stacks which are not proper, including many global quotient stacks. This leads us to revisit the definition of properness for stacks. We introduce the notion of a formally proper morphism of stacks and study its properties. We develop methods for establishing formal properness in a large class of examples. Along the way, we prove strong h-descent results which hold in the setting of derived algebraic geometry but not in classical algebraic geometry. Our main applications are algebraicity results for mapping stacks and the stack of coherent sheaves on a flat and formally proper stack.