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appendix: local appearance of blow up

Let us define a ring homomorphism $ \varphi$ as follows.

    $\displaystyle \varphi: A[X_0,$ $\displaystyle \dots,X_s]$   $\displaystyle \to$ $\displaystyle A[f_0 \cdot T ,$ $\displaystyle \dots f_s \cdot T, (f_j \cdot T)^{-1}]_0$
      $\displaystyle \mathrel{\rotatebox[origin=c]{90}{$\in$}}$       $\displaystyle \mathrel{\rotatebox[origin=c]{90}{$\in$}}$
      $\displaystyle X_i$   $\displaystyle \mapsto$ $\displaystyle (f_i \cdot T)$ $\displaystyle (f_j \cdot T)^{-1}$

It is easy to see that $ \varphi$ is a surjective homomorphism.

      $\displaystyle p(x_0,x_1,\dots x_s)\in \operatorname{Ker}(\bar\varphi)$
    $\displaystyle \iff$ $\displaystyle p\left(\dfrac{f_0 T}{f_j T},\dots, \dfrac{f_s T}{f_j T} \right)=0$
    $\displaystyle \iff$ $\displaystyle \sum_{i_0,i_1,\dots i_s} p_{i_0 i_1 \dots i_s} \left(\dfrac{f_0 T}{f_j T}\right)^{i_0}\cdot\dots \cdot \left(\dfrac{f_s T}{f_j T}\right)^{i_s} =0$
    $\displaystyle \iff$ % latex2html id marker 548
$\displaystyle \exists N>0 \quad \sum_{i_0,i_1,\dots...
...}\cdot\dots \cdot \left({f_s T}\right)^{i_s} (f_j T)^{N-(i_0+i_1+\dots +i_s)}=0$
    $\displaystyle \iff$ $\displaystyle \sum_{i_0,i_1,\dots i_s} p_{i_0 i_1 \dots i_s} f_0 ^{i_0}\cdot\dots \cdot f_s ^{i_s} f_j ^{N-(i_0+i_1+\dots +i_s)}=0$
    $\displaystyle \iff$ $\displaystyle p(f_0 f_j^{-1},\dots f_s f_j^{-1})=0$    in $\displaystyle A[f_j^{-1}].$

We conclude:

PROPOSITION 0.1   Let us denote by $ A[f_0 f_j^{-1},\dots f_s f_j^{-1}]$ the subalgebra of $ A[f_j^{-1}]$ generated by $ f_0 f_j^{-1},\dots f_s f_j^{-1}$ . (Note that this notation is ambiguous and should not be used without an explanation.) Then the ring homomorphism $ \varphi$ as above induces an algebra isomorphism

$\displaystyle \bar\varphi: A[f_0 f_j^{-1},\dots f_s f_j^{-1}]
A[f_0 T ,\dots f_s T , (f_j T)^{-1}]_0.

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