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Let $B$ be a ring which is the colimit of rings $B_\lambda$. Let $X_\lambda$ be a stack (not necessarily algebraic) over $B_\lambda$ such that $X_\lambda \times_{B_\lambda} B_\mu = X_\mu$ and let $X = X_\lambda \times_{B_\lambda} B$.

If $X$ is an algebraic stack, then does some $X_\lambda$ have to be algebraic? Are there assumptions we can add to make this true? What if the $X_\lambda$ are sheaves (so that the question becomes: if $X$ is an algebraic space, then is some $X_\lambda$ an algebraic space)?

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I fixed your LaTeX – Harry Gindi Feb 8 '10 at 11:41
You probably know about this, but David Rydh wrote a beautiful paper, Noetherian approximation of algebraic spaces and stacks. In Rydh's case, he's approximating an algebraic stack $X$ with a system $\{X_{\lambda}\}$ of others, and studying the circumstances under which one may translate useful properties between $X$ and a cofinal subsystem of the $X_{\lambda}$. But I think you're asking something slightly different: if we have a fixed system of stacks approximating an algebraic one, can we conclude that a cofinal system of those is algebraic? – Clark Barwick Feb 8 '10 at 12:27
To Harry: thanks for fixing the latex. To Clark: exactly, I was asking if some cofinal system is algebraic. – Matt Satriano Feb 10 '10 at 6:39
up vote 7 down vote accepted

The answer is no, even for the sheaf-case.

First of all, you would have to make some assumptions such as assume that Xλ→ Spec(Bλ) is locally of finite presentation. For simplicity also assume that the algebraic space X is of finite presentation over Spec(B). Then if U→X is an étale presentation, it descends to a morphism Uλ→Xλ. Now the main problem is that even if this morphism is étale after base change to X, we cannot guarantee that it eventually becomes étale. Indeed, this is related to the openness of versality in Artin's algebraization theorem.

For example, let B0 be a ring and let B=colim Bλ be a direct limit of essentially étale algebras such that B is henselian. If openness of versality holds for Xλ, then Uλ→Xλ is smooth in a neighborhood of the image of U→Uλ and (at least if B0 is noetherian) it follows that Uλ→Xλ is smooth for sufficiently large λ.

But ... openness of versality does not hold for general Xλ. Artin has given a nice bunch of examples where everything but this holds (see "The implicit function theorem in algebraic geometry", S5). For example, let B0=k[x] be the affine line and let B be the henselization (or localization) at the origin. Let X0 be a "bad" sheaf, e.g., the following (Ex. 5.10):

Let X0=colimk X0,k where X0,i=Spec(k[x,y]/y(x-1)(x-2)...(x-k)) — the union of the x-axis and k vertical lines. This is an example of an ind-space, so by definition:

  X0(T)=colimk X0,k(T)

for any scheme T. Then clearly, X=X0×B0 Spec(B) is isomorphic to Spec(B) but Xλ is not an algebraic space for any λ.

If you assume that openness of versality holds for Xλ and that the Spec(B)→Spec(Bλ) are smooth, then the answer to your question is likely "yes". However, openness of versality is the most subtle of Artin's criteria so it is probably difficult to assert.

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What's S in the first paragraph? Is it Spec(B)? – shenghao Feb 17 '10 at 19:59
Yes, and I have replaced it by Spec(B) now. Thanks. – David Rydh Feb 23 '10 at 5:57

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