Erzwingende Algebra/Primäres Ideal/Induzierter Torsor/en/Fakt/Beweis

Beweis

We compute the cohomology class $δ (f) \in H^{1} (U, Syz (f_{1}, \dots, f_{n}))$ and the cohomology class given by the forcing algebra. For the first computation we look at the short exact sequence

0 ⟶ Syz (f_{1}, \dots, f_{n}) ⟶ 𝒪_{U}^{n} \overset{f_{1}, \dots, f_{n}}{⟶} 𝒪_{U} ⟶ 0.

On $D (f_{i})$ , the element $f$ is the image of $(0, \dots, 0, \frac{f}{f_{i}}, 0, \dots, 0))$ (the non-zero entry is at the $i$ th place). The cohomology class is therefore represented by the family of differences

(0, \dots, 0, \frac{f}{f_{i}}, 0, \dots, 0, - \frac{f}{f_{j}}, 0, \dots, 0) \in Γ (D (f_{i}) \cap D (f_{j}), Syz (f_{1}, \dots, f_{n})) .

On the other hand, there are isomorphisms

V |_{D (f_{i})} ⟶ T |_{D (f_{i})}, (s_{1}, \dots, s_{n}) ⟼ (s_{1}, \dots, s_{i - 1}, s_{i} + \frac{f}{f_{i}}, s_{i + 1}, \dots, s_{n}) .

The composition of two such isomorphisms on $D (f_{i} f_{j})$ is the identity plus the same section as before.

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