## Tag `00T8`

Chapter 10: Commutative Algebra > Section 10.135: Smooth ring maps

Example 10.135.8. Let $R$ be a ring. Let $f_1, \ldots, f_c \in R[x_1, \ldots, x_n]$. Let $$ h = \det \left( \begin{matrix} \partial f_1/\partial x_1 & \partial f_2/\partial x_1 & \ldots & \partial f_c/\partial x_1 \\ \partial f_1/\partial x_2 & \partial f_2/\partial x_2 & \ldots & \partial f_c/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial f_1/\partial x_c & \partial f_2/\partial x_c & \ldots & \partial f_c/\partial x_c \end{matrix} \right). $$ Set $S = R[x_1, \ldots, x_{n + 1}]/(f_1, \ldots, f_c, x_{n + 1}h - 1)$. This is an example of a standard smooth algebra, except that the presentation is wrong and the variables should be in the following order: $x_1, \ldots, x_c, x_{n + 1}, x_{c + 1}, \ldots, x_n$.

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 35300–35331 (see updates for more information).

```
\begin{example}
\label{example-make-standard-smooth}
Let $R$ be a ring.
Let $f_1, \ldots, f_c \in R[x_1, \ldots, x_n]$.
Let
$$
h =
\det
\left(
\begin{matrix}
\partial f_1/\partial x_1 &
\partial f_2/\partial x_1 &
\ldots &
\partial f_c/\partial x_1 \\
\partial f_1/\partial x_2 &
\partial f_2/\partial x_2 &
\ldots &
\partial f_c/\partial x_2 \\
\ldots & \ldots & \ldots & \ldots \\
\partial f_1/\partial x_c &
\partial f_2/\partial x_c &
\ldots &
\partial f_c/\partial x_c
\end{matrix}
\right).
$$
Set $S = R[x_1, \ldots, x_{n + 1}]/(f_1, \ldots, f_c, x_{n + 1}h - 1)$.
This is an example of a standard smooth algebra, except that the
presentation is wrong and the variables should be in the
following order:
$x_1, \ldots, x_c, x_{n + 1}, x_{c + 1}, \ldots, x_n$.
\end{example}
```

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