The following question can basically be seen as the “spiritual sequel” to a question that I asked no less than two years ago (time flies fast, doesn’t it?). In retrospect, that question was fairly poorly formulated and fairly confused, and so it’s understandable that no clear answer could be given to it at the time. Fortunately, it turned out that for the purposes I had in mind at the time, the question was kind of irrelevant, and so I managed to get what I wanted to accomplish accomplished without having it sorted out.
Nevertheless, here I find myself going through algebraic geometry again, and so I hope that this time I will be able to ask it in a clearer way.
So, some background. In the first chapter of Hartshorne’s Algebraic Geometry, he first introduces much of the machinery of classical algebraic geometry in the framework of algebraic sets, and then moves on to the specifics of varieties. This is fairly common as far as textbooks are concerned, of course, all algebraic sets can be expressed as finite unions of varieties, so it sort of makes sense to work within the framework of varieties, and then implicitly leave it as an exercise for the reader to extend it to the more general case of algebraic sets. Fair enough.
I’m trying to perform such an extension here. Specifically, in Prop. I.3.5, Hartshorne writes that if $X,Y$ are affine varieties, then there exists a natural bijection between morphisms between the varieties $X,Y$, and between their coordinate rings $A(Y),A(X)$,
$$\text{Hom}(X,Y) \ \tilde{\rightarrow} \ \text{Hom}(A(Y),A(X)).$$
Proving this for the case of $X,Y$ being affine algebraic sets turns out to be a bit tricky though, because the proof Hartshorne uses relies explicitly on $X,Y$ being irreducible as algebraic sets, so you cannot just to a “search-and-replace variety for algebraic set everywhere in the proof”.
My fundamental idea was that if you have a algebraic sets $X,Y$, consisting of finite unions of varieties $\{ X_i \}$ and $\{ Y_j \}$ respectively, that is,
$$X = \bigcup_{i=1}^n X_i \qquad \text{and} \qquad Y = \bigcup_{j=1}^n Y_j ,$$
then you can simply think of any morphism $\phi : X \rightarrow Y$ as a set of morphisms between varieties, $\{ \phi_{1,1}, \dots , \phi_{1,n}, \dots , \phi_{m,1} , \dots, \phi_{m,n} \}$, where for given $i,j \in \{ 1 , \dots , n \}$ and $k \in \{ 1 , \dots , m \}$,
$$\phi_{i,k} : X_i \rightarrow Y_k,$$
and
$$\phi_{i,k} = \phi_{j,k} \quad \text{on} \quad X_i \cap X_j .$$
This is where I run into troubles when I try to establish a similar correspondence of morphisms between coordinate rings. For ideals of varieties $X,Y$, we do of course have that $I(X \cup Y) = I(X) \cap I(Y)$, and so therefore,
$$A(X) = A(\bigcup_{i=1}^n X_i) = \mathbb{F} [x_1 , \dots , x_n] / I(\bigcup_{i=1}^n X_i) = \mathbb{F} [x_1 , \dots , x_n] / (\bigcap_{i=1}^n I(X_i)) ,$$
but I don’t know how to proceed. It seems to me like you might be able to make some sort of argument using the fact that the $I(X_i)$ are radical ideals and somehow bring the Chinese remainder theorem into the mix, but that kind of strikes me as too contrived.
Anyone have any idea on how to establish a corresponding “set of morphisms of coordinate rings”?