  We consider the stack of coherent algebras with proper support, a moduli
problem generalizing Alexeev and Knutson's stack of branchvarieties to the case
of an Artin stack. The main results are proofs of the existence of Quot and Hom
spaces in greater generality than is currently known and several applications
to Alexeev and Knutson's original construction: a proof that the stack of
branchvarieties is always algebraic, that limits of one-dimensional families
always exist, and that the connected components of the stack of branchvarieties
are proper over the base under certain hypotheses on the ambient stack.
