Flat is enough

The stacks project now contains Artin’s Theorem Tag 06DC:

Let f : X —> Y be a 1-morphism of stacks in groupoids on (Sch/S)_{fppf}. Assume that X is representable by an algebraic space, f is representable by algebraic spaces, surjective, locally of finite presentation, and flat. Then Y is an algebraic stack.

Extremely loosely speaking this means that to verify that a stack in groupoids Y is an algebraic stack, it suffices to find a flat cover by scheme, smoothness not required!

This has some pleasing consequences which we have not yet spelled out in the stacks project (most of these are similar to the consequences to the corresponding result for algebraic spaces, see Theorem Tag 04S6 ff). For example

  1. Given a stack in groupoids X over a scheme S and an fppf covering {S_i —> S} such that X|_{S_i} is an algebraic stack for each i, then X is an algebraic stack.
  2. Given a flat, finitely presented group scheme G over S acting on a scheme X over S, then the quotient stack [X/G] is algebraic.
  3. Given a groupoid scheme (U, R, s, t, c) with s, t flat and locally of finite presentation, then the quotient stack [U/R] is an algebraic stack.

And so on and so forth. Moreover, in the process of proving the theorem stated above we proved some results on algebraicity of spaces of sections, relative morphisms, restriction of scalars, and finite Hilbert stacks, most of which can now be considerably improved.

Our next goal in the stacks project is to add more basic theory on algebraic stacks, add some material on deformation theory, and (perhaps) something on approximation theory.

A fun fact is that the graph of logical dependencies for Theorem Tag 06DC has depth 68 and has 4006 edges!

4 thoughts on “Flat is enough

  1. Johan, since this “flat is enough” result is due to M. Artin, perhaps that should be noted in the blog entry above.

  2. Pingback: Flat is not enough | Stacks Project Blog

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