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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 35324–35355 (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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