Let denote a locally free sheaf on a scheme . For a cohomology class
one can construct a geometric object: Because of
,
the class defines an extension
-
This extension is such that under the connecting homomorphism of cohomology,
is sent to
.
The extension yields a projective subbundle
-
If is the corresponding geometric vector bundle of , one may think of as which consists for every base point
of all the lines in the fiber passing through the origin. The projective subbundle has codimension one inside , for every point it is a projective space lying
(linearly)
inside a projective space of one dimension higher. The complement is then over every point an affine space. One can show that the global complement
-
is another model for the torsor given by the cohomology class. The advantage of this viewpoint is that we may work, in particular when is projective, in an entirely projective setting.