In Goetz and Wedhorn's scheme book, I don't understand the last sentence of the following text.
$\Phi_i: R[\ldots]_d \to R[\ldots]$ is the dehomogenization map that, if I understand correctly, replaces $T_i$ with $1$ in polynomials.
The last sentence utterly puzzles me. As I see it, $\Phi_i \left(\frac{f}{g}\right) = \frac{\frac{f}{X_i^d}}{\frac{g}{X^d_i}} = \frac{f}{g}$. Am I missing something?
On
You have missed the identification $K(T_0, \dotsc, \hat T_i, \dotsc, T_n) \cong K(\frac{X_0}{X_i}, \dotsc, \frac{X_n}{X_i})$, which identifies $T_j$ with $\frac{X_j}{X_i}$. Thus $$\Phi_i(f)=f(T_0, \dotsc, 1, \dotsc, T_n) = f(\frac{X_0}{X_i}, \dotsc ,1, \dotsc, \frac{X_n}{X_i}) = \frac{f}{X_i^d}= \overline f$$ and subsequently $$\Phi_i(\frac{f}{g}) = \frac{\overline f}{\overline g}.$$
Of course this is the same fraction as $\frac{f}{g}$, but $f,g \notin K[\frac{X_0}{X_i}, \dotsc, \frac{X_n}{X_i}]$ and since $\Phi_i(\frac{f}{g}) \in K(\frac{X_0}{X_i}, \dotsc, \frac{X_n}{X_i})$, it makes sense to change the fraction, such that both numerator and denominator are in that ring. Thats why one divides each by $X_i^d$.
Yes, indeed $\Phi_i(f/g)=f/g$ (where we identify $X_j/X_i$ with $T_j$), but writing it as $\tilde f/\tilde g$ exhibits it explicitly as a quotient of polynomials in the $T_j$.
I would also say $\Phi_i$ replaces $X_i$ by $1$ and each other $X_j$ by $T_j$.