Grothendieck’s Relative Point of View

/ Anton Hilado

In Varieties and Schemes Revisited we defined the notion of schemes, which is a far-reaching generalization inspired by the concept of varieties, which is essentially a kind of “shape” defined by polynomials in some way. However, the definition of schemes were but one of many innovations in algebraic geometry developed by the mathematician Alexander Grothendieck. In this post, we discuss another of these innovations, the so-called “relative point of view“, in which the focus is not just on schemes in isolation, but schemes relative to (with a morphism to) some “base scheme”.

Let S be a scheme. A scheme over S, or an S-scheme, is a scheme X with a morphism  f:X\rightarrow S called the structural morphism. If Y is another S-scheme with structural morphism g:Y\rightarrow S, a morphism of S-schemes is a morphism u:X\rightarrow Y such that f=g\circ u.

If the scheme S is the spectrum of some ring R, we may also refer to X above as a scheme over R. Every ring has a morphism from the ring of ordinary integers \mathbb{Z}, and every scheme therefore has a morphism to the scheme \text{Spec}(\mathbb{Z}), so we may think of all schemes as schemes over \mathbb{Z}.

Given two schemes X and Y over a third scheme S, we define the fiber product X\times_{S}Y to be a scheme together with projection morphisms \pi_{X}:X\times_{S}Y\rightarrow X and \pi_{Y}:X\times_{S}Y\rightarrow Y such that f\circ\pi_{X}=g\circ\pi_{Y}, and such that for any other scheme Z and morphisms p:Z\rightarrow X and q:Z\rightarrow Y, there is a unique morphism Z\rightarrow X\times_{S}Y up to isomorphism (the concept of fiber product is part of category theory – see also More Category Theory: The Grothendieck Topos).

We can use the fiber product to introduce the concept of base change. Given a scheme X over a scheme S, and a morphism S'\rightarrow S, the fiber product X\times_{S}S' is a scheme over S'. We may think of it as being “induced” by the morphism S'\rightarrow S. One of the things that can be done with this idea of base change is to look at the properties of X\times_{S}S' and see if we can use these to learn about the properties of X, which may be useful if the properties of X are difficult to determine directly compared to the properties of X\times_{S}S' (in essence we want to be able to attack a difficult problem indirectly by first attacking an easier problem related to it, which is a common strategy in mathematics).

A special case of base change is when S' is given by the spectrum of the residue field (see Localization) k corresponding to a point P of S. There is a morphism of schemes \text{Spec}(k)\rightarrow S which we may think of as the inclusion of the point P into the scheme X. Then the fiber product X\times_{S}\text{Spec}(k) is called the fiber of X at the point P. The terminology is perhaps reminiscent of fiber bundles (see Vector Fields, Vector Bundles, and Fiber Bundles), and is also rather similar to the concept of covering spaces (see Covering Spaces) in that we have some kind of space “over” every point of our “base” scheme. However, unlike those two earlier concepts, the spaces which make up our fibers may now vary as the points vary.

Actually, the concept that this special case of fiber product and base change should bring to mind is that of a moduli space (see The Moduli Space of Elliptic Curves), where every point represents a space, and the spaces vary as the points vary. Or, as we worded it in The Moduli Space of Elliptic Curves, every point of the moduli space (given by the base scheme) corresponds to a space (given by the fiber), and the moduli space tells us how these spaces vary, so that spaces which are similar to each other in some way correspond to points in the moduli space that are close together.

The lecture notes of Andreas Gathmann listed among the references below contain some nice diagrams to help visualize the idea of the fiber product and base change (these can be found in chapter 5 of the 2002 version). To see these ideas in action, one can look at the article Arithmetic on Curves by Barry Mazur (also among the references) which discusses, among other things, the approach taken by Gerd Faltings in proving the famous conjecture of Louis J. Mordell which says that there is a finite number of rational points on a curve of genus greater than 1.

References:

Grothendieck’s Relative Point of View on Wikipedia

Arithmetic on Curves by Barry Mazur

Algebraic Geometry by Andreas Gathmann

The Rising Sea: Foundations of Algebraic Geometry by Ravi Vakil

Algebraic Geometry by Robin Hartshorne

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