Derived Grothendieck existence and existence of Hom stacks                                           
-- Anatoly Preygel, February 7, 2014

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(joint w/ D. Halpern-Leistner)

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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 = BG<sub>m</sub> and Y = BG -- and here X is
certainly not a proper stack. 

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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             

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(1) "projective" => "proper"                                                                    

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(2) flat and "proper" => "mapping stack exists Hom(X, Y)
exists for most reasonable Y"            

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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 BG<sub>m</sub>, A<sup>1</sup>/G<sub>m</sub>, and
more generally X/G where X is projective-over-affine and
H<sup>0</sup>(O<sub>X</sub>)<sup>G</sup> = 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. 

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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.