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.
You’ve shown two constructions, but in what sense is it a correspondence? If you mean it’s an equivalence — that the two constructions invert each other (up to isomorphism) — then you need to show that. If you mean it’s an adjunction you need to show that too.
It is a correspondence up to isomorphism, yeah. A fairly quick way to see it is that you can take a locally free sheaf and then construct a vector bundle out of it, but then the sheaf of local sections over each trivializing set are determined, and they agree on overlaps. A standard exercise is that given an open cover and sheaves on each element which agree on overlaps, there is a unique sheaf which restricts to them. For the other direction, use the trivializing open cover for the sheaf of sections, and it glues together using the transition maps for the vector bundle.
It does require a bit of checking, but it’s not too difficult to do it.
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In the penultimate paragraph, what do you mean by g_ij(v)? g_ij both an automorphism of the sections of F on the intersection of Ui and Uj and correspondingly an nxn-matrix of regular function on the intersection. how does this act on v– a point in A^n to give another point in A^n?
Thanks for these blogs. I’ve struggled to find an explicit comparison of sheaves and bundles like this.
Have look at Hartshorne p.128 ex. 5.18.
It acts as the
matrix, think of
as vectors. So it will, for instance, identify the origins of both copies of
.
In property (2) of a vector bundle, wouldn’t you need that the isomorphism commutes with the projection?
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