The Stacks project

Lemma 53.19.5. Let $k$ be a field. Let $A = k[[x_1, \ldots , x_ n]]$. Let $I = (f_1, \ldots , f_ m) \subset A$ be an ideal. For any $r \geq 0$ the ideal in $A/I$ generated by the $r \times r$-minors of the matrix $(\partial f_ j/\partial x_ i)$ is independent of the choice of the generators of $I$ or the regular system of parameters $x_1, \ldots , x_ n$ of $A$.

Proof. The “correct” proof of this lemma is to prove that this ideal is the $(n - r)$th Fitting ideal of a module of continuous differentials of $A/I$ over $k$. Here is a direct proof. If $g_1, \ldots g_ l$ is a second set of generators of $I$, then we can write $g_ s = \sum a_{sj}f_ j$ and we have the equality of matrices

\[ (\partial g_ s/\partial x_ i) = (a_{sj}) (\partial f_ j/\partial x_ i) + (\partial a_{sj}/\partial x_ i f_ j) \]

The final term is zero in $A/I$. By the Cauchy-Binet formula we see that the ideal of minors for the $g_ s$ is contained in the ideal for the $f_ j$. By symmetry these ideals are the same. If $y_1, \ldots , y_ n \in \mathfrak m_ A$ is a second regular system of parameters, then the matrix $(\partial y_ j/\partial x_ i)$ is invertible and we can use the chain rule for differentiation. Some details omitted. $\square$

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