Do quotients of representable sheaves represent quotients? - MathOverflow

Do quotients of representable sheaves represent quotients?

2009-11-12T05:59:24Z

Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $\mathcal{F}$ (on the site Et(X)) is constructible if and only if it is the coequalizer of an etale equivalence relation $\mathcal{R}\rightrightarrows \mathcal{Y}$, where $\mathcal{R}$ and $\mathcal{Y}$ are representable sheaves. Here an etale equivalence relation is defined exactly as you would expect: $\mathcal{R}\rightarrow \mathcal{Y}\times \mathcal{Y}$ is injective, and for every etale $U\rightarrow X$, $\mathcal{R}(U)\subset \mathcal{Y}(U)\times \mathcal{Y}(U)$ is an equivalence relation (of sets).

Now if the quotient $\mathcal{Y}/\mathcal{R}$ "should" be represented by the quotient $Y/R$ (where $Y$ represents $\mathcal{Y}$ and $R$ represents $\mathcal{R}$), well, it sounds like constructible sheaves should be algebraic spaces, or at least there should be some relationship. On the other hand, I don't think this could be right.

So is the problem that you can't take sheafy quotients like this, or is it something more subtle?

Answer by Anton Geraschenko for Do quotients of representable sheaves represent quotients?

2009-11-12T07:53:51Z

It seems like your interpretation is correct. The bottom of page 39 reads

As we have already seen in § 1, for every sheaf of sets $\mathcal F$ on the scheme $X$ there is a family $X_\alpha$ of etale $X$-schemes and a surjective sheaf mapping $\coprod \tilde X_\alpha\to \mathcal F$ (where $\tilde X_\alpha$ is the sheaf represented by $X_\alpha$). The sheaf $\mathcal F$ is called constructible if one can get by with a finite family. In this case $\mathcal F$ actually turns out to be the quotient of a representable sheaf by a representable equivalence relation (not separated, in general). Thus one can interpret $\mathcal F$ as an etale algebraic space over $X$ (not necessarily separated) in the sense of M. Artin, Knutson [91].

Since the disjoint union of an arbitrary collection of etale $X$-schemes is an etale $X$-scheme, they must be implicitly assuming each $X_\alpha$ is finite type or something.

Answer by Timo Schürg for Do quotients of representable sheaves represent quotients?

2009-11-16T13:42:04Z

This is just a little side remark that in general it does make a difference whether you take quotients as representable functors or as schemes. There is a counterexample in section 4 of http://www.math.univ-toulouse.fr/~toen/cours1.pdf.

The counter example is as follows. Let R be the real number and Q the rationals. Then Q acts on R by translation. The quotiens R/Q is nothing reasonably geometric. But if we take h_R to be the functor represented by R, then the quotient h_R / Q is an algebraic space!