Lemma 110.56.1. 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.
110.56 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
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
(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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