This is a question about the following proof of Lemma 3.8 in the paper Lovely pairs of models of Itay Ben-Yaacov, Anand Pillay and Evgueni Vassiliev. The paper can be found here.
Let $f:a→b$ be the partial $L_P$-isomorphism given by our hypothesis. It is enough (by back-and-forth, and symmetry) to show that any $c\in M$ is included in the domain of a partial $L_P$-isomorphism extending $f$. So choose $c$. Extending to a suitable small tuple, we may assume that $c$ is $P$-independent. Let $c_1=P(c)$ and let $c_2$ be the rest of $c$. Let $p$ be the $L$-type of $c_1$ over $a$, and let $p′$ be its copy over $b$. Then by $P$-independence of $a$ and $b$ in $(M;P(M))$ and $(N;P(N))$, respectively, and the axiom (ii) of lovely pairs, $p′$ is realized in $P(N)$ by some $d_1$. Now let $q$ be the $L$-type of $c_2$ over $c_1 a$, and $q′$ the copy over $d_1 b$. Then by the axiom (i) of lovely pairs, some nonforking extension of $q′$ over $P(N)\cup a$ is realized in $N$, by say $d_2$. Note that all coordinates of $d_2$ are outside $P(N)$. Let $g$ extend $f$ by taking $c_1$ to $d_1$ and $c_2$ to $d_2$. Then $g$ is a partial $L_P$-isomorphism. The proof is complete.
Questions:
- I think that they meant "some nonforking extension of $q′$ over $P(N)\cup b$" because $q'$ is a type over $d_1 b$, so not over $a$?
- Why are all coordinates of $d_2$ outside $P(N)$? I can only deduce this if no coordinate of $d_2$ is algebraic over $d_1 b$.