example of functor of points of a scheme
Let be some field extension of . What are the -points of ? Recall that a -point of is by definition a morphism (observe that since we have an embedding we have a morphism , so is natuarlly a -scheme). Since is affine, this must come from a ring homomorphism
which takes elements of to themselves inside . Such a homomorphism is completely specified by specifying the images of ; for it to be a homomorphism, these images must satisfy . In other words, a -point on is identified with an element of satisfying all the polynomials .
If is an algebraically closed field, a point on corresponds uniquely to a point on an affine variety defined by the same equations as . If is just any extension of , then we have simply found which new points belong on 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 . Then has no points over , its natural base field. Over , it has two points, corresponding to and .
This suggests that schemes may be the appropriate adaptation of varieties to deal with non-algebraically closed fields.
Observe that we never used the fact that (or in fact ) was a field. One often chooses as something other than a field in order to solve a problem. For example, one can take . Then specifying a -point on amounts to choosing an image for each . It is clear that the must satisfy the . But upon reflection, we see that the must specify a tangent vector to at the point specified by the . So the -points tell us about the tangent bundle to . Observe that we made no assumption about the field — we can extract these “tangent vectors” in positive characteristic or over a non-complete field.
The ring 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|
|Date of creation||2013-03-22 14:11:07|
|Last modified on||2013-03-22 14:11:07|
|Last modified by||archibal (4430)|