Definition 16.2.3. Let $R \to A$ be a ring map of finite presentation. We say an element $a \in A$ is *elementary standard in $A$ over $R$* if there exists a presentation $A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ and $0 \leq c \leq \min (n, m)$ such that

16.2.3.1
\begin{equation} \label{smoothing-equation-elementary-standard-one} a = a' \det (\partial f_ j/\partial x_ i)_{i, j = 1, \ldots , c} \end{equation}

for some $a' \in A$ and

16.2.3.2
\begin{equation} \label{smoothing-equation-elementary-standard-two} a f_{c + j} \in (f_1, \ldots , f_ c) + (f_1, \ldots , f_ m)^2 \end{equation}

for $j = 1, \ldots , m - c$. We say $a \in A$ is *strictly standard in $A$ over $R$* if there exists a presentation $A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ and $0 \leq c \leq \min (n, m)$ such that

16.2.3.3
\begin{equation} \label{smoothing-equation-strictly-standard-one} a = \sum \nolimits _{I \subset \{ 1, \ldots , n\} ,\ |I| = c} a_ I \det (\partial f_ j/\partial x_ i)_{j = 1, \ldots , c,\ i \in I} \end{equation}

for some $a_ I \in A$ and

16.2.3.4
\begin{equation} \label{smoothing-equation-strictly-standard-two} a f_{c + j} \in (f_1, \ldots , f_ c) + (f_1, \ldots , f_ m)^2 \end{equation}

for $j = 1, \ldots , m - c$.

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