Definition 10.137.6. 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 $S$.

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