# Vektorbündel/Torsor/Kohomologische Klassifikation/Einführung/Textabschnitt

We have seen that
    $V=\operatorname {Spec} {\left(R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}\right)}\right)}\,$
acts on the spectrum of a forcing algebra $T=\operatorname {Spec} {\left(R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}\right)}$ by addition. The restriction of $V$ to $U=D({\left(f_{1},\ldots ,f_{n}\right)}$ is a vector bundle, and $T$ restricted to $U$ becomes a $V$-torsor.

## Definition

Let $V$ denote a geometric vector bundle over a scheme $X$. A scheme $T\rightarrow X$ together with an action
    $\beta \colon V\times _{X}T\longrightarrow T$
is called a geometric (Zariski)-torsor for $V$
(or a $V$-principal fiber bundle or a principal homogeneous space) if there exists an open covering $X=\bigcup _{i\in I}U_{i}$ and isomorphisms
    $\varphi _{i}\colon T{|}_{U_{i}}\longrightarrow V{|}_{U_{i}}$
such that the diagrams (we set $U=U_{i}$ and $\varphi =\varphi _{i}$)
    ${\begin{matrix}V{|}_{U}\times _{U}T{|}_{U}&{\stackrel {\beta }{\longrightarrow }}&T{|}_{U}&\\\!\!\!\!\!\operatorname {Id} \times \varphi \downarrow &&\downarrow \varphi \!\!\!\!\!&\\V{|}_{U}\times _{U}V{|}_{U}&{\stackrel {\alpha }{\longrightarrow }}&V{|}_{U}&\!\!\!\!\!\\\end{matrix}}$
commute, where $\alpha$ is the addition on the vector bundle.
The torsors of vector bundles can be classified in the following way.

## Proposition

Let $X$ denote a noetherian separated scheme and let
    $p\colon V\longrightarrow X$
denote a geometric vector bundle on $X$ with sheaf of sections ${\mathcal {S}}$.
Then there exists a correspondence between first cohomology classes $c\in H^{1}(X,{\mathcal {S}})$ and geometric $V$-torsors.

### Proof

We describe only the correspondence. Let $T$ denote a $V$-torsor. Then there exists by definition an open covering $X=\bigcup _{i\in I}U_{i}$ such that there exist isomorphisms
    $\varphi _{i}\colon T{|}_{U_{i}}\longrightarrow V{|}_{U_{i}}$
which are compatible with the action of $V{|}_{U_{i}}$ on itself. The isomorphisms $\varphi _{i}$ induce automorphisms
    $\psi _{ij}=\varphi _{j}\circ \varphi _{i}^{-1}\colon V{|}_{U_{i}\cap U_{j}}\longrightarrow V{|}_{U_{i}\cap U_{j}}.$
These automorphisms are compatible with the action of $V$ on itself, and this means that they are of the form
    $\psi _{ij}=\operatorname {Id} _{V}{|}_{U_{i}\cap U_{j}}+s_{ij}\,$
with suitable sections $s_{ij}\in \Gamma (U_{i}\cap U_{j},{\mathcal {S}})$. This family defines a Čech cocycle for the covering and gives therefore a cohomology class in $H^{1}(X,{\mathcal {S}})$.
For the reverse direction, suppose that the cohomology class $c\in H^{1}(X,{\mathcal {S}})$ is represented by a Čech cocycle
$s_{ij}\in \Gamma (U_{i}\cap U_{j},{\mathcal {S}})$ for an open covering $X=\bigcup _{i\in I}U_{i}$. Set $T_{i}:=V{|}_{U_{i}}$. We take the morphisms
    $\psi _{ij}\colon T_{i}{|}_{U_{i}\cap U_{j}}=V{|}_{U_{i}\cap U_{j}}\longrightarrow V{|}_{U_{i}\cap U_{j}}=T_{j}{|}_{U_{i}\cap U_{j}}$
given by $\psi _{ij}:=\operatorname {Id} _{V}{|}_{U_{i}\cap U_{j}}+s_{ij}$ to glue the $T_{i}$ together to a scheme $T$ over $X$. This is possible since the cocycle condition guarantees the glueing condition for schemes.
The action of $T_{i}=V{|}_{U_{i}}$ on itself glues also together to give an action on $T$.
$\Box$
It follows immediately that for an affine scheme (i.e. a scheme of type $\operatorname {Spec} {\left(R\right)}$) there is no non-trivial torsor for any vector bundle. There will however be in general many non-trivial torsors on the punctured spectrum (and on a projective variety).