Quot functors for Deligne-Mumford stacks

Given a separated and locally finitely-presented Deligne-Mumford stack script x sign over an algebraic space S, and a locally finitely-presented script O sign_{script x sign}-module ℱ, we prove that the Quot functor Quot(ℱ/script x sign/S) is represented by a separated and locally finitely-presented algebraic space over S. Under additional hypotheses, we prove that the connected components of Quot(ℱ/script x sign/S) are quasi-projective over S.