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 .

定义 A (principle) -bundle is a scheme over , where is fppf, with an -action , such that the map given by , is isomorphism, where .

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).

定义 A map of -bundles is a map of -schemes such that

定义 Let be a site and a sheaf of groups. A -torsor is a sheaf with such that:


for all , there exists such that for all .


we have where the map is given by .

定义 A -torsor is trivial if as -torsor.

命题 If is representable on by a group scheme , thenis fully faithful. And if is affine then is equivalence.

证明. Yoneda says is fully faithful. Why is -torsor? In other word, in def of torsor, holds by definition, but why we have ?

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 .

注 If is smooth, then is also smooth. This is by fppf descent, i.e. we getand hence the arrow is smooth as well. Note here smooth does nothing and it can be changed to basically any property.

命题 Let be a scheme, be a sheaf on , and . Then there is an equivalence of categoriesgiven by mapwhere means -equivariant maps.

证明. We first show is -torsor. We deinfe a map by by , i.e. we get a diagram

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.

注 If is an -module, and , then -torsor are equivalent to -modules which are locally isomorphic to as -modules.

例 Take as -modules. Then we are shown that line bundles are isomorphic to -torsors, where , where is representable by . Hence, we see the line bundles are the same as -bundles.

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.

例 We can also talk about Brauer-Sever varieties, which are locally isomorphic to . Those are -torsors, and it is related to Azumaya algebras and the Brauer groups, where the Brauer groups are deeply related to class field theory in number theory.