Torsor (algebraic geometry)


In algebraic geometry, given a smooth algebraic group G, a G-torsor or a principal G-bundle P over a scheme X is a scheme with an action of G that is locally trivial in the given Grothendieck topology in the sense that the base change along "some" covering map is the trivial torsor . Equivalently, a G-torsor P on X is a principal homogeneous space for the group scheme
The definition may be formulated in the sheaf-theoretic language: a sheaf P on the category of X-schemes with some Grothendieck topology is a G-torsor if there is a covering in the topology, called the local trivialization, such that the restriction of P to each is a trivial -torsor.
A line bundle is nothing but a -bundle, and, like a line bundle, the two points of views of torsors, geometric and sheaf-theoretic, are used interchangeably.
It is common to consider a torsor for not just a group scheme but more generally for a group sheaf.

Examples and basic properties

Examples
Remark: A G-torsor P over X is isomorphic to a trivial torsor if and only if is nonempty.
Let P be a G-torsor with a local trivialization in étale topology. A trivial torsor admits a section: thus, there are elements. Fixing such sections, we can write uniquely on with. Different choices of amount to 1-coboundaries in cohomology; that is, the define a cohomology class in the sheaf cohomology group. A trivial torsor corresponds to the identity element. Conversely, it is easy to see any class in defines a G-torsor on X, unique up to an isomorphism.
If G is a connected algebraic group over a finite field, then any G-bundle over is trivial.

Reduction of a structure group

Most of constructions and terminology regarding principal bundles in algebraic topology carry over in verbatim to G-bundles. For example, if is a G-bundle and G acts from the left on a scheme F, then one can form the associated bundle with fiber F. In particular, if H is a closed subgroup of G, then for any H-bundle P, is a G-bundle called the induced bundle.
If P is a G-bundle that is isomorphic to the induced bundle for some H-bundle P', then P is said to admit a reduction of structure group from G to H.
Let X be a smooth projective curve over an algebraically closed field k, G a semisimple algebraic group and P a G-bundle on a relative curve, R a finitely generated k-algebra. Then a theorem of Drinfeld and Simpson states that, if G is simply connected and split, there is an étale morphism such that admits a reduction of structure group to a Borel subgroup of G.

Invariants

If P is a parabolic subgroup of a smooth affine group scheme G with connected fibers, then its degree of instability, denoted by, is the degree of its Lie algebra as a vector bundle on X. The degree of instability of G is then. If G is an algebraic group and E is a G-torsor, then the degree of instability of E is the degree of the inner form of G induced by E ; i.e.,. E is said to be semi-stable if and is stable if.