Do quotients of representable sheaves represent quotients? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T22:57:35Z http://mathoverflow.net/feeds/question/5162 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5162/do-quotients-of-representable-sheaves-represent-quotients Do quotients of representable sheaves represent quotients? Rebecca Bellovin 2009-11-12T05:59:24Z 2009-11-25T04:57:12Z <p>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).</p> <p>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.</p> <p>So is the problem that you can't take sheafy quotients like this, or is it something more subtle?</p> http://mathoverflow.net/questions/5162/do-quotients-of-representable-sheaves-represent-quotients/5169#5169 Answer by Anton Geraschenko for Do quotients of representable sheaves represent quotients? Anton Geraschenko 2009-11-12T07:53:51Z 2009-11-12T07:53:51Z <p>It seems like your interpretation is correct. The bottom of page 39 reads</p> <blockquote> <p>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].</p> </blockquote> <p>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.</p> http://mathoverflow.net/questions/5162/do-quotients-of-representable-sheaves-represent-quotients/5699#5699 Answer by Timo Schürg for Do quotients of representable sheaves represent quotients? Timo Schürg 2009-11-16T13:42:04Z 2009-11-16T13:42:04Z <p>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 <a href="http://www.math.univ-toulouse.fr/~toen/cours1.pdf" rel="nofollow">http://www.math.univ-toulouse.fr/~toen/cours1.pdf</a>. </p> <p>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!</p>