So now that we have abstract varieties on hand, we’re going to do a bit more with sheaves, leading to some of the intimate connections between sheaf theory and geometry. Sadly, this often gives students a lot of trouble (I know I had a bit of trouble with it at first) because things are presented very algebraically and the geometry gets lost. So we’ll be making a point of the connections between the geometry and the algebra.

As usual, we start out with a definition. We take $X$ to be a ringed space (say, a variety) with structure sheaf $\mathscr{O}_X$. Then we say that a sheaf $\mathscr{F}$ is an $\mathscr{O}_X$-module (or sometimes a sheaf of $\mathscr{O}_X$-modules) if for each $U$, we have $\mathscr{F}(U)$ is an $\mathscr{O}_X(U)$-module, and the restriction maps on $\mathscr{F}$ are compatible with the module structures induces by the restriction maps in $\mathscr{O}_X$. A morphism $\mathscr{F}\to\mathscr{G}$ is a homomorphism of $\mathscr{O}_X$-modules if over each open set it is a module homomorphism.

Now, pretty much anything we can do with $R$-modules we can do with $\mathscr{O}_X$-modules. We can take kernels, cokernels, images, quotients and the like and all we get are more $\mathscr{O}_X$-moduels. We can talk about exact sequences of $\mathscr{O}_X$-modules by just looking to see if they are exact as sequences of sheaves. In fact, we can even make a sheaf out of morphisms of $\mathscr{O}_X$-modules so that this sheaf is one too, by defining it as $U\mapsto Hom_{\mathscr{O}_X|_U}(\mathscr{F}|_U,\mathscr{G}|_U)$, where restriction to $U$ just means that we consider it as a sheaf on $U$ and so only care about open sets contained in $U$.

We can even define the tensor product of two sheaves, though we do have to perform a sheafification, because it won’t generally be a sheaf itself.

However, now, we’re going to focus for a moment on what these can tell use geometrically. To start with, we will look at an affine variety $V$, which has coordinate ring $k[V]$. Now, any closed subset of $V$ is defined by an ideal $I\subset k[V]$. In fact, this gives an ideal in every localization of $k[V]$, which is to say, for each open subset of $V$, we have an ideal in the coordinate ring of that subset. Now, ideals are always modules, so in fact, this collection of ideals forms a sheaf of modules!

We can, in fact, generalize this. Let $Y\subset X$ be a closed subset of a variety. Then, to each $U\subset X$ open, we assign $\mathscr{I}_Y(U)=\{f\in\mathscr{O}_X(U)|f(Y\cap U)=0\}$, that is, on each open set, we assign the regular functions that are zero on $Y\cap U$. This gives a sheaf of modules (in fact, this example is so important that we call these sheaves of ideals) and it is determined by the closed subset. So if we took a subvariety, that is, an irreducible closed subset of a variety, we’d even get a sheaf of prime ideals.

Now, we should note that the correspondence doesn’t quite go both ways. A closed subset of a variety defines a unique sheaf of ideals. However, if we are given a sheaf of ideals $\mathscr{I}$, and then we take the set of points in each $U$ where the elements of $\mathscr{I}(U)$ vanish, we might not even get a closed subset. There are sheaves of ideals which bear little to no resemblance to the example of an ideal for an affine variety, so we will fix this with a condition called quasi-coherence. We say that a sheaf of modules is quasicoherent if we can cover $X$ with open affine subsets such that when we restrict the sheaf to each of these sets we get particularly nice sheaves of modules, which we will now describe:

Given an affine variety $V$ and a module $M$ over its coordinate ring, we get a sheaf of modules called $\tilde{M}$ on $V$, by looking at the localizations of the modules. We will call a sheaf of modules quasi-coherent if it is locally isomorphic to sheaves of this form. If we can even take the modules to be finitely generated, we will call the $\mathscr{O}_X$-module coherent.

So what makes the ideals we want special? We only care about quasi-coherent sheaves of ideals, that is, sheaves of ideals that locally look like they come from the localizations of actual ideals. In fact, we can do better, because the Hilbert Basis Theorem tells us that ideals are finitely generated for polynomial rings and their quotients. Thus, the sheaves of ideals that we want are coherent.

So now given a coherent sheaf of ideals, we do in fact get a closed subset. However, many coherent sheaves of ideals give the same one, just as many ideals give the same closed subset of an affine variety. We can fix this by taking the radical ideal over each open set. This still gives us a coherent sheaf of ideals, and one which is equal to its radical will be called reduced. But now we get the same correspondence we used to have: closed subsets of $V$ are in one-to-one correspondence with reduced coherent sheaves of ideals. Subvarieties, in fact, correspond to reduced coherent sheaves of prime ideals, and any other operation on ideals can still be performed, taking all the notions from affine geometry with us into this new land of abstract algebraic geometry.