Let $A\subset B$ be rings and $P$ a projective left $A$-module. Write $\rho:A\rightarrow B$ for the canonical injection. Suppose $\rho^*(P)=\rho_*(B)\otimes_A P$ is finitely generated. The author says,
Since $\rho^*(P)$ is finitely generated, there exists a finite family $(y_i)_{i\in I}$ such that $(1\otimes y_i)_{i\in I}$ is a generating system of $\rho^*(P).$
I am not sure how this follows. The fact that $\rho^*(P)$ is finitely generated implies that there exists an exact sequence $$B^{\oplus n}\xrightarrow{k}\rho^*(P)\rightarrow0.$$ Writing $(e_i)_{1\leq i\leq n}$ for the canonical basis of $B^{\oplus n}$, we have that $(k(e_i))_{1\leq i\leq n}$ is a generating system for $\rho^*(P)$. On the other hand, we have a canonical injection $\phi:P\rightarrow\rho^*(P)$ sending $x$ to $1\otimes x$. Then $\rho^*(P)$ is generated by $(1\otimes x)_{x\in P}$. So for each $1\leq i\leq n$, we have $$k(e_i)=\sum_{x\in P}\xi^i_x(1\otimes x)$$ for some $\xi^i\in B^{(P)}$. I don't know how to get $k(e_i)$ in the form $1\otimes y_i$ for some $y_i\in P$. Any suggestions?
The sums in your last equation are finite. That is for each $i$ there is a finite set $P_i \subseteq P$ such that $$ k(e_i) = \sum_{x \in P_i} \xi_x^i (1 \otimes x) $$ Then $\rho^*(P)$ is generated by $\{1 \otimes x \mid x \in \bigcup_i P_i\}$, a finite set.