Disjoint union of two affine schemes

Question

Say I have two commutative rings with unity, $R$ and $S$. What does the sheaf of disjoint union of $\DeclareMathOperator{Spec}{Spec}(\Spec(R), \mathscr O_{\Spec(R)})$ and $(\Spec(S), \mathscr O_{\Spec(S)})$ look like? Thanks!

Answer 1

It looks a lot like $(\operatorname{Spec}(R\times S), \mathcal{O}_{\operatorname{Spec}(R\times S)})$.

Much easier than the bracketed argument is this: The ring of global sections of a disjoint union of ringed spaces is exactly the product of the global sections of the components. So your disjoint union has $R\times S$ as its global sections—from there, it's an easy guess that it's the affine scheme $\operatorname{Spec}(R\times S)$.

[In fact, we can intuit this very quickly: the category of affine schemes is dual to the category of commutative rings, i.e. they are essentially the same category, with the arrows reversed. So $R\times S$, the binary product of $R$ and $S$, should correspond to the binary coproduct of their spectra.]

(It is necessary to verify that the disjoint union really is the binary coproduct in said category, but this turns out to be a good guess. This is false for infinite disjoint unions, however.)