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

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

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

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

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

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