Approximation of stacks / algebraic spaces

Question

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)?

Answer 1

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

First of all, you would have to make some assumptions such as assume that X_λ→ Spec(B_λ) 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 U_λ→X_λ. 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 B_λ be a direct limit of essentially étale algebras such that B is henselian. If openness of versality holds for X_λ, then U_λ→X_λ is smooth in a neighborhood of the image of U→U_λ and (at least if B₀ is noetherian) it follows that U_λ→X_λ is smooth for sufficiently large λ.

But ... openness of versality does not hold for general X_λ. 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=colim_k X_0,k where X_0,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)=colim_k X_0,k(T)

for any scheme T. Then clearly, X=X₀×_B0 Spec(B) is isomorphic to Spec(B) but X_λ is not an algebraic space for any λ.

If you assume that openness of versality holds for X_λ and that the Spec(B)→Spec(B_λ) 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.