On affine scheme

Question

Let $(X,O_X)$ be an affine scheme. By definition we know that $(X,O_X) \cong (Spec \ R , O_{Spec \ R})$ for some ring $R$. In fact we can show that we could take $R$ to be the global section $\Gamma(X, O_X)$. What I was wondering was that can we express the isomorphism $(X,O_X) \cong (Spec \ \Gamma(X, O_X),O_{ Spec \ \Gamma(X, O_X) } )$ explicitly? (For example, do we know where $p \in X$ gets mapped to? Is there a natural way to do this? and similarly for sheaf etc) Thanks!

Answer 1

Maybe I don't understand your question, but by definition $\Gamma(X, \mathcal{O}_X) = \mathcal{O}_X(X) = R$ for some ring $R$. This is the coordinate ring of your affine scheme $(X, \mathcal{O}_X)$. This coordinate ring contains the same information as the affine scheme (in fact there is an equivalence of categories $(Rings)^{op} \simeq (Affine\, Schemes)$ ). The functor ${\rm Spec}$ applied to the ring $R$ gives you the underlying topological space ${\rm Spec}(R)$ with the usual topology (the Zariski topology). So the pair you denote by $({\rm Spec}\Gamma(X, \mathcal{O}_X), \mathcal{O}_{{\rm Spec} \Gamma(X,\mathcal{O}_X)})$ is nothing but the pair $({\rm Spec}(R), R)$ defining your affine scheme. You should probably read the first couple of pages in some algebraic geometry lecture notes.

Answer 2

If you take a point $p \in X$ then the corresponding prime ideal is the set of functions that vanish, in the sense that they map to $0$ in then residue field $\kappa(p)$. In the other direction, if you have a prime $\mathfrak{p} \subset \Gamma(X, \mathscr O_X)$ then the set of its common zeros in $X$ is irreducible; take the generic point.