Lemma 16.3.6. Let $R \to A$ be a standard smooth ring map. Let $E \subset A$ be a finite subset of order $|E| = n$. Then there exists a presentation $A = R[x_1, \ldots , x_{n + m}]/(f_1, \ldots , f_ c)$ with $c \geq n$, with $\det (\partial f_ j/\partial x_ i)_{i, j = 1, \ldots , c}$ invertible in $A$, and such that $E$ is the set of congruence classes of $x_1, \ldots , x_ n$.

Proof. Choose a presentation $A = R[y_1, \ldots , y_ m]/(g_1, \ldots , g_ d)$ such that the image of $\det (\partial g_ j/\partial y_ i)_{i, j = 1, \ldots , d}$ is invertible in $A$. Choose an enumerations $E = \{ a_1, \ldots , a_ n\}$ and choose $h_ i \in R[y_1, \ldots , y_ m]$ whose image in $A$ is $a_ i$. Consider the presentation

$A = R[x_1, \ldots , x_ n, y_1, \ldots , y_ m]/ (x_1 - h_1, \ldots , x_ n - h_ n, g_1, \ldots , g_ d)$

and set $c = n + d$. $\square$

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