As I understand, one of the reasons for "bootstrapping" to the category of algebraic spaces before constructing the category of Artin stacks is that algebraic spaces form a stack in the etale (at least) topology, while schemes do not, even though one frequently has (at least in other contexts) to use the fact that, say, affine or quasi-affine schemes do form a stack even in the fpqc topology (by descent theory for quasi-coherent sheaves). As a result, I'm curious: what is the simplest example of non-gluable (say, etale) descent data for schemes?

To clarify, I'm looking for an example of an fpqc morphism $Y' \to Y$, a scheme $X' \to Y'$ together with the usual patching after pull-back to $Y' \times_Y Y'$ that does not come from a scheme over $Y$.

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    $\begingroup$ Isn't this just the same as an algebraic space which is not a scheme? $\endgroup$ – Martin Brandenburg Aug 5 '11 at 8:46
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    $\begingroup$ The descent for affine schemes is not the same as the descent for quasicoherent sheaves; it is a harder fact and while the descent for qcoh sheaves generalizes to the noncommutative case, the descent for affine schemes does not. $\endgroup$ – Zoran Skoda Aug 5 '11 at 12:00
  • $\begingroup$ @Martin: It depends what your base site is. I interpreted the question as "schemes do not form a stack on the etale topology on Sch (or Aff)?" In that case, non-gluable descent data consists of an etale sheaf $F$, together with a morphism to a scheme (or affine scheme) $X$, and an (affine) etale cover $U\to X$ such that $F\times_X U$ and $F \times_X U\times_X U$ are schemes. $F$ has to be a scheme etale locally on some (affine) scheme. If you replace Sch (or Aff) by the topos with the canonical topology, then you're correct since $F$ only needs to be a scheme etale locally on itself. $\endgroup$ – Anton Geraschenko Aug 5 '11 at 12:39
  • $\begingroup$ Akhil -- Just to clarify: are you looking for a scheme $B$, an étale cover $B'\to B$, a scheme over $B', $f':X'\to B'$, and an isomorphism of the two pullbacks to $B'\times_B B'$ satisfying the cocycle condition, but such that the descent datum is not effective (in schemes)? This seems to be a little different than the question some posters are answering. $\endgroup$ – Jason Starr Aug 5 '11 at 15:25
  • $\begingroup$ Hmm -- I must have forgotten some dollar signs. Sorry for the ugly output. $\endgroup$ – Jason Starr Aug 5 '11 at 15:27

I'm pretty sure this example works. I do not include any proof that this is the simplest example, and it may not be, but it's not too complicated.

Let $L_1$ and $L_2$ be two rational curves in $\def\P{\mathbb P}\P^3$ which intersect in two points. A standard example of a proper non-projective variety $X$ is obtained by blowing up $L_1$ and $L_2$, but doing it in one order at one intersection point and in the other order at the other intersection point (I think this example is explained at the end of Hartshorne).

There is an involution $\sigma$ of $\P^3$ which switches the two lines and the two intersection points. Let $U\subseteq \P^3$ be the open locus where $\sigma$ acts freely, and let $Y=U\times_{\P^3}X$. Then $Y/\sigma$ is an algebraic space (over the scheme $U/\sigma$) which is not a scheme. It becomes a scheme after the etale base change $U\to U/\sigma$.

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    $\begingroup$ Mumford's book on Geometric Invariant Theory devotes a few pages to a discussion of this example. $\endgroup$ – Kevin Buzzard Aug 5 '11 at 8:16
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    $\begingroup$ Raynaud's thesis (LNM 119) contains an example (XIII.3.2, and remark XIII.3.1(b) following it) where the base $Y$ is local, noetherian, 2-dimensional and normal, $Y'\to Y$ is finite étale, and $X'\to Y'$ is a relative curve of genus $1$ (a torsor under an elliptic curve). $\endgroup$ – Laurent Moret-Bailly Aug 7 '11 at 10:04
  • $\begingroup$ Is $Y/\sigma\to U/\sigma$ fpqc? (I think the OP is requiring this? not sure) $\endgroup$ – Qfwfq Mar 22 '19 at 14:54

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