Let $n$ be a positive integer and $A$ be a commutative ring. The ring $A$ is said to be of Bass stable range $\mathrm{sr}(A)\leq n$ if for $a, a_1, \dots, a_n \in A$ one has the following implication:
$$1 \in \langle a, a_1, \dots, a_{n}\rangle \implies \exists \ x_1, \dots, x_n \in A, 1 \in \langle a_1+x_1a,\ \dots, a_n + x_n a\rangle.$$
(Above, $\langle \cdot \rangle$ denotes the ideal generated by the elements inside).
Of course, one says that $A$ is of stable range $n$ if $\mathrm{sr}(A) \leq n$ and $\mathrm{sr}(A) \not \leq n-1$.
Bass proved that if $A$ is noetherian of Krull dimension $d$ then $\mathrm{sr}(A)\leq d+1$.
Examples are known; for example Vaserstein proved that $\mathrm{sr}(k[x_1,\dots,x_n]) = n+1$ when $k$ is a subfield of the real numbers.
My question is : is the stable range of the ring of integer polynomials $\mathbf Z[X]$ known?
What I wrote before shows that $\mathrm{sr}(\mathbf Z[X]) \leq 3$ and it seems very likely that $\mathrm{sr}( \mathbf Z[X])=3$.
A refinement of my previous question is : could you provide an explicit unimodular triplet of polynomials $(P_1,P_2,P_3) \in \mathbf Z[X]$ showing that $\mathrm{sr}(\mathbf Z[X]) \not \leq 2$?
Best Answer
Sorry for putting this in a separate answer, but I think it will be cleaner this way.
I believe I now understand Vaserstein's intended argument that it's $\ge 3$:
(1) There are rings of the form $A={\mathbb Z}[x]/(h)$ such that $SK_1(A)\neq 0$. One way to get such a ring is to start with the ring of integers in a quadratic field, let $I$ be a "sufficiently small" ideal (I confess to not being exactly sure what this means) and look at the subring generated by $1$ and $I$. For some definition of "sufficiently small", Bass has shown that this gives us $SK_1(A)\neq 0$.
(2) Take a non-zero element of $SK_1(A)$ and represent it by a Mennicke symbol $[\overline{f},\overline{g}]$.
(3) Then $(f,g,h)$ is a unimodular row over $\mathbb{Z}[x]$.
(4) Clearly, the Mennicke symbol $[\overline{f},\overline{g}]$ does not lift to $K_1({\mathbb Z}[x])$.
(5) It follows from Lemma 17.1 of the paper referenced by Jeremy Rickard that the row $(f,g,h)$ is not reducible.
I'm still just a tad unclear on why it is, in point (1), that we can take the kernel of ${\mathbb Z}[x]\rightarrow A$ to be principal. Is this obvious? I'll add a comment if I nail this down.