We have seen that the fibers of the spectrum of a forcing algebra are
(empty or)
affine spaces. However, this is not only fiberwise true, but more general: If we localize the forcing algebra at we get
-
since we can write
-
So over every the spectrum of the forcing algebra is an -dimensional affine space over the base. So locally, restricted to , we have isomorphisms
-
On the intersections we get two identifications with affine space, and the transition morphisms are linear if
,
but only affine-linear in general
(because of the translation with ).
So the forcing algebra has locally the form and its spectrum has locally the form . This description holds on the union
.
Moreover, in the homogeneous case
()
the transition mappings are linear. Hence , where is the spectrum of a homogeneous forcing algebra, is a geometric vector bundle according to the following definition.