Vector bundle/Torsor/Definition

Geometric torsor

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.