The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Example 10.141.12. Let $n , m \geq 1$ be integers. Consider the ring map

\begin{eqnarray*} R = \mathbf{Z}[a_1, \ldots , a_{n + m}] & \longrightarrow & S = \mathbf{Z}[b_1, \ldots , b_ n, c_1, \ldots , c_ m] \\ a_1 & \longmapsto & b_1 + c_1 \\ a_2 & \longmapsto & b_2 + b_1 c_1 + c_2 \\ \ldots & \ldots & \ldots \\ a_{n + m} & \longmapsto & b_ n c_ m \end{eqnarray*}

of Example 10.134.7. Write symbolically

\[ S = R[b_1, \ldots , c_ m]/(\{ a_ k(b_ i, c_ j) - a_ k\} _{k = 1, \ldots , n + m}) \]

where for example $a_1(b_ i, c_ j) = b_1 + c_1$. The matrix of partial derivatives is

\[ \left( \begin{matrix} 1 & c_1 & \ldots & c_ m & 0 & \ldots & 0 \\ 0 & 1 & c_1 & \ldots & c_ m & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & 0 & 1 & c_1 & \ldots & c_ m \\ 1 & b_1 & \ldots & b_ n & 0 & \ldots & 0 \\ 0 & 1 & b_1 & \ldots & b_ n & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & 0 & 1 & b_1 & \ldots & b_ n \\ \end{matrix} \right) \]

The determinant $\Delta $ of this matrix is better known as the resultant of the polynomials $g = x^ n + b_1 x^{n - 1} + \ldots + b_ n$ and $h = x^ m + c_1 x^{m - 1} + \ldots + c_ m$, and the matrix above is known as the Sylvester matrix associated to $g, h$. In a formula $\Delta = \text{Res}_ x(g, h)$. The Sylvester matrix is the transpose of the matrix of the linear map

\begin{eqnarray*} S[x]_{< m} \oplus S[x]_{< n} & \longrightarrow & S[x]_{< n + m} \\ a \oplus b & \longmapsto & ag + bh \end{eqnarray*}

Let $\mathfrak q \subset S$ be any prime. By the above the following are equivalent:

  1. $R \to S$ is étale at $\mathfrak q$,

  2. $\Delta = \text{Res}_ x(g, h) \not\in \mathfrak q$,

  3. the images $\overline{g}, \overline{h} \in \kappa (\mathfrak q)[x]$ of the polynomials $g, h$ are relatively prime in $\kappa (\mathfrak q)[x]$.

The equivalence of (2) and (3) holds because the image of the Sylvester matrix in $\text{Mat}(n + m, \kappa (\mathfrak q))$ has a kernel if and only if the polynomials $\overline{g}, \overline{h}$ have a factor in common. We conclude that the ring map

\[ R \longrightarrow S[\frac{1}{\Delta }] = S[\frac{1}{\text{Res}_ x(g, h)}] \]

is étale.


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