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I have a technical question coming from reading Toen's master course on stacks.

If we view schemes as locally ringed spaces then there we could define a morphism to be surjective if it the underlying morphism of topological spaces is surjective. I believe this is the same as saying that for a morphism $f: X \to Y$ and every $k$-point $Spec k \to Y$ (with k a field) there exists a field extension $Spec l \to Spec k$ and an $l$-point of $X$ $Spec l \to X$ such that $Spec l \to X \to Y = Spec l \to Spec k \to Y$. (I believe one can equivalently work with geometric points, i.e. only with algebraically closed fields) So let's take the latter as a definition of surjective, as it is more natural if we work functorially (and also because it makes sense for arbitrary presheaves on Aff, the category of affine schemes)

Now, Toen uses the etale topology on affine schemes and defines schemes (and more general beasts) as particular sheaves on the etale site. His notion of surjectivity is being an epimorphism of sheaves.

My question is: do these notions coincide? What happens if we take a different topology (like fpqc)?

To be honest, the case I care about the most is when defining atlases, i.e. for $\coprod U_i \to X$, with $U_i$ affine schemes and $U_i \to \coprod_i U_i \to X$ an open immersion (or etale, or smooth, or more generally flat I guess). Nevertheless I would like to understand how much epis of sheaves capture the notion of being surjective for arbitrary morphisms.

EDIT: The most concrete description I know for an epi of sheaves $X \to Y$ is that given any affine $T \to Y$ there exists a covering (etale, or of the chosen topology) $T_i \to T$ such that there exist maps $T_i \to X$ such that $T_i \to T \to Y = T_i \to X \to Y$.

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  • $\begingroup$ Perhaps Lemma 2.1 of Cours 4 of Toen's master course on stacks can be modified to encompass more general cases? I'll give it a try. $\endgroup$
    – Yosemite Sam
    Commented Oct 27, 2011 at 21:29
  • $\begingroup$ A morphism of schemes $X \to Y$ is surjective as sheaves iff for every morphism $T \to Y$ there is some étale cover $E \to T$, such that $E \to T \to Y$ lifts to some morphism $E \to X$. Is this correct? $\endgroup$
    – Martin Brandenburg
    Commented Oct 28, 2011 at 5:12
  • $\begingroup$ I edited to answer your question. $\endgroup$
    – Yosemite Sam
    Commented Oct 28, 2011 at 12:47
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    $\begingroup$ It is necessary that $X\to Y$ has sections locally, so you get counterexamples for the etale topology when there are inseparable or infinite residue field extensions, or counterexamples like $(\mathbf A^1\setminus\{0\})\sqcup\{0\}\to\mathbf A^1$ for any of the mentioned topologies. $\endgroup$
    – user2035
    Commented Oct 28, 2011 at 15:32
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    $\begingroup$ Yes, $X\to Y$ smooth and surjective works, essentially by EGA IV, 17.16.3. Generally, you cannot expect morphisms of one class to always have local sections with respect to a coarser topology, as explained in Vistoli's article in "FGA explained", example 2.51. $\endgroup$
    – user2035
    Commented Oct 28, 2011 at 18:30

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I think (what currently is) Lemma 5.9 (tag 05VM) of the Algebraic Spaces chapter of the stacks project is exactly what we want.

Let F,G fppf sheaves and let $F \to G$ be a schematic, flat, locally of finite presentation and surjective (as in: surjective on fields). Then it is an epimorphism of sheaves.

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