Universal adequate moduli space morphisms to Deligne-Mumford stacks exist from algebraic stacks under mild assumptions, with ascending chains stabilizing when cohomologically affine or targeting DM stacks, but failing in general as shown by an explicit Noetherian DM stack counterexample whose limit
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Factorizations of Moduli Morphisms and Universal Maps to Deligne-Mumford Stacks
Universal adequate moduli space morphisms to Deligne-Mumford stacks exist from algebraic stacks under mild assumptions, with ascending chains stabilizing when cohomologically affine or targeting DM stacks, but failing in general as shown by an explicit Noetherian DM stack counterexample whose limit