The notion of a surjective map of schemes is somewhat confusing. For example, consider an elliptic curve over of positive rank. The multiplication-by- map is obviously not surjective as a map , 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 1Let be a map of schemes of finite type over , and an algebraic closure for . The induced map is surjective if and only if is surjective on closed points.

I think one direction is reasonably straightforward. Given a closed point , take any geometric point with image , for example the point obtained from for some affine containing (since we are of finite type over and is maximal, is a field of finite type over , so finite over by the Nullstellensatz). Then by assumption, there is a with , so in particular any closed point in the closure of is a closed point of lying above .

Let’s try to prove the other direction. So now we are given a geometric point and we want to lift it to such a point of . Well, such an determines a unique pair with and , and must be closed by a dimension argument, so let us choose a closed point with . There is an induced map in the wrong direction . However, since is closed, is a finite extension of , so in particular we can generate a map . Furthermore, it is possible to do so in such a way as to ensure that is the map .

But then the pair determine a geometric point in the obvious fashion, and it is clear from our construction that , 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 can be replaced with , a separable closure? There is some kind of condition needed to say that induced maps are separable (so I guess insisting that is unramified) but also that the extensions 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 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 on an elliptic curve follows (on the closed points) immediately, because given any geometric point I can divide it by by solving an appropriate degree polynomial, hence I get a geometric point with co-ordinates in .

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November 27, 2011 at 5:35 pm

Zhen LinI don’t think it’s at all clear whether it should be possible to replace algebraic closures by separable closures. Consider, for example, A = K[x], K = k(t), char k = p > 0. Then, one of the closed points of X = Spec A – a perfectly good integral scheme of finite type! – is the point corresponding to the maximal ideal (x^p – t), and its residue field is purely inseparable over K.

But why worry about separability, when you could assume your base field is a nice perfect field, like say Q or F_p?

November 27, 2011 at 8:55 pm

tloveringSeparability is a happy condition in my mind because then the closed points become Galois orbits of geometric points. I agree that in most cases arising from `nature’ it is not of any concern, but it would be nice to identify less trivial circumstances in which all the residue fields could be guaranteed to be separable.

November 27, 2011 at 9:12 pm

Zhen LinAren’t closed points always Galois orbits of geometric points, in an appropriately general sense? (Replace Gal(k) with the automorphism group of the algebraic closure.)

It seems that the notion of unramified morphism is very nearly what you want… unfortunately the usual definition also specifies that each residue field upstairs should be a _finite_ separable extension of the residue field downstairs. Otherwise we could just ask for the scheme to be unramified over the base field (there are problems at the generic points).

On the other hand, returning to affine schemes, Noether normalisation tells us that every affine integral scheme X of finite type over a field k of dimension n is a finite over affine n-space over k. So if n > 0 and k is imperfect, there seems to be no hope of having all the residue fields of X separable over k…