Invertible Jacobian minor gives regular parameters in a polynomial fibre

The field $k$ is a regular Noetherian ring. Its ideals are $0$ and $k$: a nonzero ideal contains a nonzero element, which is a unit by [F12], hence contains $1$ and equals $k$. Both ideals are finitely generated, so $k$ is Noetherian by [F11]. The only prime ideal of $k$ is $(0)$: it is prime because $k\mathrm{/}(0)\cong k$ is a domain and every nonzero ideal equals $k$, which is not prime; so $k$ has exactly one maximal ideal, namely $(0)$, and is a local ring in the sense of [F14] with residue field $k$. There is no strict chain of primes, so $\dim k=0$ by [F13]; and $\mathrm{edim}k={\dim }_k((0)\mathrm{/}(0{)}^2)={\dim }_{k}0=0=\dim k$, so $k$ is regular local by [F3]. Every prime localisation of $k$ is $k$ itself, hence regular local, so $k$ is a regular Noetherian ring by [F4].

The local ring $A$ is regular. By [F2] applied to the regular Noetherian ring $k$ of step 1.1, the polynomial ring $P=k[x_1,\dots ,x_n]$ is regular, and its localisation $A=P_{\mathfrak{q}}$ at the prime $\mathfrak{q}$ is regular as well; by [F4] this says that every prime localisation of the Noetherian ring $A$ is a regular local ring, in particular $A$ itself, whose only maximal ideal is $\mathfrak{m}=\mathfrak{q}A$ by [F16]. Hence $A$ is a regular local ring with residue field $\kappa =A\mathrm{/}\mathfrak{m}$ [F17], and $\dim A=\mathrm{edim}A={\dim }_{\kappa }(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)$ by [F3].

The differential map $\delta \text{\hspace{0.17em}⁣}:\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2\to {\kappa }^n$. Localising the exact sequence of $P$-modules ${\mathfrak{q}}^2\to \mathfrak{q}\to \mathfrak{q}\mathrm{/}{\mathfrak{q}}^2\to 0$ at $P\setminus \mathfrak{q}$ and using [F9] identifies $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2=(\mathfrak{q}\mathrm{/}{\mathfrak{q}}^2){\otimes }_{P}A$. The assignment $(F<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>{\mathfrak{q}}^2,a)\mapsto a\cdot ({\mathrm{\partial }}_{i}f<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathfrak{q}{)}_i$ is $P$-balanced: for $F\in \mathfrak{q}$ and $G\in P$ the Leibniz rule of [F1] gives ${\mathrm{\partial }}_i(FG)-G{\mathrm{\partial }}_{i}F=F{\mathrm{\partial }}_{i}G\in \mathfrak{q}$, and for $F\in {\mathfrak{q}}^2$ one has ${\mathrm{\partial }}_{i}F\in \mathfrak{q}$ by the same rule applied to a product of two elements of $\mathfrak{q}$. Hence it induces an $A$-linear map $(\mathfrak{q}\mathrm{/}{\mathfrak{q}}^2){\otimes }_{P}A\to {\kappa }^n$, that is, a $\kappa$-linear map $\delta$ on $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$, which sends the class of $f_j$ to the $j$-th Jacobian column $({\mathrm{\partial }}_i{f}_j<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathfrak{q}{)}_i$.

The classes of $f_1,\dots ,f_c$ are linearly independent. Let ${\lambda }_1,\dots ,{\lambda }_c\in A$ satisfy ${\sum }_j{\lambda }_j\overline{f_j}=0$ in $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$. Applying $\delta$ of step 3.1 and using its $\kappa$-linearity gives ${\sum }_j\overline{{\lambda }_j}({\mathrm{\partial }}_i{f}_j<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathfrak{q}{)}_i=0$ in ${\kappa }^n$. The first $c$ coordinates are the matrix equation ${\bar{M}}^{\mathsf{T}}\bar{\lambda }=0$, where $\bar{M}$ is the image in $\kappa$ of the $c\times c$ matrix $({\mathrm{\partial }}_i{f}_j{)}_{1\le i,j\le c}$; its determinant is the image of $h$, which is nonzero because $h\mathrm{\notin }\mathfrak{q}$ and $\kappa =k[x{]}_{\mathfrak{q}}\mathrm{/}\mathfrak{q}k[x{]}_{\mathfrak{q}}$ has kernel exactly $\mathfrak{q}$ on $P$. Hence $\bar{M}$ is invertible over the field $\kappa$ and $\bar{\lambda }=0$, that is, every ${\lambda }_j\in \mathfrak{m}$. Therefore no nontrivial $\kappa$-linear relation exists among the classes of $f_1,\dots ,f_c$.

A regular system of parameters. Put $d:=\dim A={\dim }_{\kappa }(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)$ by step 2.1. By step 4.1 the family of classes $(\overline{f_1},\dots ,\overline{f_c})$ in the $\kappa$-vector space $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$ is linearly independent, so by [F8] it extends to a $\kappa$-basis $(\overline{f_1},\dots ,\overline{f_c},\overline{z_{c+1}},\dots ,\overline{z_d})$ with $z_i\in \mathfrak{m}$; here $c\le d$ because the independent family has at most ${\dim }_{\kappa }(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)=d$ members. Define $y_i:=f_i$ for $1\le i\le c$ and $y_i:=z_i$ for $c<i\le d$. The classes of $y_1,\dots ,y_d$ form a $\kappa$-basis of $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$, so $(y_1,\dots ,y_d)$ is a regular system of parameters of $A$ by [F5], of length $d=\dim A$ as required by [F7].

Conclusion. By [F6] applied to the regular local ring $A$ of step 2.1 and its regular system of parameters $(y_1,\dots ,y_d)$ of step 5.1, the tuple $(y_1,\dots ,y_d)$ is an $A$-regular sequence and $A\mathrm{/}(y_1,\dots ,y_c)$ is a regular local ring of dimension $d-c=\dim A-c$. Since $y_i=f_i$ for $i\le c$, the quotient $A\mathrm{/}(y_1,\dots ,y_c)$ is $A\mathrm{/}(f_1,\dots ,f_c)$ and the initial segment $(f_1,\dots ,f_c)$ is a regular sequence. This proves both assertions.