Philosophy : seeking examples illuminating deeper geometric ideas behind base change of schemes

Question

So for certain base changes, it's clear that 'base change' really means base change from high school : for example, if a curve is defined over a field, it will be of course defined over any extension of that field, and 'base change' as defined for schemes agrees with that, essentially because of properties of tensor product. Similarly, if we define projective space over $\mathbb{Z}$, or the general linear group scheme over $\mathbb{Z}$, then changing base by the unique $Spec(A)\to Spec(Z)$ (for any ring $A$) gives projective space over $A$ and the general linear group scheme over $A$ as well.

It would be nice to see some examples of base change of $X/S$ through some stranger change of base maps $S'\to S$, and to see how $X/S$ is deformed by $S'\to S$ in ways that are 'fundamentally' not like the above ; i.e., the examples I listed above don't really capture the 'spirit' of base change in general. I'm looking for a list of other central geometric ideas behind the notion of base change that go beyond the above ideas.

For example, if $X/S$ is finite etale, $s$ in the image, then changing base by the strict localization at $s$ gives a trivial finite étale covering (i.e. the base change is a finite union of copies of the strict localization at $s$), thus showing that (surjective?), finite étale maps are locally trivial for the étale topology (and therefore the name 'étale covering' is meaningful in this sense).

Can someone give a motivated discussion of the geometry of base change ? E.g., examples-based observation and pondering.

Answer 1

Dear Giovanni, here is a point of view orthogonal to the examples you mention (although I am not sure they will satisfy your wish that they be "stranger"... ).

Given a morphism $X \to S$, instead of enlarging $S$ to some bigger scheme $S'$, take instead a subscheme $S'\subset S$ : base change will then correspond to scheme-theoretic inverse image. In the affine case, $SpecB \to SpecA$, base change will lead to $Spec B/IB \to Spec A/I$ : extension corresponds to quotient.

In particular for a closed point $P \in S$ you will get the fibre over $P$. And if the base scheme is integral the fibre over the generic point $\eta$ of $S$ will give you the generic fibre of the morphism, which is a scheme over the rational function field $Rat(S)= k(\eta)$ of $S$.

Probably you know all this but I thought it might be useful to have a reminder that contrary to our unconscious bias due to the terminology , base change and ring extensions may lead to smaller objects : subschemes and quotient rings.

Answer 2

I think that thinking about nontrivial pullbacks in the category of topological spaces might help you out. The answer might be below you, but hopefully it will be of some use to people thinking about this stuff for the first time.

In topological spaces, the pullback of $X \overset{f}{\rightarrow} S$ and $Y \overset{g}{\rightarrow} S$ is just that subspace of $X \times Y$ where $f(x) = g(y)$. A similar description in all other categories is possible: The pull back always has a natural monic arrow to the product induced by the projection maps.

Examples:

1) The pullback of $X \rightarrow 1$ and $Y \to 1$ is just the ordinary product. In general the pullback of two maps to the terminal object of a category is just the categorical product of the domains of those maps. This is why we define products of schemes to be their fiber product as $\mathbb{Z}-schemes$.

2) If $1 \overset{s}{\to} S$ just picks out an element $s$ of $S$ and $X \overset{f}{\to} S$ is any morphism, then the pullback of $s$ with $f$ is just the fiber $f^{-1}(s)$ considered as a subspace of $X$.

Really nontrivial pullbacks are kind of hard to actually visualize. Here is one example where it is possible, and might be useful to think about:

3) It is difficult to draw in this format, but consider the map $\pi: S^1 \to [-1,1]$ from the unit circle centered at the origin to the interval given by just projecting down to the x-axis, and the map $\rho: D^1 \to [-1,1]$ from the unit disk to the interval also given by projection.

The product space $D \times S$ is just a solid torus. The pullback in this case will be a two-dimensional surface sitting inside this torus with two cusp points corresponding to the endpoints of the interval. You will have to just draw this out and see what subset of $D \times S$ I am talking about.

So it is a bit weird that "extending the base" from the line segment to the disk results in this singular surface contained in the solid torus, but it is really the data which the projection maps are carrying around that matters. The geometry seems to get a bit lost here compared with the first two examples.

Pullbacks in the category of schemes are even crazier because you also have morphisms of sheaves to worry about, and so all of this algebraic data enters the mix, and actually kills the geometric interpretation we developed for topological spaces (The product of two schemes is not defined on the product of the topological spaces!). But hopefully getting a handle on the topological space first will help you think about schemes.