Example 10.143.12 (00UA)—The Stacks project

Example 10.143.12. 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 10.136.7. 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

\[ \left( \begin{matrix} 1 & c_1 & \ldots & c_ m & 0 & \ldots & \ldots & 0 \\ 0 & 1 & c_1 & \ldots & c_ m & 0 & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & 0 & 1 & c_1 & c_2 & \ldots & c_ m \\ 1 & b_1 & \ldots & b_{n - 1} & b_ n & 0 & \ldots & 0 \\ 0 & 1 & b_1 & \ldots & b_{n - 1} & b_ n & \ldots & 0 \\ \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & \ldots & 0 & 1 & b_1 & \ldots & b_ n \end{matrix} \right) \]

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:

$R \to S$ is étale at $\mathfrak q$,
$\Delta = \text{Res}_ x(g, h) \not\in \mathfrak q$,
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.

Comments (2)

Comment #3967 by Manuel Hoff on January 27, 2019 at 05:22

I am a bit confused about the Sylvester matrix. In my opinion, in the middle of the matrix, the entry $b_n$ should be right below $c_1$ . I see that it is difficult to display the whole matrix correctly, but maybe change this detail, because like this, it doesn't look like a quadratic matrix. (sorry fot the bad explanation..)

Comment #4102 by Johan on April 04, 2019 at 19:32

As you say it cannot be done perfectly, but I tried to fix the issue you mentioned. Thanks. See this commit.

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