# 4.2. G-丛

In this section, we are going to start with -bundles/torsors.

Thus, let be a scheme and . We are going to start with a group scheme over , say , which is flat and locally of finite presentation over .

This condition is equivalent to saying if and , then the group action of acts on is simple and transitive, i.e. acts on has no stabilizers and for all there exists so , i.e. for all there exists unique so .

So, before we give examples, we talk about the idea of -bundle. In particular, we can think of as a group without a choice of identity. Here, if we give two elements, we only care about the difference, not which particular we are working with(so it is similar to the idea of potential functions in physics, i.e. we only care about diff of potentials, not the initial value).

 1 for all , there exists such that for all . 2 we have where the map is given by .

ConsiderWe want fppf cover such that . However, note we haveHence we are done.

Now assume is affine and is -torsor, we want where is a -bundle.

Note by assumption, we can find fppf cover such that . But then since acts on simple and transitive where , i.e. is locally representable.

Since is a sheaf over , have canonical descent data. On the other hand, is scheme affine over . So, this yields descent data for the . But last result we showed last class is descent data is effective for affine morphisms. Therefore, we get a scheme that also defines a sheaf which must agree with because they are sheaves with same descent data.

Last thing we need to check is that why is a -bundle.

We need to check there exists action such that the graph of the action is isomorphism.

Why exists? Well, we have by Yoneda because we have action such that so by Yoneda we get with .

Why is this simple and transitive? If , then for we geti.e. for a unique .

Why is a sheaf? GivenThen for with -equivariant map such that , since sheaves satisfy descent so we get a unique such that .

Why is -equivariant? Viz, we want commutative diagramHowever, diagram commutes fppf locally over , hence it commutes over .

Why locally?

Locally we get -equivariant isomorphism . So we claim is locally isomorphic to . Well, suppose we have . Then we see and hence given by . This concludes the proof.

Clearly we don’t have to restrict to . In other word, we can take , then , which is representable by scheme . We can also define this as and then is a polynomial and we look at , which is affine. In this case, we just get rank vector bundles are equivalent to -bundles.