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Erzwingende Algebra/Übergangsabbildung/Motivation/Bemerkung

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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.