Last time, we talked about sheaves of modules, and focused on the correspondence between sheaves of ideals and subvarieties. We were talking about the internal geometry of the variety. Today, we’ll talk a bit about more external geometry. Specifically, we’ll talk about how sheaves give us new varieties with maps to whose fibers are all vector spaces. In fact, they’ll look locally like open sets of times a vector space. Such objects are called vector bundles, and are rather closely tied to the theory of sheaves of modules on .
We call a sheaf of modules free if it is isomorphic to a direct sum of copies of . We call the number of copies of the rank of the free sheaf. Now, we will call a sheaf locally free if there exists a cover of by open sets (we don’t require them to be affine) such that is a free -module. Note that a locally free module is automatically quasi-coherent, and is coherent if it has finite rank. If is connected, then the rank is the same everywhere.
A quick algebraic comment is in order: the coherent sheaf associated to a finitely generated module is locally free if and only if it is a projective module, that is, there exists such that is free. Though beware, if we were working with schemes instead of varieties, we need to be more careful about this. But we’re focusing on varieties right now, so everything’s ok.
Now let’s look at things from the other direction. As mentioned earlier, if is a variety, a vector bundle over is a variety with a map such that the following conditions hold:
- For each , we have is isomorphic to for some
- There exists a cover of such that is isomorphic to .
We note quickly that if is connected, then we can use the same at each point, and say that the vector bundle has rank .
So now lets do some examples of each. For any variety , there is a vector bundle for each positive . This is called the trivial vector bundle. Other vector bundles are easiest to describe by using locally free sheaves, once we’ve described the correspondence. But first, some examples of locally free sheaves. Fix an integer . Then we define the locally free sheaf of rank one on called to be the one taking each open set to the set of ratios of homogeneous functions such that is nonzero on and . This is locally free, because if we restrict to a standard copy of affine space, we are then looking at is assigned rational functions defined everywhere on , that is, we get precisely .
So on to the correspondence. If we take a vector bundle, there is a relatively simple way to construct a sheaf. Assign to the collection of morphisms such that is the identity on . This sheaf will be locally trivial, and the open sets we use can be the ones for which . This is because on these sets, we’re looking at functions which compose with the projection to give the identity. This is the same as looking at morphisms , which is just a list of regular functions on . So a vector bundle defines a locally free sheaf.
So now we start with a locally free sheaf of rank . Pick an open cover of such that is free for each in it. We can choose the open cover to be finite, because varieties are quasi-compact. So now we take the disjoint union of for all . So now we have one isomorphism and another . Restricting each of these to (we will use this convention from here on out) we get two different isomorphisms , and we will denote them by . We then get , an automorphism of . Now, by the isomorphism with , we can identify this with an matrix of regular functions on .
So now we glue. We take and and identify them along by the map which takes to . So now we perform this for all , and call this object , and it comes with a map by forgetting the vector coordinate on any point. So the fibers are now copies of and by construction, around each point there’s a neighborhood on which the space is . So all we need to do in order to check that this is a vector bundle is to check that it is a variety. It certainly has an open cover by affine varieties, again by construction, and in fact this cover is finite. The rest follows from the fact that is the identity map. So we’ve now established a correspondence between locally free sheaves and vector bundles.
Next time, we’ll further investigate a special class of these, the line bundles. That is, vector bundles of rank one. The associated locally free sheaves are called invertible sheaves, because we’re going to make a group out of them, which means that each will have an inverse.