Erzwingende Algebra/Ideal/Operation des Syzygiengruppenschemas/en/Fakt

Let ${}R$ denote a commutative ring, ${}I=(f_{1},\ldots ,f_{n})$ an ideal with the syzygy group scheme given by ${}G=\operatorname {Spec} {\left(R[S_{1},\ldots ,S_{n}]/(f_{1}S_{1}+\cdots +f_{n}S_{n})\right)}$. Let ${}f\in R$ be another element and let ${}Z=\operatorname {Spec} {\left(R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}-f\right)}\right)}$ be the spectrum of the corresponding forcing algebra.

Then there is a natural action of ${}G$ on ${}Z$. The restriction of this action to the open subset ${}U=D(I)$ makes ${}Z{|}_{U}$ to a torsor for the vector bundle ${}G{|}_{U}=\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}{|}_{U}$.

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