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[1]
-
![{\displaystyle {}{\mathbb {P} }({\mathcal {S}}^{\vee })\subset {\mathbb {P} }({{\mathcal {S}}'}^{\vee })\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f43c1cf6fb34e5484ad7fccfb05489872d114131)
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
-
![{\displaystyle {}T={\mathbb {P} }({{\mathcal {S}}'}^{\vee })\setminus {\mathbb {P} }({\mathcal {S}}^{\vee })\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f752a107326156ea259b319314ff4acddfab5bb0)
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.
- ↑
denotes the dual bundle. According to our convention, the geometric vector bundle corresponding to a locally free sheaf
is given by
and the projective bundle is
, where
denotes the
th symmetric power.