The Keel–mori Theorem via Stacks

Let X be an Artin stack (always assumed to have quasi-compact and separated diagonal over SpecZ; cf. [2, §1.3]). A coarse moduli space for X is a map π : X → X to an algebraic space such that (i) π is initial among maps from X to algebraic spaces (note that the category of maps from X to an algebraic space is discrete), and (ii) for every algebraically closed field k the map [X (k)]→ X(k) is bijective (where [X (k)] denotes the set of isomorphism classes of objects in the small category X (k)). If X is equipped with a map to a scheme S then X has a unique compatible map to S, and so it is equivalent to require the universal property for algebraic spaces over S. In [5], Keel and Mori used a close study of groupoids to prove that if X is locally of finite type over a locally noetherian scheme S and its inertia stack IS(X ) = X ×X ×SX X is finite over X then there exists a coarse moduli space π : X → X with X locally of finite type over S (and separated over S when X is separated over S). They also proved that π is a proper universal homeomorphism, that for any flat (locally finite type) base change X ′ → X in the category of algebraic spaces the map π×X X ′ is a coarse moduli space, and that OX → π∗(OX ) is an isomorphism. The finiteness hypothesis on IS(X ) is weaker than finiteness of ∆X /S and is stronger than quasi-finiteness of ∆X /S . The purpose of this note is to explain how to systematically use stacks instead of groupoids to give a more transparent version of the Keel–Mori method and to eliminate noetherian assumptions. Since the formation of coarse spaces does not generally commute with (non-flat) base change, it does not seem possible to immediately reduce existence problems to the locally noetherian case. If we wish to avoid separatedness hypotheses on X over S but want π : X → X to be a universal homeomorphism then it is unreasonable to consider X for which ∆X /S is not quasi-finite (e.g., the non-separated stack Q = A/Gm that classifies line bundles equipped with a section has k-points specializing to other k-points and so it cannot admit a coarse moduli space Q → Q that is a universal homeomorphism).