Vektorbündel/A^1/Punktierte affine Fläche/Erzwingende Algebra/Beispiel/en

We continue with Beispiel, so let ${}(R,{\mathfrak {m}})$ denote a two-dimensional normal local noetherian domain and let ${}f$ and ${}g$ be two parameters in ${}R$. The torsor given by a cohomology class ${}c={\frac {h}{fg}}\in H^{1}(U,{\mathcal {O}}_{X})$ can be realized by the forcing algebra

$R[T_{1},T_{2}]/(fT_{1}+gT_{2}-h).$

Note that different forcing algebras may give the same torsor, because the torsor depends only on the spectrum of the forcing algebra restricted to the punctured spectrum of ${}R$. For example, the cohomology class ${}{\frac {1}{fg}}={\frac {fg}{f^{2}g^{2}}}$ defines one torsor, but the two quotients yield the two forcing algebras ${}R[T_{1},T_{2}]/(fT_{1}+gT_{2}+1)$ and ${}R[T_{1},T_{2}]/(f^{2}T_{1}+g^{2}T_{2}+fg)$, which are quite different. The fiber over the maximal ideal of the first one is empty, whereas the fiber over the maximal ideal of the second one is a plane.

If ${}R$ is regular, say ${}R=K[X,Y]$ (or the localization of this at $(X,Y)$ or the corresponding power series ring) then the first cohomology classes are linear combinations of ${}{\frac {1}{x^{i}y^{j}}}$, ${}i,j\geq 1$. They are realized by the forcing algebras ${}K[X,Y]/(X^{i}T_{1}+Y^{j}T_{2}-1)$. Since the fiber over the maximal ideal is empty, the spectrum of the forcing algebra equals the torsor. Or, the other way round, the torsor is itself an affine scheme.