example of functor of points of a scheme

Let X be an affine schemeMathworldPlanetmath of finite type over a field k. Then we must have


with the structure morphism XSpeck induced from the natural embedding kk[X1,,Xn].

Let k be some field extension of k. What are the k-points of X? Recall that a k-point of X is by definition a morphism SpeckX (observe that since we have an embedding kk we have a morphism SpeckSpeck, so Speck is natuarlly a k-scheme). Since X is affine, this must come from a ring homomorphismMathworldPlanetmath


which takes elements of k to themselves inside k. Such a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is completely specified by specifying the images of X1,,Xn; for it to be a homomorphism, these images must satisfy f1,,fm. In other words, a k-point on X is identified with an element of (k)n satisfying all the polynomials fi.

If k is an algebraically closed field, a point on X corresponds uniquely to a point on an affine varietyMathworldPlanetmath defined by the same equations as X. If k is just any extensionPlanetmathPlanetmathPlanetmath of k, then we have simply found which new points belong on X when we extend the base field. T

For an example of why schemes contain much more information than the list of points over their base field, take X=Spec[X]/X2+1. Then X has no points over , its natural base field. Over , it has two points, corresponding to i and -i.

This suggests that schemes may be the appropriate adaptation of varietiesMathworldPlanetmathPlanetmathPlanetmath to deal with non-algebraically closed fields.

Observe that we never used the fact that k (or in fact k) was a field. One often chooses k as something other than a field in order to solve a problem. For example, one can take k=k[ϵ]/ϵ2. Then specifying a k-point on X amounts to choosing an image κi+λiϵ for each Xi. It is clear that the κi must satisfy the fj. But upon reflection, we see that the λi must specify a tangent vector to X at the point specified by the κi. So the k[ϵ]/ϵ2-points tell us about the tangent bundle to X. Observe that we made no assumptionPlanetmathPlanetmath about the field k — we can extract these “tangent vectors” in positive characteristic or over a non-complete field.

The ring k[ϵ]/ϵ2 and rings like it (often any Artinian ring) can be used to define and study infinitesimal deformations of schemes, as a simple case of the study of families of schemes.

Title example of functor of points of a scheme
Canonical name ExampleOfFunctorOfPointsOfAScheme
Date of creation 2013-03-22 14:11:07
Last modified on 2013-03-22 14:11:07
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 4
Author archibal (4430)
Entry type Example
Classification msc 14A15