Conditions on diagonal not needed

In a recent contribution of Jonathan Wang to the stacks project we find the following criterion of algebraicity of stacks (see Lemma Tag 05UL):

If X is a stack in groupoids over (Sch/S)_{fppf} such that there exists an algebraic space U and a morphism u : U —> X which is representable by algebraic spaces, surjective, and smooth, then X is an algebraic stack.

In other words, you do not need to check that the diagonal is representable by algebraic spaces. The analogue of this statement for algebraic spaces is Lemma Tag 046K (for etale maps) and Theorem Tag 04S6 (for smooth maps).

The quoted result is closely related to the statement that the stack associated to a smooth groupoid in algebraic spaces is an algebraic stack (Theorem Tag 04TK). Namely, given u : U —> X as above you can construct a groupoid by taking R = U x_X U and show that X is equivalent to [U/R] as a stack. But somehow the statements have different flavors. Finally, the result as quoted above is often how one comes about it in moduli theory: Namely, given a moduli stack M we often already have a scheme U and a representable smooth surjective morphism u : U —> M. Please try this out on your favorite moduli problem!

5 thoughts on “Conditions on diagonal not needed

  1. That is beautiful. But I do have one (obvious) observation. If you try to prove that a “moduli” stack is algebraic by using Artin’s axioms, then almost certainly you will prove the diagonal is representable along the way.

    • Yes. Also note that, since you have to check that “u is representable by algebraic spaces”, you do have to show something about the Isom sheaves. But the thing you have to show involves the family of objects which defines u : U —> M and it may be that it is sometimes easier to prove it for that family.

  2. Pingback: Space over Stack « Stacks Project Blog

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