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?
Best Answer
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.