Lemma 110.55.1. There exists a functor $F : \mathit{Sch}^{opp} \to \textit{Sets}$ which satisfies the sheaf condition for the fpqc topology, has representable diagonal $\Delta : F \to F \times F$, and such that there exists a surjective, flat, universally open, quasi-compact morphism $U \to F$ where $U$ is a scheme, but such that $F$ is not an algebraic space.
110.55 Sheaf with quasi-compact flat covering which is not algebraic
Consider the functor $F = (\mathbf{P}^1)^\infty $, i.e., for a scheme $T$ the value $F(T)$ is the set of $f = (f_1, f_2, f_3, \ldots )$ where each $f_ i : T \to \mathbf{P}^1$ is a morphism of schemes. Note that $\mathbf{P}^1$ satisfies the sheaf property for fpqc coverings, see Descent, Lemma 35.13.7. A product of sheaves is a sheaf, so $F$ also satisfies the sheaf property for the fpqc topology. The diagonal of $F$ is representable: if $f : T \to F$ and $g : S \to F$ are morphisms, then $T \times _ F S$ is the scheme theoretic intersection of the closed subschemes $T \times _{f_ i, \mathbf{P}^1, g_ i} S$ inside the scheme $T \times S$. Consider the group scheme $\text{SL}_2$ which comes with a surjective smooth affine morphism $\text{SL}_2 \to \mathbf{P}^1$. Next, consider $U = (\text{SL}_2)^\infty $ with its canonical (product) morphism $U \to F$. Note that $U$ is an affine scheme. We claim the morphism $U \to F$ is flat, surjective, and universally open. Namely, suppose $f : T \to F$ is a morphism. Then $Z = T \times _ F U$ is the infinite fibre product of the schemes $Z_ i = T \times _{f_ i, \mathbf{P}^1} \text{SL}_2$ over $T$. Each of the morphisms $Z_ i \to T$ is surjective smooth and affine which implies that
\[ Z = Z_1 \times _ T Z_2 \times _ T Z_3 \times _ T \ldots \]
is a scheme flat and affine over $Z$. A simple limit argument shows that $Z \to T$ is open as well.
On the other hand, we claim that $F$ isn't an algebraic space. Namely, if $F$ where an algebraic space it would be a quasi-compact and separated (by our description of fibre products over $F$) algebraic space. Hence cohomology of quasi-coherent sheaves would vanish above a certain cutoff (see Cohomology of Spaces, Proposition 69.7.2 and remarks preceding it). But clearly by taking the pullback of $\mathcal{O}(-2, -2, \ldots , -2)$ under the projection
\[ (\mathbf{P}^1)^\infty \longrightarrow (\mathbf{P}^1)^ n \]
(which has a section) we can obtain a quasi-coherent sheaf whose cohomology is nonzero in degree $n$. Altogether we obtain an answer to a question asked by Anton Geraschenko on mathoverflow.
Proof. See discussion above. $\square$
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