Torsor (algebraic geometry)

Let ${\displaystyle {\mathcal {T}}}$ be a Grothendieck topology and ${\displaystyle X}$ a scheme. Moreover let ${\displaystyle G}$ be a group scheme over ${\displaystyle X}$, a ${\displaystyle G}$-torsor (or principal ${\displaystyle G}$-bundle) over ${\displaystyle X}$ is the data of a scheme ${\displaystyle P}$ and a morphism ${\displaystyle f:P\to X}$ with a ${\displaystyle G}$-invariant action on ${\displaystyle P}$ that is locally trivial in ${\displaystyle {\mathcal {T}}}$ i.e. there exists a covering ${\displaystyle \{U_{i}\to X\}}$ in the sense that the base change ${\displaystyle U_{i}\times _{X}P}$ is isomorphic to the trivial torsor ${\displaystyle U_{i}\times G\to U_{i}}$ [2]

A line bundle can be seen as a ${\displaystyle \mathbb {G} _{m}}$-torsor, and, more in general, a vector bundle can be seen as a ${\displaystyle {GL}_{n}}$-torsor, for some ${\displaystyle n\in \mathbb {N} }$.

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).

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).

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.)

In this context torsors have to be taken in the fpqc topology. Let ${\displaystyle S}$ be a Dedekind scheme (e.g. the spectrum of a field) and ${\displaystyle f:X\to S}$ a faithfully flat morphism, locally of finite type. Assume ${\displaystyle f}$ has a section ${\displaystyle x\in X(S)}$. We say that ${\displaystyle X}$ has a fundamental group scheme ${\displaystyle \pi _{1}(X,x)}$ if there exist a pro-finite and flat ${\displaystyle \pi _{1}(X,x)}$-torsor ${\displaystyle {\hat {X}}\to X}$, called the universal torsor of ${\displaystyle X}$, with a section ${\displaystyle {\hat {x}}\in {\hat {X}}_{x}(S)}$ such that for any finite ${\displaystyle G}$-torsor ${\displaystyle Y\to X}$ with a section ${\displaystyle y\in Y_{x}(S)}$ there is a unique morphism of torsors ${\displaystyle {\hat {X}}\to Y}$ sending ${\displaystyle {\hat {x}}}$ to ${\displaystyle y}$. Its existence has been proved by Madhav V. Nori[4][5][6] for ${\displaystyle S}$ the spectrum of a field and by Marco Antei, Michel Emsalem and Carlo Gasbarri when ${\displaystyle S}$ is a Dedekind scheme of dimension 1.[7][8]

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 or contracted product (from the French produit contracté).

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.[9][10]

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 ${\displaystyle \deg _{i}(E)<0}$.

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.[11]

In basic calculus, he cites indefinite integrals as being examples of torsors.[11]

• Beauville–Laszlo theorem
• Moduli stack of principal bundles

• 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, vol. 33, Princeton University Press, ISBN 978-0-691-08238-7, MR 0559531

• Brian Conrad, Finiteness theorems for algebraic groups over function �fields