0

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?

Math Lover
  • 3,832
  • 1
  • 10
  • 26
  • Can you prove that there is a polynomial $f$ (not necessarily monic) such that $P$ becomes free after inverting $f$? – user26857 Apr 12 '17 at 07:32
  • @user26857: Sorry i'm unable to see the existence of such $f$ – Math Lover Apr 12 '17 at 07:49
  • Then please read this thread: http://math.stackexchange.com/questions/16814/finitely-generated-projective-modules-are-locally-free – user26857 Apr 12 '17 at 07:51
  • @user26857: Sorry i can't see how does one uses this to prove the required result.( I know that projective modules are locally free) – Math Lover Apr 12 '17 at 07:57
  • I've asked if you know to prove that $P_f$ is $A_f$-free (here $A$ denotes the polynomial ring) for some polynomial $f$ and you said "Sorry I'm unable to see the existence of such $f$". As far as I can see the linked thread deals with a similar question requiring a little more: "There are elements $f_1,\dots,f_n \in A$ such that $(f_1,\dots,f_n) = 1$, and such that $M_{f_i}$ is a free $A_{f_i}$-module for all $1\le i \le n$." – user26857 Apr 12 '17 at 08:00
  • Last but not least, I think it's a crucial step to prove that $P_f$ is $A_f$-free for some $f\in A$. – user26857 Apr 12 '17 at 08:04
  • @user26857: I think i'm unable to think much because its already late night here. Whatever you said is obvious from the thread. Do we get this as a consequence of "P is free after inverting monic polynomials" ? (Because author said this implies..) and how do we proceed to get monic polynomial? – Math Lover Apr 12 '17 at 08:24
  • Step 1. "There exists a non-zero polynomial f such that P becomes free after we invert f." (Done) Step 2. "A classical lemma of Noether shows that making an appropriate change of variables we may assume that f is monic in Xn." (Yet to be done, but now check the proof of Noether normalization and see there what change of variable they use. Notice that this can make your $f$ monic in Xn.) – user26857 Apr 12 '17 at 08:46
  • @user26857 Thank you,give me some time to verify this! – Math Lover Apr 12 '17 at 08:58

0 Answers0