Q:

Projective, Quasi-Projective

X is a projective variety, W is a quasi-projective variety over the algebraically closed ﬁeld k.
I would like to construct a k-algebra isomorphism between O(XxW) and O(W) (the rings of regular functions). 
On the level of varieties I know that the projection XxW->W maps closed sets to closed sets. I feel like this should lead me to the algebra, but I have no idea how I would define the map going the other way.

A:

Any regular function on a projective variety (assumed irreducible) is constant. Recall that this is  a consequence of the valuative criterion for properness (among other things; alternatively, the image would be proper, and no non-finite subset of the affine line is proper). 
So given any regular function $f$ on $X \times W$, it is constant on every fiber over $W$. Thus if $W \to X \times W$ is any section of the projection, the pull-back of $f$ to $W$ pulls back to $f$ via the projection $X \times W \to W$. Thus the map $\mathcal{O}(W) \to \mathcal{O}(X \times W)$ is surjective. Injectivity is clear because $X \times W \to W$ is surjective.

