Schemes do not form a stack in the etale topology?

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 $Y$.

• Isn't this just the same as an algebraic space which is not a scheme? – Martin Brandenburg Aug 5 '11 at 8:46
• 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. – Zoran Skoda Aug 5 '11 at 12:00
• @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. – Anton Geraschenko Aug 5 '11 at 12:39