The notion of a surjective map of schemes is somewhat confusing. For example, consider an elliptic curve ${E}$ over ${{\mathbb Q}}$ of positive rank. The multiplication-by-${m}$ map is obviously not surjective as a map ${E({\mathbb Q}) \rightarrow E({\mathbb Q})}$, but in fact it is surjective as a map of schemes, because schemes carry baggage associated to the points we haven’t yet `added in’ by passing to an algebraic closure. In this post we investigate the condition of a map being surjective on schemes as related to its inducing a surjection on the geometric points. I looked briefly in Hartshorne and did not seem to find anything about this, and I still do not have the whole picture (I have only considered what happens on closed points), so if readers can suggest a proper reference I would be grateful. Also, this post is likely to contain errors which might need fixing, so I urge readers not to assume I know what I’m doing.

Proposition 1 Let ${\phi: Y \rightarrow X}$ be a map of schemes of finite type over ${k}$, and ${k^a}$ an algebraic closure for ${k}$. The induced map ${Hom_k(Spec \ k^a, Y) \rightarrow Hom_k(Spec \ k^a, X)}$ is surjective if and only if ${Y \rightarrow X}$ is surjective on closed points.

I think one direction is reasonably straightforward. Given a closed point ${x \in X}$, take any geometric point ${f:Spec\ k^a \rightarrow X}$ with image ${x}$, for example the point obtained from ${O_U \rightarrow O_{U,x} \rightarrow k(x) \rightarrow k^a}$ for some affine ${U}$ containing ${x}$ (since we are of finite type over ${k}$ and ${x}$ is maximal, ${k(x)}$ is a field of finite type over ${k}$, so finite over ${k}$ by the Nullstellensatz). Then by assumption, there is a ${g:Spec\ k^a \rightarrow Y}$ with ${\phi g = f}$, so in particular any closed point in the closure of ${g(0)}$ is a closed point of ${Y}$ lying above ${x}$.

Let’s try to prove the other direction. So now we are given a geometric point ${f:Spec\ k^a \rightarrow X}$ and we want to lift it to such a point of ${Y}$. Well, such an ${f}$ determines a unique pair ${(x,\phi)}$ with ${x\in X}$ and ${\psi: k(x) \rightarrow k^a}$, and ${x}$ must be closed by a dimension argument, so let us choose a closed point ${y \in Y}$ with ${\phi(y)=x}$. There is an induced map in the wrong direction ${k(x) \rightarrow k(y)}$. However, since ${y}$ is closed, ${k(y)}$ is a finite extension of ${k}$, so in particular we can generate a map ${\psi'k(y) \rightarrow k^a}$. Furthermore, it is possible to do so in such a way as to ensure that ${k(x) \rightarrow k(y) \rightarrow k^a}$ is the map ${\psi}$.

But then the pair ${(y,\psi')}$ determine a geometric point ${g:Spec\ k^a \rightarrow Y}$ in the obvious fashion, and it is clear from our construction that ${\phi g = f}$, so we have proved that the map on geometric points is a surjection.

There are at least two things I still don’t know which would be good to know. Firstly, what conditions do I need to impose to guarantee that ${k^a}$ can be replaced with ${k^s}$, a separable closure? There is some kind of condition needed to say that induced maps are separable (so I guess insisting that ${\phi}$ is unramified) but also that the extensions ${k(x)/k}$ are separable, which is probably some simple condition but I haven’t yet worked out what. Secondly, can I extend the result to surjectivity of the map ${Y \rightarrow X}$ on all points? Again, hopefully this is fairly easy messing around with taking closures, etc. but I still don’t seem to have managed it.

It would also be nice to know if the above argument is actually correct and/or necessary. It’s been assembled in a very ad-hoc way to convince me that things I was doing with étale covers of elliptic curves weren’t nonsense, so I can believe my sketchy knowledge of scheme theory could have allowed a flaw to pass. Note that the result claimed about multiplication by ${m}$ on an elliptic curve follows (on the closed points) immediately, because given any geometric point I can divide it by ${m}$ by solving an appropriate degree ${m^2}$ polynomial, hence I get a geometric point with co-ordinates in ${k^a}$.