TITLE: Why is the coordinate ring of a projective variety not determined by the isomorphism class of the variety?
QUESTION [25 upvotes]: I know that there are isomorphic projective varieties which have nonisomorphic coordinate rings, but I'm a little mystified as to "why" this is the case. Why doesn't a usual functoriality proof go through to prove this, and is there any insight into this other than that the definitions just work out this way?

REPLY [5 votes]: Here's a slightly tangential elaboration on the above two answers.
Let $X = \mathrm{Proj} A$ for $A$ a noetherian ring. So here $A$ should fade into the background, but $X$ is the important thing; it turns out that the $\mathrm{Proj}$ depends only on the asymptotic part of $A$ in high dimensions. Then we can consider the category of coherent sheaves on $\mathrm{Proj} A$. The sheaf of regular functions is the canonical example.
There is a functor from finitely graded $A$-modules to coherent sheaves on $\mathrm{Proj} A$, denoted by a tilde. (This is as in Hartshorne II.5.) Unlike the case for an affine scheme, however, it is not an equivalence of categories. It is not too hard to show that the functor is essentially surjective, since given a coherent sheaf $\mathcal{F}$ on $\mathrm{Proj} A$ you can construct the graded $A$-module $\bigoplus \Gamma(X, \mathcal{F}(n))$ of the global sections of all the twists. Then one can show using elementary methods that this recovers $\mathcal{F}$. (In fact, this is true even if $A$ is not noetherian and $\mathcal{F}$ is simply assumed quasi-coherent.)
However, if we have a graded $A$-module $M$, we cannot recover $M$ from the associated sheaf $\tilde{M}$. For instance, if $M$ has finitely many nonzero components, then the homogeneous localizations $M_{(f)}$ vanish (for any homogeneous element $f$), so the associated sheaf is zero. But $M$ may not be zero.
This suggests that if we consider the abelian category of finitely generated graded $A$-modules, and quotient out by the Serre class of modules that are "asymptotically zero," then we will get the category of coherent sheaves on $\mathrm{Proj} A$. This is in fact true, though the proof uses a bit of cohomological machinery (see EGA III.2).
So the moral of the story is that the $\mathrm{Proj}$ sees only asymptotic data. If we have  a morphism $A \to A'$ that induces an isomorphism in large enough degrees, then $\mathrm{Proj} A, \mathrm{Proj} A'$ will be isomorphic (as one example).