Erzwingende Algebra/Übergangsabbildung/Motivation/Bemerkung

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 $f_{i}$ we get

{(R [T_{1}, \dots, T_{n}] / (f_{1} T_{1} + \dots + f_{n} T_{n} - f))}_{f_{i}} ≅ R_{f_{i}} [T_{1}, \dots, T_{i - 1}, T_{i + 1}, \dots, T_{n}],

since we can write

T_{i} = - \sum_{j \neq i} \frac{f_{j}}{f_{i}} T_{j} + \frac{f}{f_{i}} .

So over every $D (f_{i})$ the spectrum of the forcing algebra is an $(n - 1)$ -dimensional affine space over the base. So locally, restricted to $D (f_{i})$ , we have isomorphisms

T |_{D (f_{i})} ≅ D (f_{i}) \times 𝔸_{}^{n - 1} .

On the intersections $D (f_{i}) \cap D (f_{j})$ we get two identifications with affine space, and the transition morphisms are linear if $f = 0$ , but only affine-linear in general (because of the translation with $\frac{f}{f_{i}}$ ).

So the forcing algebra has locally the form $R_{f_{i}} [T_{1}, \dots, T_{i - 1}, T_{i + 1}, \dots, T_{n}]$ and its spectrum $Spec (B)$ has locally the form $D (f_{i}) \times 𝔸_{}^{n - 1}$ . This description holds on the union $U = ⋃_{i = 1}^{n} D (f_{i})$ . Moreover, in the homogeneous case ( $f = 0$ ) the transition mappings are linear. Hence $V |_{U}$ , where $V$ is the spectrum of a homogeneous forcing algebra, is a geometric vector bundle according to the following definition.