TITLE: Finitely generated projective modules over polynomial rings with integral coefficients
QUESTION [5 upvotes]: There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free.
I have never carefully read a proof of this theorem, which is for example in the Lang's Algebra. Probably it is based on Quillen's original ideas.- this is not true as it was pointed out in the answer below.
Questions: Is every finitely generated projective modules over $\mathbb{Z}[x_1,...,x_n]$ free? 
If yes, then is the proof modification of the one given in Lang's Algebra?
And if yes, then how about polynomial rings over other Dedekind domains or number rings?

REPLY [4 votes]: Ok Slup, here goes.
Let $R$ be any commutative ring and let $A$ be a polynomial ring over $R$. Let $P$ be any projective module over $R$. Then Quillen (and Suslin a bit later in this generality) proved that if for every maximal ideal $\mathfrak{m}$ of $R$, $P_{\mathfrak{m}}$ is of the form $Q\otimes_{R_{\mathfrak{m}}} A_{\mathfrak{m}}$ for some projective $R_{\mathfrak{m}}$ module $Q$, then there exists a projective module $Q$ over $R$ such that $P=Q\otimes_R A$. Using this, they deduced that this always happens when $R$ is a Dedekind domain. The last part is done by the following observation. If for such a $P$, $P_f$ is free for a polynomial which is monic in one of the variables, then $P$ is free. As you can see, since any non-zero polynomial after a change of variables can be made into a monic polynomial in one of the variables if $R$ is a field, one immediately deduces Serre conjecture from this.
Many years later, Lindel generalized this for $R$ any regular ring containing a field.