Summary: Lecture 4
December 1, 2008
Recall: functor of points.
If K is a field and a k[x1, ..., xn] is an ideal then if k L, L a field. Given
f1, ..., fn k[x1, ..., xn] we can ask for the zero in Ln. the solutions to a
over L are in bijection with the k-algebra homomorphisms k[x1, ..., xn] L,
k-algebra homomorphisms are morphisms of schemes.
Spec k[x1, . . . , xn]/a Spec L
In particular, take L = ¯k and letting X = Z(a) An
we get X =
homsch(k)(¯k, Spec A(X)). So we can recover X from it's "associated scheme"
Spec A(x). This is a better way of thinking of a variety as a scheme. Also,
Spec A(x) carries more information than X. We can also recover solutions
over any field extension of k. This might explain the terminology "scheme".
It motivates the following definition:
If X is a scheme over some field k, and L is an extension field, then