Derived Grothendieck existence and existence of Hom stacks -- Anatoly Preygel, February 7, 2014
(joint w/ D. Halpern-Leistner)
Given two algebraic stacks X and Y, when is Hom(X, Y) representable by an algebraic stack and how do you prove it? The original "Weil restriction" (i.e., X finite) constructions were explicit, and one still has explicit constructions in the case that X is projective via Quot-schemes. The more general cases where X is a proper scheme, or a proper stack, seems to inevitably require an appeal to Artin's representability theorem aided by Grothendieck existence. The classical literature does this in case X is a proper stack with finite diagonal [Olsson] or X a proper stack but with restrictions on Y [Aoki, Lieblich]; but one also finds the ingredients needed to prove the case X = BGm and Y = BG -- and here X is certainly not a proper stack.
In this talk, we'll give a common generalization of these results. To do this, we will offer up new definitions of "proper" and "projective" for stacks so that the implications
(1) "projective" => "proper"
(2) flat and "proper" => "mapping stack exists Hom(X, Y) exists for most reasonable Y"
hold and where now "proper" is phrased in terms of sheaves and is not much stronger than just the conclusion of the Grothendieck existence theorem. We'll give a laundry-list of examples of "projective" stacks, including BGm, A1/Gm, and more generally X/G where X is projective-over-affine and H0(OX)G = k; and, all stacks with enough vector bundles and admitting a projective good moduli space, and quotients X/G where X is projective-over-affine with G acting on the affinization.
Beyond definitions, we'll also explain the salient points of a few key proofs: J. Lurie's beautiful Tannakian argument that essentially proves the implication (2); a "derived h-descent" result that gives a different view on classical Chow's lemma arguments for showing that ordinary proper implies "proper"; and some words on the rest.