Forcing algebra/Transformation behavior/Motivation/Remark

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

${\left(R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}-f\right)}\right)}_{f_{i}}\cong R_{f_{i}}[T_{1},\ldots ,T_{i-1},T_{i+1},\ldots ,T_{n}]\,,$

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

${}T{|}_{D(f_{i})}\cong D(f_{i})\times {{\mathbb {A} }_{}^{n-1}}\,.$

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 ${}R_{f_{i}}[T_{1},\ldots ,T_{i-1},T_{i+1},\ldots ,T_{n}]$ and its spectrum ${}\operatorname {Spec} {\left(B\right)}$ has locally the form ${}D(f_{i})\times {{\mathbb {A} }_{}^{n-1}}$. This description holds on the union ${}U=\bigcup _{i=1}^{n}D(f_{i})$. 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.