# Vector bundle/Torsor/Cohomological classification/Introduction/Section

We have seen that

${\displaystyle {}V=\operatorname {Spec} {\left(R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}\right)}\right)}\,}$

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

## Definition

Let ${\displaystyle {}V}$ denote a geometric vector bundle over a scheme ${\displaystyle {}X}$. A scheme ${\displaystyle {}T\rightarrow X}$ together with an action

${\displaystyle \beta \colon V\times _{X}T\longrightarrow T}$

is called a geometric (Zariski)-torsor for ${\displaystyle {}V}$ (or a principal fiber bundle or a principal homogeneous space) if there exists an open covering ${\displaystyle {}X=\bigcup _{i\in I}U_{i}}$ and isomorphisms

${\displaystyle \varphi _{i}\colon T{|}_{U_{i}}\longrightarrow V{|}_{U_{i}}}$

such that the diagrams (we set ${\displaystyle {}U=U_{i}}$ and ${\displaystyle {}\varphi =\varphi _{i}}$)

${\displaystyle {\begin{matrix}V{|}_{U}\times _{U}T{|}_{U}&{\stackrel {\beta }{\longrightarrow }}&T{|}_{U}&\\\!\!\!\!\!\operatorname {Id} \times \varphi \downarrow &&\downarrow \varphi \!\!\!\!\!&\\V{|}_{U}\times _{U}V{|}_{U}&{\stackrel {\alpha }{\longrightarrow }}&V{|}_{U}&\!\!\!\!\!\\\end{matrix}}}$
commute, where ${\displaystyle {}\alpha }$ is the addition on the vector bundle.

The torsors of vector bundles can be classified in the following way.

## Proposition

Let ${\displaystyle {}X}$ denote a noetherian separated scheme and let

${\displaystyle p\colon V\longrightarrow X}$

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

### Proof

We describe only the correspondence. Let ${\displaystyle {}T}$ denote a ${\displaystyle {}V}$-torsor. Then there exists by definition an open covering ${\displaystyle {}X=\bigcup _{i\in I}U_{i}}$ such that there exist isomorphisms

${\displaystyle \varphi _{i}\colon T{|}_{U_{i}}\longrightarrow V{|}_{U_{i}}}$

which are compatible with the action of ${\displaystyle {}V{|}_{U_{i}}}$ on itself. The isomorphisms ${\displaystyle {}\varphi _{i}}$ induce automorphisms

${\displaystyle \psi _{ij}=\varphi _{j}\circ \varphi _{i}^{-1}\colon V{|}_{U_{i}\cap U_{j}}\longrightarrow V{|}_{U_{i}\cap U_{j}}.}$

These automorphisms are compatible with the action of ${\displaystyle {}V}$ on itself, and this means that they are of the form

${\displaystyle {}\psi _{ij}=\operatorname {Id} _{V}{|}_{U_{i}\cap U_{j}}+s_{ij}\,}$

with suitable sections ${\displaystyle {}s_{ij}\in \Gamma (U_{i}\cap U_{j},{\mathcal {S}})}$. This family defines a Čech cocycle for the covering and gives therefore a cohomology class in ${\displaystyle {}H^{1}(X,{\mathcal {S}})}$.
For the reverse direction, suppose that the cohomology class ${\displaystyle {}c\in H^{1}(X,{\mathcal {S}})}$ is represented by a Čech cocycle ${\displaystyle {}s_{ij}\in \Gamma (U_{i}\cap U_{j},{\mathcal {S}})}$ for an open covering ${\displaystyle {}X=\bigcup _{i\in I}U_{i}}$. Set ${\displaystyle {}T_{i}:=V{|}_{U_{i}}}$. We take the morphisms

${\displaystyle \psi _{ij}\colon T_{i}{|}_{U_{i}\cap U_{j}}=V{|}_{U_{i}\cap U_{j}}\longrightarrow V{|}_{U_{i}\cap U_{j}}=T_{j}{|}_{U_{i}\cap U_{j}}}$

given by ${\displaystyle {}\psi _{ij}:=\operatorname {Id} _{V}{|}_{U_{i}\cap U_{j}}+s_{ij}}$ to glue the ${\displaystyle {}T_{i}}$ together to a scheme ${\displaystyle {}T}$ over ${\displaystyle {}X}$. This is possible since the cocycle condition guarantees the glueing condition for schemes.

The action of ${\displaystyle {}T_{i}=V{|}_{U_{i}}}$ on itself glues also together to give an action on ${\displaystyle {}T}$.

${\displaystyle \Box }$

It follows immediately that for an affine scheme (i.e. a scheme of type ${\displaystyle \operatorname {Spec} {\left(R\right)}}$) 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).