The assumption for Lemma A.4.1 is $A \to B$ is flat. The second assumption is that $A$ and $B$ are Artinian rings. From this Lemma A.4.1 states that $l_B(B) = l_A(A) \cdot l_B(B/mB)$ where $m$ is the maximal ideal of $A$. The proof is given as follows :

Take a chain of ideals $I_i$ of $A$ such that $A \subset I_1 \subset I_2 \cdots \subset I_r = 0$ where $I_{i-1}/I_i$ is isomorphic to $A/m$. Then we have the chain $B \subset I_1B \subset I_2B \subset \cdots \subset I_rB = 0$ with $I_{i-1}B/I_iB \approx (I_{i-1}/I_i) \otimes_A B \approx B/mB$.

However I think we do not need flatness condition for the proof since taking tensor product is right exact and hence we have $I_{i-1}B/I_iB \approx (I_{i-1}/I_i) \otimes_A B \approx B/mB$ without any assumption. Am I right? If so, then why does Fulton use this condition for the construction of pull back of algebraic cycles? For the pull back of algebraic cycles I guess we only need surjectivity between the generic points of both sides of the map.