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 the 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]).

Examples and basic properties

Examples

A $\operatorname{GL}_n$ -torsor on X is a vector bundle of rank n (i.e., a locally free sheaf) 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 canonical. (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 $\operatorname{deg}_i (E) \le 0$ 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
Milne, James S. (1980), Étale cohomology, Princeton Mathematical Series 33, Princeton University Press, ISBN 978-0-691-08238-7, MR 559531