Torsor (algebraic geometry)

In algebraic geometry, a torsor or a principal bundle is an analog of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in étale topology or some other flat topologies. The notion also generalizes a Galois extension in abstract algebra.

The category of torsors over a fixed base forms a stack. Conversely, a prestack can be stackified by taking the category of torsors (over the prestack).

Definition

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 isomorphic to 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 ${\displaystyle \deg _{i}(P)}$ , 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 ${\displaystyle \deg _{i}(G)=\max\{\deg _{i}(P)\mid P\subset G{\text{ parabolic subgroups}}\}}$ . 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., ${\displaystyle \deg _{i}(E)=\deg _{i}({}^{E}G)}$ . E is said to be semi-stable if ${\displaystyle \deg _{i}(E)\leq 0}$  and is stable if ${\displaystyle \deg _{i}(E)<0}$ .

Examples of torsors in applied mathematics

According to John Baez, energy, voltage, position and the phase of a quantum-mechanical wavefunction are all examples of torsors in everyday physics; in each case, only relative comparisons can be measured, but a reference point must be chosen arbitrarily to make absolute values meaningful. However, the comparative values of relative energy, voltage difference, displacements and phase differences are not torsors, but can be represented by simpler structures such as real numbers, vectors or angles.^[6]

In basic calculus, he cites indefinite integrals as being examples of torsors.^[6]

See also

• Beauville–Laszlo theorem
• Moduli stack of principal bundles
• Fundamental group scheme

Notes

1. Algebraic stacks, Example 2.3. harvnb error: no target: CITEREFAlgebraic_stacks (help)
2. Behrend 1993, Lemma 4.3.1 harvnb error: no target: CITEREFBehrend1993 (help)
3. Milne 1980, The discussion preceding Proposition 4.6.
4. http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Oct27(Higgs).pdf^{[bare URL PDF]}
5. http://www.math.harvard.edu/~lurie/282ynotes/LectureXIV-Borel.pdf^{[bare URL PDF]}
6. Baez, John (December 27, 2009). "Torsors Made Easy". math.ucr.edu. Retrieved 2022-11-22.

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), , archived from the original on 2008-05-05
• Milne, James S. (1980), Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, ISBN 978-0-691-08238-7, MR 0559531

Further reading

• Brian Conrad, [http://math.stanford.edu/~conrad/papers/cosetfinite.pdf Finiteness theorems for algebraic groups over function �fields]