Isomorphism between ring of regular functions and coordinate ring of an affine variety.

Question

I'm reading Hartshorne's Algebraic geometry book and there is a theorem which states that the ring of global functions $\mathcal{C}(Y)$ is isomorphic to the coordinate ring $A(Y)$ of an affine variety $Y$. I get it that there is natural injective homomorphism from $A(Y)$ to $\mathcal{C}(Y)$ but it is not clear to me the surjective part. Can you please help me on this? Thanks.

Answer 1

This is the key fact that Hartshorne uses: If $B$ is an integral domain, then $B = \bigcap_\mathfrak{m} B_\mathfrak{m}$, where $\mathfrak{m}$ ranges over all maximal ideals, and each $B_\mathfrak{m}$ is considered as a subset of the field of fractions of $B$.

To prove this fact, take some $t\in K(B)$, and let $I = \{x\in B\ \mid tx\in B \}$. If $I=B$, then $t\in B$, and otherwise $I\subset\mathfrak{m}$ for some maximal ideal $\mathfrak{m}$, and $t\notin B_\mathfrak{m}$.

The result follows from the other parts in the same theorem, that establish a correspondence between points/regular functions of an affine variety and maximal ideals/localizations of its coordinate ring.