Vektorbündel auf Schema/Torsor und H^1/Korrespondenz/en/Fakt/Beweis

Beweis

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 ${}T$.

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