Mannheimer Manuskripte 177 gk-mp-9403/3 SOME CONCEPTS OF ...

18 MARTIN SCHLICHENMAIERleft ideals of R. The kernel P has to be maximal otherwise the image of a maximal ideallying between P and R would be a non-trivial submodule of M. Hence, M ∼ = R/P. □From this point of view the maximal ideals of R(V ) correspond to R(V )−modulehomomorphisms to simple R(V )−modules. If R(V ) is a algebra over the field K, thena simple module M is of course a vector space over K. By the above, we saw that it iseven a field extension of K. (Recall that M ∼ = R/P with P a maximal ideal). BecauseR(V ) is finitely generated as K−algebra it is a finite dimensional vector space over K(see [Ku,S.56]) hence, a finite (algebraic) field extension.Observation. The maximal ideals (the “points”) of R = K[X 1 ,X 2 ,... ,X n ]/I correspondto the K−algebra homomorphism from R to arbitrary finite (algebraic) fieldextensions L of the base field K. We call these homomorphisms L−valued points.In particular, if the field K is algebraically closed there are no nontrivial algebraic fieldextensions. Hence, there are only K−valued points. If we consider reduced varieties (i.e.varieties whose coordinate rings are reduced rings) we get a complete dictionary. Let Vbe a variety, P = I(V ) the associated prime ideal generated as P = (f 1 ,f 2 ,... ,f r ) withf i ∈ K[X 1 ,X 2 ,... ,X n ] suitable polynomials, and R(V ) the coordinate ring R n /P.The points can be given in 3 ways:(1) As classical points. α = (α 1 ,α 2 ,... ,α n ) ∈ K n withf 1 (α) = f 2 (α) = · · · = f r (α) = 0.(2) As maximal ideals in R(V ). They in turn can be identified with the maximalideals in K[X 1 ,X 2 ,... ,X n ] which contain the prime ideal P. In an explicitmanner these can be given as (X 1 − α 1 , X 2 − α 2 , ... , X n − α n ) with thecondition f 1 (α) = f 2 (α) = · · · = f r (α) = 0.(3) As surjective algebra homomorphisms φ : R(V ) → K. They are fixed by defining¯X i ↦→ φ( ¯X i ) = α i ,i = 1,... ,n in such a way thatφ(f 1 ) = φ(f 2 ) = · · · = φ(f r ) = 0.The situation is different if we drop the assumption that K is algebraically closed. Thetypical changes can already be seen if we take the real numbers R and the real affine line.The associated coordinate ring is R[X]. There are only two finite extension fields of R,either R itself or the complex number field C. If we consider R−algebra homomorphismfrom R[X] to C then they are given by prescribing X ↦→ α ∈ C. If α ∈ R we are againin the same situation as above (this gives us the type (i) maximal ideals). If α ∉ R thenthe kernel I of the map is a maximal ideal of type (ii) I = (f) where f is a quadraticpolynomial. f has α and ᾱ as zeros. This says that the homomorphism Ψᾱ : X ↦→ ᾱwhich is clearly different from Ψ α : X ↦→ α has the same kernel. In particular, forone maximal ideal of type (ii) we have two different homomorphisms. Note that themap α → ᾱ is an element of the Galois group G(C/R) = {id,τ} where τ is complexconjugation. The two homomorphisms Ψ α and Ψᾱ are related as Ψᾱ = τ ◦ Ψ α .This is indeed the general situation for R(V ), a finitely generated K−algebra. In