>>12155229Just to correct you, a -rational point is not just a ring morphism, but a -algebra morphism.
This is an important distinction for the following reason:
Suppose you have an algebraic variety over a field , and let be a field extension. A morphism is equivalent to specifying a point of (obvious, since a field only has one prime ideal), and a field extension , where is the residue field at a point. The latter comes from the fact that the morphism on the sheaves induce a local ring homomorphism , ie the ideal corresponding to gets mapped to the zero ideal of the field , so in particular it induces the field homomorphism (which is by definition a field extension).
In particular, if we take and further, that is a morphism of -schemes/varieties, then we force that the extension canonically takes to itself, so that . And that's the definition of a -rational point.
Translating all this to the language of algebra, locally, the morphism looks like , and the kernel of this morphism is a maximal ideal. Forcing the morphism to be that of a -algebra, then the kernel is a maximal ideal corresponding to points that are a solution over to the polynomial equations defining . Taking in particular gives us exactly one point for every such morphism.
Therefore, the -rational points are in bijection with the solutions to your polynomial equations.
