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 $Y\times _{X}P$ along "some" covering map $Y\to X$ is the trivial torsor $Y\times G\to Y$ (G acts only on the second factor).[1] Equivalently, a G-torsor P on X is a principal homogeneous space for the group scheme $G_{X}=X\times G$ (i.e., $G_{X}$ acts simply transitively on $P$ .)

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 $\{U_{i}\to X\}$ in the topology, called the local trivialization, such that the restriction of P to each $U_{i}$ is a trivial $G_{U_{i}}$ -torsor.

A line bundle is nothing but a $\mathbb {G} _{m}$ -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 $\operatorname {GL} _{n}$ -torsor on X is a principal $\operatorname {GL} _{n}$ -bundle on X.
If $L/K$ is a finite Galois extension, then $\operatorname {Spec} L\to \operatorname {Spec} K$ is a $\operatorname {Gal} (L/K)$ -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 $P(X)=\operatorname {Mor} (X,P)$ is nonempty. (Proof: if there is an $s:X\to P$ , then $X\times G\to P,(x,g)\mapsto s(x)g$ is an isomorphism.)

Let P be a G-torsor with a local trivialization $\{U_{i}\to X\}$ in étale topology. A trivial torsor admits a section: thus, there are elements $s_{i}\in P(U_{i})$ . Fixing such sections $s_{i}$ , we can write uniquely $s_{i}g_{ij}=s_{j}$ on $U_{ij}$ with $g_{ij}\in G(U_{ij})$ . Different choices of $s_{i}$ amount to 1-coboundaries in cohomology; that is, the $g_{ij}$ define a cohomology class in the sheaf cohomology (more precisely Čech cohomology with sheaf coefficient) group $H^{1}(X,G)$ .[3] A trivial torsor corresponds to the identity element. Conversely, it is easy to see any class in $H^{1}(X,G)$ defines a G-torsor on X, unique up to an isomorphism.

If G is a connected algebraic group over a finite field $\mathbf {F} _{q}$ , then any G-bundle over $\operatorname {Spec} \mathbf {F} _{q}$ 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 $P\to X$ is a G-bundle and G acts from the left on a scheme F, then one can form the associated bundle $P\times ^{G}F\to X$ with fiber F. In particular, if H is a closed subgroup of G, then for any H-bundle P, $P\times ^{H}G$ is a G-bundle called the induced bundle.

If P is a G-bundle that is isomorphic to the induced bundle $P'\times ^{H}G$ 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 $X_{R}=X\times _{\operatorname {Spec} k}\operatorname {Spec} R$ , 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 $R\to R'$ such that $P\times _{X_{R}}X_{R'}$ 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 $\operatorname {deg} _{i}(P)$ , is the degree of its Lie algebra $\operatorname {Lie} (P)$ as a vector bundle on X. The degree of instability of G is then $\operatorname {deg} _{i}(G)=\max\{\operatorname {deg} _{i}(P)|P\subset G{\text{ parabolic subgroups}}\}$ . 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 ${}^{E}G=\operatorname {Aut} _{G}(E)$ of G induced by E (which is a group scheme over X); i.e., $\operatorname {deg} _{i}(E)=\operatorname {deg} _{i}({}^{E}G)$ . E is said to be semi-stable if $\operatorname {deg} _{i}(E)\leq 0$ and is stable if $\operatorname {deg} _{i}(E)<0$ .

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