say a B-algebra A satisfies (†) if for each finitely generated A-module M, there exists a nonzero f ∈ B such that $M_f$ is a free $B_f$-module.
Grothendieck's generic Freeness lemma States that if B is a Noetherian domain and A is a finitely generated B algebra, then A satisfies (†).
Its an exercise (7.4.G) in Ravi Vakil's algebraic geometry notes to reduce this statement to following claim: if A is a finitely-generated B-algebra satisfying (†), then A[T] does too. that is the exercise is to assume this claim to be true and prove the generic freeness lemma. ** My question is how to do this?**
In The exercise preceeding this one I proved that B itself satisfies (†). Thus if I could prove (†) is Preserved under passing to a quotient I could just Adjoin the generators to B as indeterminates (using the claim inductively) and then take a quotient to get A. but I don't know how to prove that (†) is preserved by passing to a quotient.