appendix: local appearance of blow up

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

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

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

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