Let denote a two-dimensional normal local noetherian domain and let
and
be two parameters in . On
we have the short exact sequence
-
and its corresponding long exact sequence of cohomology,
-
The connecting homomorphism sends an element
to . The torsor given by such a cohomology class
can be realized by the forcing algebra
-
Note that different forcing algebras may give the same torsor, because the torsor depends only on the spectrum of the forcing algebra restricted to the punctured spectrum of . For example, the cohomology class
defines one torsor, but the two fractions yield the two forcing algebras
and ,
which are quite different. The fiber over the maximal ideal of the first one is empty, whereas the fiber over the maximal ideal of the second one is a plane.
If is regular, say
(or the localization of this at or the corresponding power series ring)
then the first cohomology classes are -linear combinations of
, .
They are realized by the forcing algebras
-
Since the fiber over the maximal ideal is empty, the spectrum of the forcing algebra equals the torsor. Or, the other way round, the torsor is itself an affine scheme.