Formal Schemes Mittag-Leffler

Here is a question that is similar to my last one. I've been trying to learn about Grothendieck's Existence Theorem, but it seems that there aren't very many places that talk about formal schemes and even less that come up with examples.

Suppose $(\mathfrak{X}, \mathcal{O}_\mathfrak{X})$ is a Noetherian formal scheme and let $\mathcal{I}$ be an ideal of definition. Then we have a system of schemes $X_n=(|\mathfrak{X}|, \mathcal{O}_\mathfrak{X}/\mathcal{I}^n)$.

If the inverse system $\Gamma(X_n, \mathcal{O}_{X_n})\to \Gamma(X_{n-1}, \mathcal{O}_{X_{n-1}})$ satisfies the Mittag-Leffler condition (the images eventually stabilize), then we get some particularly nice properties such as $Pic(\mathfrak{X})=\lim Pic(X_n)$.

More generally, we don't have to be worried about converting between thinking about coherent sheaves on the formal scheme and thinking about them as compatible systems of coherent sheaves on actual schemes.

My question is, is there a known example of a formal scheme for which that system of global sections does not satisfy the Mittag-Leffler condition? One thing to note is that it can't be affine (the maps are all surjective) or projective (finite dimensionality forces the images to stabilize).

A subquestion is whether or not there is a general reason to believe such an example exists. People I talk to usually say things along the lines of: you definitely have to be careful here because in principle this could happen. But no one seems to have ever thought up an example.

Lastly (still related...I think), is there a known example where you can't think of coherent (or maybe just invertible) sheaves as systems because the two aren't the same?

Solution 1:

I don't have enough reputation to comment, so I must submit an answer which I do not really have. There is one class of varieties you must consider when searching for a counterexample: quasiprojective varieties. That is, open subsets of projective varieties which are not affine, e.g a surface minus a point. You also want the maps in question to fail to be surjective, so you should look for an ideal $\mathcal I$ which satisfies $H^1(X, \mathcal I^r / \mathcal I^{r+1}) \neq 0$ for all $r>>0$. This sounds like it's harder to do, but not impossible if you know examples.