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

Let $(R, 𝔪)$ denote a two-dimensional normal local noetherian domain and let $f$ and $g$ be two parameters in $R$ . On $U = D (𝔪)$ we have the short exact sequence

0 ⟶ 𝒪_{U} ≅ Syz (f, g) ⟶ 𝒪_{U}^{2} \overset{f, g}{⟶} 𝒪_{U} ⟶ 0

and its corresponding long exact sequence of cohomology,

0 ⟶ R ⟶ R^{2} \overset{f, g}{⟶} R \overset{δ}{⟶} H^{1} (U, 𝒪_{X}) ⟶ \dots .

The connecting homomorphism $δ$ sends an element $h \in R$ to $\frac{h}{f g}$ . The torsor given by such a cohomology class $c = \frac{h}{f g} \in H^{1} (U, 𝒪_{X})$ can be realized by the forcing algebra

R [T_{1}, T_{2}] / (f T_{1} + g T_{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 $\frac{1}{f g} = \frac{f g}{f^{2} g^{2}}$ defines one torsor, but the two fractions yield the two forcing algebras $R [T_{1}, T_{2}] / (f T_{1} + g T_{2} - 1)$ and $R [T_{1}, T_{2}] / (f^{2} T_{1} + g^{2} T_{2} - f g)$ , 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 $K$ -linear combinations of $\frac{1}{x^{i} y^{j}}$ , $i, j \geq 1$ .

They are realized by the forcing algebras

K [X, Y, T_{1}, T_{2}] / (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.