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

Beweis

The action is induced by the co-operation which sends ${}T_{i}\mapsto S_{i}+T_{i}$. In terms of points this is just the mapping which sends a syzygy ${}(s_{1},\ldots ,s_{n})$ and a solution ${}t_{1},\ldots ,t_{n}$ of the forcing equation to the new solution ${}(s_{1}+t_{1},\ldots ,s_{n}+t_{n})$ of the forcing equation.

For the second statement let ${}V=G{|}_{U}$ denote the syzygy bundle over ${}U$. We may consider the situation on ${}U_{i}=D(f_{i})$, where we have the isomorphism

$V{|}_{U_{i}}\longrightarrow T_{U_{i}},\,(s_{1},\ldots ,s_{n})\longmapsto (s_{1},\ldots ,s_{i-1},s_{i}+{\frac {f}{f_{i}}},s_{i+1},\ldots ,s_{n}).$

With these isomorphisms the natural diagram

${\begin{matrix}V{|}_{U_{i}}\times _{U_{i}}V{|}_{U_{i}}&{\stackrel {}{\longrightarrow }}&V{|}_{U_{i}}&\\\!\!\!\!\!\operatorname {Id} \times \varphi \downarrow &&\downarrow \varphi \!\!\!\!\!&\\V{|}_{U_{i}}\times _{U_{i}}T{|}_{U_{i}}&{\stackrel {}{\longrightarrow }}&T{|}_{U_{i}}&\!\!\!\!\!\\\end{matrix}}$

commutes, so locally the natural action of the vector bundle on the restricted spectrum of the forcing algebra is isomorphic to the addition of the vector bundle on itself.

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