Artin's criterion for representability

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[Todo: write this page]

We work over an excellent base $\ast$ (need not be a Dedekind domain).

A functor $F:Sch^{op}\to Set$ is a locally finitely presented algebraic space if and only if the following are true:

0. $F$ is locally finitely presented (i.e. it commutes with filtered(?) projective limits); ref to theorem that for schemes/algspaces this agrees with usual notion.

1. $F$ is an fppf sheaf (is etale sheaf enough?)

2. $F$ has representable diagonal (can this be removed in light of that result in the stacks project? i.e. is it any easier to check representability of the cover constructed in the subsequent steps?)

3a. ("prorepresentability") For any field $k$ of finite type over $\ast$, and any point $\xi_0\in F(k)$, there is a formal versal deformation of $\xi_0$. That is, there is a complete local noetherian ring $R$ with residue field $k$ and a compatible family $\xi_i\in F(R/m^{i+1})$ (such that $\xi_0=\xi_0$) such that the "formal morphism" $Spec(R)\to F$ is formally smooth at the closed point.

This is done with Schlessinger's criteria.[Todo: explain criteria] Only need H1-3 to get a formal versal deformation, but may as well mention that H4 will give you a universal deformation.

You verify Schelessinger's conditions by using deformation theory to get your hands on $F(A)$. If $F$ parameterizes flat families of foos, then use deformation theory of foos.

3b. ("effectivization") Given a compatible family $\xi_i\in F(R/m^{i+1})$ which is formally smooth at the closed point, show that it is actually induced by some $\xi\in F(R)$ which is formally smoth at the closed point.

If $F$ parameterizes things that are controlled by coherent sheaves, this is where you can use Grothendieck's existence theorem. [Todo: once this is written, add a link from Research#Grothendieck's Existence Theorem for good moduli spaces]

But GET gives you more that you need. There are other approaches to effectivization; see §3[Todo: ?] of

[Todo: Matt tells me that Martin uses a trick where he shows effectivizes (and algebraizes?) by showing that there is some other functor with the same completed local ring (?)]

3c. ("algebraization") Given $\xi\in F(R)$ with $R$ a complete local noetherian ring, show that there is a finite type scheme $X$, $f\in F(X)$, and a closed point $x\in X$ such that $\widehat\O_{X,x}\cong R$ and the $\xi_i\in F(R/m^{i+1})$ are induced by $f$.

You don't actually need to check this, as Artin's theorem (Theorem 1.6 of ) says it always works. (note: $\xi$ need not itself be induced by $f$; it's possible to have multiple algebraizations) The result is supposed to boil down to Artin's approximation theorem (Theorem 1.12 of , but I don't understand how.

3d. ("openness of versality") Given a finite type scheme $X$ and $f\in F(X)$ which is formally smooth at a closed point $x\in X$, show there is an open neighborhood $U$ of $x\in X$ so that $f|_U$ is (formally?) smooth.

If the deformation theory is unobstructed, openness of versality should be essentially automatic.

Putting these steps together produces a smooth surjection from a scheme to $F$ (note that $F$ must be locally finitely presented to know that the map is a surjection). This shows that $F$ is an algebraic stack, but since it's a functor, it's an algebraic space.

If you want to check that $F$ is an algebraic stack, the same criterion works (in particular, Artin's theorem in step 3c applies to the functor of isomorphism classes of the stack), but you have to allow finite separable field extensions in step