Torsor (algebraic geometry)

In algebraic geometry, a torsor or a principal bundle is an analogue 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. Though other notions of torsors are known in more general context (e.g. over stacks) this article will focus on torsors over schemes, the original setting where torsors have been thought for. The word torsor comes from the French torseur. They are indeed widely discussed, for instance, in Michel Demazure's and Pierre Gabriel's famous book Groupes algébriques, Tome I.^[1]

Definition

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 ${\displaystyle X}$ for the topology ${\displaystyle {\mathcal {T}}}$ (or simply a ${\displaystyle G}$-torsor when the topology is clear from the context) 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 ${\displaystyle \{U_{i}\to X\}}$ such that the

${\displaystyle U_{i}\times _{X}P}$

${\displaystyle P}$

${\displaystyle U_{i}\times G\to U_{i}}$

Notations

When ${\displaystyle {\mathcal {T}}}$ is the étale topology (resp. fpqc, etc.) instead of a torsor for the étale topology we can also say an étale-torsor (resp. fpqc-torsor etc.).

Étale, fpqc and fppf topologies

Unlike in the Zariski topology in many Grothendieck topologies a torsor can be itself a covering. This happens in some of the most common Grothendieck topologies, such as the fpqc-topology the fppf-topology but also the étale topology (and many less famous ones). So let ${\displaystyle {\mathcal {T}}}$ be any of those topologies (étale, fpqc, fppf). Let ${\displaystyle X}$ be a scheme and ${\displaystyle G}$ a group scheme over ${\displaystyle X}$. Then ${\displaystyle P\to X}$ is a ${\displaystyle G}$-torsor if and only if ${\displaystyle P\times _{X}P}$ over ${\displaystyle P}$ is isomorphic to the trivial torsor ${\displaystyle P\times G}$ over ${\displaystyle P}$. In this case we often say that a torsor trivializes itself (as it becomes a trivial torsor when pulled back over itself).

Correspondence vector bundles-${\displaystyle {GL}_{n}}$-torsors

Over a given scheme ${\displaystyle X}$ there is a bijection, between

${\displaystyle X}$

${\displaystyle {GL}_{n}}$

${\displaystyle n=rk(V)\in \mathbb {N} }$

${\displaystyle V}$

${\displaystyle V}$

${\displaystyle Isom(V,{\mathcal {O}}_{X}^{\oplus n})}$

${\displaystyle Isom({\mathcal {O}}_{X}^{\oplus n},{\mathcal {O}}_{X}^{\oplus n})}$

${\displaystyle Isom({\mathcal {O}}_{X}^{\oplus n},{\mathcal {O}}_{X}^{\oplus n})\simeq GL_{n,X}}$

Trivial torsors and sections

A ${\displaystyle G}$-torsor ${\displaystyle f:P\to X}$ is isomorphic to a trivial torsor if and only if ${\displaystyle P(X)=\operatorname {Mor} (X,P)}$ is nonempty, i.e. the morphism ${\displaystyle f}$ admits at least a section ${\displaystyle s:X\to P}$. Indeed, if there exists a section ${\displaystyle s:X\to P}$, then ${\displaystyle X\times G\to P,(x,g)\mapsto s(x)g}$ is an isomorphism. On the other hand if ${\displaystyle f:P\to X}$ is isomorphic to a trivial ${\displaystyle G}$-torsor, then ${\displaystyle P\simeq X\times G}$; the identity lement ${\displaystyle 1_{G}\in G}$ gives the required section ${\displaystyle s=id_{X}\times 1_{G}}$.

Examples and basic properties

• 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.) By abuse of notation we have still denoted by ${\displaystyle \operatorname {Gal} (L/K)}$ the finite constant group scheme over ${\displaystyle K}$ associated to the abstract group ${\displaystyle \operatorname {Gal} (L/K)}$. This fact is a basis for
  integral extension
  for a generalization.
• If ${\displaystyle X}$ is an abelian variety over a field ${\displaystyle k}$ then the multiplication by ${\displaystyle n\in \mathbb {N} }$, ${\displaystyle n_{X}:X\to X}$ is a torsor for the fpqc-topology under the action of the finite ${\displaystyle k}$-group scheme ${\displaystyle ker(n_{X})}$. That happens for instance when ${\displaystyle X}$ is an elliptic curve.

