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 .

See also

Notes

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

Further reading

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