Lemma 16.2.2. Let $R$ be a ring. Let $A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$. Let $\mathfrak q \subset A$ be a prime ideal. Assume $R \to A$ is smooth at $\mathfrak q$. Then there exists an $a \in A$, $a \not\in \mathfrak q$, an integer $c$, $0 \leq c \leq \min (n, m)$, subsets $U \subset \{ 1, \ldots , n\} $, $V \subset \{ 1, \ldots , m\} $ of cardinality $c$ such that

\[ a = a' \det (\partial f_ j/\partial x_ i)_{j \in V, i \in U} \]

for some $a' \in A$ and

\[ a f_\ell \in (f_ j, j \in V) + (f_1, \ldots , f_ m)^2 \]

for all $\ell \in \{ 1, \ldots , m\} $.

**Proof.**
Set $I = (f_1, \ldots , f_ m)$ so that the naive cotangent complex of $A$ over $R$ is homotopy equivalent to $I/I^2 \to \bigoplus A\text{d}x_ i$, see Algebra, Lemma 10.134.2. We will use the formation of the naive cotangent complex commutes with localization, see Algebra, Section 10.134, especially Algebra, Lemma 10.134.13. By Algebra, Definitions 10.137.1 and 10.137.11 we see that $(I/I^2)_ a \to \bigoplus A_ a\text{d}x_ i$ is a split injection for some $a \in A$, $a \not\in \mathfrak q$. After renumbering $x_1, \ldots , x_ n$ and $f_1, \ldots , f_ m$ we may assume that $f_1, \ldots , f_ c$ form a basis for the vector space $I/I^2 \otimes _ A \kappa (\mathfrak q)$ and that $\text{d}x_{c + 1}, \ldots , \text{d}x_ n$ map to a basis of $\Omega _{A/R} \otimes _ A \kappa (\mathfrak q)$. Hence after replacing $a$ by $aa'$ for some $a' \in A$, $a' \not\in \mathfrak q$ we may assume $f_1, \ldots , f_ c$ form a basis for $(I/I^2)_ a$ and that $\text{d}x_{c + 1}, \ldots , \text{d}x_ n$ map to a basis of $(\Omega _{A/R})_ a$. In this situation $a^ N$ for some large integer $N$ satisfies the conditions of the lemma (with $U = V = \{ 1, \ldots , c\} $).
$\square$

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