Example 10.137.8. 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: $x_1, \ldots , x_ c, x_{n + 1}, x_{c + 1}, \ldots , x_ n$.

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