On Mathoverflow Anton Geraschenko asks the following question:<\/p>\n
Suppose F : Sch^{opp}\u2192Set is a sheaf in the fpqc topology, has quasi-compact representable diagonal, and has an fpqc cover by a scheme. Must F be an algebraic space? That is, must F have an \u00e9tale cover by a scheme?<\/p><\/blockquote>\n
The question isn’t well posed as the question does not specify _exactly_ what is meant by an “fpqc cover by a scheme”. Does it mean a morphism which is a surjection of sheaves in the fpqc topology? In that case you get counter examples by looking at ind schemes (for example the functor of morphisms A^1 —> A^1). Does it mean a flat morphism which is a surjection of sheaves in the fpqc topology? In that case, I think there is a counter example by taking an ind-scheme where the transition morphisms are flat closed immersions (I can explain but it isn’t interesting). Does it mean a flat, surjective, quasi-compact morphism? In this case it is more difficult to give a counter example, but I think I have one. Before I get into it, note that algebraic spaces in general do not have such coverings, so that the resulting category of “fpqc-spaces” does not contain the category of algebraic spaces.<\/p>\n
Here is my idea for a counter example. (I’m having a kind of deja vu here, so it is perhaps somebody else’s idea? Please let me know if so.) Consider the functor F = (P^1)^∞, i.e., for a scheme T the value F(T) is the set of f = (f_1, f_2, f_3, …) where each f_i : T —> P^1 is a morphism. A product of sheaves is a sheaf, so F is a sheaf. The diagonal is representable: if f : T —> F and g : S —> F, then T ×_F S is the scheme theoretic intersection of the closed subschemes T ×_{f_i, P^1, g_i} S inside the scheme T × S. Consider U = (SL_2)^∞ with its canonical morphism U —> F. Note that U is an affine scheme. OK, and now you can show that the morphism U —> F is flat, surjective, and even open. Without giving all the details, if f : T —> F is a morphism, then you show that Z = T ×_F U is the infinite fibre product of the schemes Z_i = T ×_{f_i, P^1} SL_2 over T. Each of the morphisms Z_i —> T is surjective, smooth, and affine which implies the assertions. In particular, 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 Proposition Tag 072B<\/a> and remarks preceding it). But clearly by taking O(-2,…,-2,0,…) on F = (P^1)^∞ we get a quasi-coherent sheaf whose cohomology is nonzero in an arbirary positive degree.<\/p>\n","protected":false},"excerpt":{"rendered":"
On Mathoverflow Anton Geraschenko asks the following question: Suppose F : Sch^{opp}\u2192Set is a sheaf in the fpqc topology, has quasi-compact representable diagonal, and has an fpqc cover by a scheme. Must F be an algebraic space? That is, must … Continue reading