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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