# 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 (or even algebraic space) 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 (

*G*acts only on the second factor).[1] Equivalently, a

*G*-torsor

*P*on

*X*is a principal homogeneous space for the group scheme (i.e., acts simply transitively on .)

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 (by permitting *P* to be a stack like an algebraic space if necessary[2]).

It is common to consider a torsor for not just a group scheme but more generally for a group sheaf (e.g., fppf group sheaf).

## Examples and basic properties

Examples

- A -torsor on
*X*is a principal -bundle on*X*. - If is a finite Galois extension, then is a -torsor (roughly because the Galois group acts simply transitively on the roots.) This fact is a basis for Galois descent. See integral extension for a generalization.

Remark: A *G*-torsor *P* over *X* is isomorphic to a trivial torsor if and only if is nonempty. (Proof: if there is an , then is an isomorphism.)

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 (more precisely Čech cohomology with sheaf coefficient) group .[3] 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. (Lang's theorem.)

## 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*.[4][5]

## 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* (which is a group scheme over *X*); i.e., . *E* is said to be *semi-stable* if and is *stable* if .

## Notes

- Algebraic stacks, Example 2.3.
- Behrend 1993, Lemma 4.3.1
- Milne 1980, The discussion preceding Proposition 4.6.
- http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Oct27(Higgs).pdf
- http://www.math.harvard.edu/~lurie/282ynotes/LectureXIV-Borel.pdf

## References

- Behrend, K. The Lefschetz Trace Formula for the Moduli Stack of Principal Bundles. PhD dissertation.
- Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006),
*Algebraic stacks*, archived from the original on 2008-05-05 - Milne, James S. (1980),
*Étale cohomology*, Princeton Mathematical Series,**33**, Princeton University Press, ISBN 978-0-691-08238-7, MR 0559531