Tent & Ziegler, Lemma 7.2.11

Question

I try to understand the proof of Lemma 7.2.11 in Tent&Ziegler book: "If $(a_i)_{i\in I}$ is independet over A and $J<K$ are subsets of I, then $tp((a_k)_{k\in K}/A\cup\{a_j|j\in J\})$ does not divide over A." Proof in the book says "We may assume that $K$ is finite" - it's clear. And the last sentence says, "The claim now follow from Proposition 7.1.6 by induction on $|K|$." - that is not clear at all, besause:

1. Proposition 7.1.6: If $tp(a/B)$ does not divide over $A\subset B$ and $tp(c/Ba)$ does not divide over $Aa$, then $tp(ac/B)$ does not divide over $A$.
2. If $(a_i)_{i\in I}$ is independent over $A$, then in particular $tp(a_0/A)$ does not divide over $A$ and $tp(a_1/Aa_0)$ does not divide $A$.
3. Now, we want to show that $tp(a_0a_1/A)$ does not divide over $A$. But it's impossible to use Propositions 7.1.6, because it's not clear, that $tp(a_1/Aa_0)$ does not divide over $Aa_0$...

Answer 1

The fact that $\text{tp}(a_1/Aa_0)$ does not divide over $Aa_0$ is clear from the definition of dividing. No type in $S(B)$ divides over $B$, since any sequence of realizations of $\text{tp}(B/B)$ is constant.

What you're missing to do the general inductive step is that if a type over $C$ does not divide over $A$, then it also does not divide over $AB$, for any set $B$. This also follows directly from the definition of dividing, since any sequence of realizations of $\text{tp}(C/AB)$ is also a sequence of realizations of $\text{tp}(C/A)$.

So the hypothesis "$\text{tp}(c/Ba)$ does not divide over $Aa$" in Proposition 7.1.6 is weaker than "$\text{tp}(c/Ba)$ does not divide over $A$". In this application, the stronger hypothesis holds.

To do the inductive step, we want to show that $\text{tp}(a_0\dots a_n/A(a_j)_{j\in J})$ does not divide over $A$. By induction, we know that $\text{tp}(a_0\dots a_{n-1}/A(a_j)_{j\in J})$ does not divide over $A$. And by independence, $\text{tp}(a_n/A(a_j)_{j\in J}a_0\dots a_{n-1})$ does not divide over $A$, and hence does not divide over $Aa_0\dots a_{n-1}$. So by Proposition 7.1.6, $\text{tp}(a_0\dots a_n/A(a_j)_{j\in J})$ does not divide over $A$.