## Tag `00T6`

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

Definition 10.135.6. Let $R$ be a ring. Given integers $n \geq c \geq 0$ and $f_1, \ldots, f_c \in R[x_1, \ldots, x_n]$ we say $$ S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c) $$ is a

standard smooth algebra over $R$if the polynomial $$ g = \det \left( \begin{matrix} \partial f_1/\partial x_1 & \partial f_2/\partial x_1 & \ldots & \partial f_c/\partial x_1 \\ \partial f_1/\partial x_2 & \partial f_2/\partial x_2 & \ldots & \partial f_c/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial f_1/\partial x_c & \partial f_2/\partial x_c & \ldots & \partial f_c/\partial x_c \end{matrix} \right) $$ maps to an invertible element in $S$.

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 35254–35284 (see updates for more information).

```
\begin{definition}
\label{definition-standard-smooth}
Let $R$ be a ring. Given integers $n \geq c \geq 0$ and
$f_1, \ldots, f_c \in R[x_1, \ldots, x_n]$ we say
$$
S = R[x_1, \ldots, x_n]/(f_1, \ldots, f_c)
$$
is a {\it standard smooth algebra over $R$} if the polynomial
$$
g =
\det
\left(
\begin{matrix}
\partial f_1/\partial x_1 &
\partial f_2/\partial x_1 &
\ldots &
\partial f_c/\partial x_1 \\
\partial f_1/\partial x_2 &
\partial f_2/\partial x_2 &
\ldots &
\partial f_c/\partial x_2 \\
\ldots & \ldots & \ldots & \ldots \\
\partial f_1/\partial x_c &
\partial f_2/\partial x_c &
\ldots &
\partial f_c/\partial x_c
\end{matrix}
\right)
$$
maps to an invertible element in $S$.
\end{definition}
```

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