Torsors and cohomology

Let ${\displaystyle P}$ be a ${\displaystyle G}$-torsor for the étale topology and let ${\displaystyle \{U_{i}\to X\}}$ be a covering trivializing ${\displaystyle P}$, as in the definition. 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 ${\displaystyle G}$-torsor over ${\displaystyle X}$, unique up to a unique isomorphism.

The universal torsor of a scheme X and the fundamental group scheme

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]

The contracted product

The contracted product is an operation allowing to build a new torsor from a given one, inflating or deflating its structure with some particular procedure also known as push forward. Though the construction can be presented in a wider generality we are only presenting here the following, easier and very common situation: we are given a right ${\displaystyle G}$-torsor ${\displaystyle f:P\to X}$ and a group scheme morphism ${\displaystyle u:G\to M}$. Then ${\displaystyle G}$ acts to the left on ${\displaystyle M}$ via left multiplication: ${\displaystyle g\star m:=u(g)m}$. We say that two elements ${\displaystyle (p,m)\in P\times M}$ and ${\displaystyle (p',m')\in P\times M}$ are equivalent if there exists ${\displaystyle g\in G}$ such that ${\displaystyle (pg^{-1},g\star m)=(p',m')}$. The space of orbits ${\displaystyle P\times ^{G}M:={\frac {P\times M}{G}}}$ is called the contracted product of ${\displaystyle P}$ through ${\displaystyle u:G\to M}$. Elements are denoted as ${\displaystyle p\wedge m}$. The contracted product is a scheme and has a structure of a right ${\displaystyle M}$-torsor when provided with the action ${\displaystyle (p\wedge m)*m':=p\wedge (mm')}$. Of course all the operations have to be intended functorially and not set theoretically. The name contracted product comes from the French produit contracté and in algebraic geometry it is preferred to its topological equivalent push forward.

Morphisms of torsors and reduction of structure group scheme

Let ${\displaystyle f:Y\to X}$ and ${\displaystyle h:T\to X}$ be respectively a (right) ${\displaystyle G}$-torsor and a (right) ${\displaystyle H}$-torsor in some Grothendieck topology ${\displaystyle {\mathcal {T}}}$ where ${\displaystyle G}$ and ${\displaystyle H}$ are ${\displaystyle X}$-group schemes. A morphism (of torsors) from ${\displaystyle Y}$ to ${\displaystyle T}$ is a pair of morphisms ${\displaystyle (a,b)}$ where ${\displaystyle a:Y\to T}$ is a ${\displaystyle X}$-morphism and ${\displaystyle b:G\to H}$ is group-scheme morphism such that ${\displaystyle \sigma _{H}\circ (a\times b)=a\circ \sigma _{G}}$ where ${\displaystyle \sigma _{G}}$ and ${\displaystyle \sigma _{H}}$ are respectively the action of ${\displaystyle G}$ on ${\displaystyle Y}$ and of ${\displaystyle H}$ on ${\displaystyle T}$.

In this way ${\displaystyle T}$ can be proved to be isomorphic to the contracted product ${\displaystyle Y\times ^{G}H}$. If the morpshims ${\displaystyle b:G\to H}$ is a closed immersion then ${\displaystyle Y}$ is said to be a sub-torsor of ${\displaystyle T}$. We can also say, inheriting the language from topology, that ${\displaystyle T}$ admits a

${\displaystyle H}$

${\displaystyle G}$

Structure reduction theorem

An important result by Vladimir Drinfeld and Carlos Simpson goes as follows: let ${\displaystyle X}$ be a smooth projective curve over an algebraically closed field ${\displaystyle k}$, ${\displaystyle G}$ a semisimple, split and simply connected algebraic group (then a group scheme) and ${\displaystyle P}$ a ${\displaystyle G}$-torsor on ${\displaystyle X_{R}=X\times _{\operatorname {Spec} k}\operatorname {Spec} R}$, ${\displaystyle R}$ being a finitely generated ${\displaystyle k}$-algebra. Then there is an étale morphism ${\displaystyle R\to R'}$ such that ${\displaystyle P\times _{X_{R}}X_{R'}}$ admits a reduction of structure group scheme to a Borel subgroup-scheme of ${\displaystyle G}$.^[9][10]

Further remarks

• 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).
• If ${\displaystyle G}$ is a connected algebraic group over a finite field ${\displaystyle \mathbf {F} _{q}}$, then any ${\displaystyle G}$-torsor over ${\displaystyle \operatorname {Spec} \mathbf {F} _{q}}$ is trivial. (Lang's theorem.)

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

• Beauville–Laszlo theorem
• Moduli stack of principal bundles

Notes