Recently I saw that an open subset of an affine scheme need not be affine. (See here for details). This led me to the following train of thought:
“Take an open subset U of a scheme X. Every point x of U has an affine neighborhood, taking the intersection of U with these neighborhoods shows that this is a scheme.”
But this is not true because the intersection of U and an affine neighborhood need not be affine, as the above link implies. Throughout the following, the main goal is to reassure myself that an open subset of a scheme is a scheme.
So why is an open subset of a scheme a scheme? First off, we have that if is an open subset, then is a locally ringed topological space. This is just because the restriction sheaf has identical stalks to those of its image in .
Now we will follow the above train of thought. Every point has an affine neighborhood . Taking the intersection gives an open subset of . Since this is an open subset of an affine scheme, we may regard it as the union of principal open subsets where the index is essentially arbitrary, but each , so maybe it would be nicer to use the index .
Anyways, we do have that the set is affine. (In fact, if is the ring associated to then the affine structure is given by the localization ; alas our indices are disadvantageous now. Depending on where you learn this from, this could be from definition as in Liu’s book or a theorem as in Hartshorne).
Finally, we have and as these are just restrictions of the structure sheaf of , we have that can be covered by affine neighborhoods which proves an open subset of a scheme is a scheme.