In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology.
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.
-torsor when the topology is clear from the context) is the data of a scheme
instead of a torsor for the étale topology we can also say an étale-torsor (resp.
be any of those topologies (étale, fpqc, fppf).
one can take the (representable) sheaf of local isomorphisms
A trivial torsor admits a section: thus, there are elements
[3] A trivial torsor corresponds to the identity element.
In this context torsors have to be taken in the fpqc topology.
be a Dedekind scheme (e.g. the spectrum of a field) and
a faithfully flat morphism, locally of finite type.
Its existence, conjectured by Alexander Grothendieck, has been proved by Madhav V. Nori[4][5][6] for
the spectrum of a field and by Marco Antei, Michel Emsalem and Carlo Gasbarri when
[7][8] 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
The contracted product is a scheme and has a structure of a right
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.
can be proved to be isomorphic to the contracted product
admits a reduction of structure group scheme from
An important result by Vladimir Drinfeld and Carlos Simpson goes as follows: let
be a smooth projective curve over an algebraically closed field
admits a reduction of structure group scheme to a Borel subgroup-scheme of
[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
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.