Vector bundle/Torsor/Cohomological classification/Introduction/Section

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 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).