Lemma 10.141.19. Let $R$ be a ring. Let $f \in R[x]$ be a monic polynomial. Let $\mathfrak p$ be a prime of $R$. Let $f \bmod \mathfrak p = \overline{g} \overline{h}$ be a factorization of the image of $f$ in $\kappa (\mathfrak p)[x]$. If $\gcd (\overline{g}, \overline{h}) = 1$, then there exist

1. an étale ring map $R \to R'$,

2. a prime $\mathfrak p' \subset R'$ lying over $\mathfrak p$, and

3. a factorization $f = g h$ in $R'[x]$

such that

1. $\kappa (\mathfrak p) = \kappa (\mathfrak p')$,

2. $\overline{g} = g \bmod \mathfrak p'$, $\overline{h} = h \bmod \mathfrak p'$, and

3. the polynomials $g, h$ generate the unit ideal in $R'[x]$.

Proof. Suppose $\overline{g} = \overline{b}_0 x^ n + \overline{b}_1 x^{n - 1} + \ldots + \overline{b}_ n$, and $\overline{h} = \overline{c}_0 x^ m + \overline{c}_1 x^{m - 1} + \ldots + \overline{c}_ m$ with $\overline{b}_0, \overline{c}_0 \in \kappa (\mathfrak p)$ nonzero. After localizing $R$ at some element of $R$ not contained in $\mathfrak p$ we may assume $\overline{b}_0$ is the image of an invertible element $b_0 \in R$. Replacing $\overline{g}$ by $\overline{g}/b_0$ and $\overline{h}$ by $b_0\overline{h}$ we reduce to the case where $\overline{g}$, $\overline{h}$ are monic (verification omitted). Say $\overline{g} = x^ n + \overline{b}_1 x^{n - 1} + \ldots + \overline{b}_ n$, and $\overline{h} = x^ m + \overline{c}_1 x^{m - 1} + \ldots + \overline{c}_ m$. Write $f = x^{n + m} + a_1 x^{n - 1} + \ldots + a_{n + m}$. Consider the fibre product

$R' = R \otimes _{\mathbf{Z}[a_1, \ldots , a_{n + m}]} \mathbf{Z}[b_1, \ldots , b_ n, c_1, \ldots , c_ m]$

where the map $\mathbf{Z}[a_ k] \to \mathbf{Z}[b_ i, c_ j]$ is as in Examples 10.134.7 and 10.141.12. By construction there is an $R$-algebra map

$R' = R \otimes _{\mathbf{Z}[a_1, \ldots , a_{n + m}]} \mathbf{Z}[b_1, \ldots , b_ n, c_1, \ldots , c_ m] \longrightarrow \kappa (\mathfrak p)$

which maps $b_ i$ to $\overline{b}_ i$ and $c_ j$ to $\overline{c}_ j$. Denote $\mathfrak p' \subset R'$ the kernel of this map. Since by assumption the polynomials $\overline{g}, \overline{h}$ are relatively prime we see that the element $\Delta = \text{Res}_ x(g, h) \in \mathbf{Z}[b_ i, c_ j]$ (see Example 10.141.12) does not map to zero in $\kappa (\mathfrak p)$ under the displayed map. We conclude that $R \to R'$ is étale at $\mathfrak p'$. In fact a solution to the problem posed in the lemma is the ring map $R \to R'[1/\Delta ]$ and the prime $\mathfrak p' R'[1/\Delta ]$. Because $\text{Res}_ x(f, g)$ is invertible in this ring the Sylvester matrix is invertible over $R'[1/\Delta ]$ and hence $1 = a g + b h$ for some $a, b \in R'[1/\Delta ][x]$ see Example 10.141.12. $\square$

Comment #3965 by Manuel Hoff on

Typo: In the sentence of the proof, $R'$ should be replaced by $R'[1/ \Delta]$ two times.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).