I'm reading the paper "Quillen's solution of Serre's Problem" in which i am unable to follow one statement:
Suppose $P$ be a finitely generated projective $k[x_1,...,x_n]$ module.We first note that $P$ obviously becomes free after we invert all non zero polynomial. Then the author writes this implies:
By a classical lemma of Noether there exists a polynomial $f$ which is monic in $x_n$ such that $P$ becomes free after we invert $f$.
I'm unable to see this.Could someone explain how does one see the above claim?