Proof
We describe only the correspondence.
Let
denote a
-torsor. Then there exists by definition an open covering
such that there exist isomorphisms
-
which are compatible with the action of
on itself. The isomorphisms
induce automorphisms
-
These automorphisms are compatible with the action of
on itself, and this
means that they are of the form
-
![{\displaystyle {}\psi _{ij}=\operatorname {Id} _{V}{|}_{U_{i}\cap U_{j}}+s_{ij}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/097e9e25c3d3d382c2c6a176d25ae105328bedbe)
with suitable sections
.
This family defines a Čech cocycle for the covering and gives therefore a cohomology class in
.
For the reverse direction, suppose that the cohomology class
is represented by a Čech cocycle
for an open covering
.
Set
.
We take the morphisms
-
given by
to glue the
together to a scheme
over
. This is possible since the cocycle condition guarantees the glueing condition for schemes.
The action of
on itself glues also together to give an action on
.