Jacobian criterion and openness of the regular locus over a perfect field

The closed-point criterion. Let $\mathfrak{m}\subseteq A$ be maximal with $\kappa =A\mathrm{/}\mathfrak{m}$ finite separable over $k$. The local ring $A_{\mathfrak{m}}$ is a Noetherian local $k$-algebra with residue field $\kappa$, which is finitely generated and separably generated over the perfect field $k$ by [F3]; since $\kappa \mathrm{/}k$ is finite separable, [F2] gives $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2\cong {\mathrm{\Omega }}_{A_{\mathfrak{m}}\mathrm{/}k}{\otimes }_{A_{\mathfrak{m}}}\kappa$, and ${\mathrm{\Omega }}_{A_{\mathfrak{m}}\mathrm{/}k}\otimes \kappa \cong {\mathrm{\Omega }}_{A\mathrm{/}k}{\otimes }_A\kappa$. By [F1] and [F4] the latter is the cokernel of the matrix $J(\mathfrak{m})$ acting from ${\kappa }^r$ to ${\kappa }^n$, so its dimension is $n-{\mathrm{rank}}_{\kappa }J(\mathfrak{m})$. Hence $\mathrm{edim}{A}_{\mathfrak{m}}=n-{\mathrm{rank}}_{\kappa }J(\mathfrak{m})$ by [F8], and by [F8] again $A_{\mathfrak{m}}$ is regular if and only if $\mathrm{edim}{A}_{\mathfrak{m}}=\dim A_{\mathfrak{m}}$, that is, if and only if $\mathrm{rank}J(\mathfrak{m})=n-\dim A_{\mathfrak{m}}$; both $\mathrm{edim}$ and $\dim$ are intrinsic, so the rank is independent of the generating set of $I$.

Local charts at regular points. Let $\mathfrak{q}\in \mathrm{Spec}A$ with $A_{\mathfrak{q}}$ regular, let $\mathfrak{p}=\mathfrak{q}\cap P$ correspond to $\mathfrak{q}$, and put $R:=P_{\mathfrak{p}}$, a regular local ring of dimension $d:=\mathrm{ht}(\mathfrak{p})$ by [F7], with ideal $I^{\mathrm{\prime }}:=IR$ and quotient $R\mathrm{/}{I}^{\mathrm{\prime }}=A_{\mathfrak{q}}$ of dimension $\dim A_{\mathfrak{q}}$. By [F6] there are $y_1,\dots ,y_c\in I^{\mathrm{\prime }}$ forming an initial part of a regular system of parameters of $R$, with $c=d-\dim A_{\mathfrak{q}}$, and generating $I^{\mathrm{\prime }}$. Put $\mathfrak{n}:=\mathfrak{p}R$. The classes of the $y_j$ form a basis of $(I^{\mathrm{\prime }}+{\mathfrak{n}}^2)\mathrm{/}{\mathfrak{n}}^2$: they span this subspace because they generate $I^{\mathrm{\prime }}$, and are linearly independent as part of a cotangent basis by [F5], with dimension $c$ by [F6]. Write $y_j=g_j\mathrm{/}{u}_j$ with $g_j\in I$ and $u_j\in P\setminus \mathfrak{p}$. Since each $u_j$ is a unit of $R$, the $g_j$ still generate $I^{\mathrm{\prime }}$ and their classes are nonzero scalar multiples of the corresponding $y_j$ classes, hence linearly independent. The map $\mathfrak{n}\mathrm{/}{\mathfrak{n}}^2\to {\mathrm{\Omega }}_{R\mathrm{/}k}{\otimes }_R\kappa (\mathfrak{p})\cong \kappa (\mathfrak{p}{)}^n$ is injective by [F2], the residue field $\kappa (\mathfrak{p})$ being finitely generated and separably generated over the perfect field $k$ by [F3], and the images of the classes of the $g_j$ are the vectors $(\mathrm{\partial }{g}_j\mathrm{/}\mathrm{\partial }{x}_i<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathfrak{p}{)}_i$ by [F1]; these vectors are therefore linearly independent over $\kappa (\mathfrak{p})$, so some $c\times c$ minor $h$ of the Jacobian matrix of $g_1,\dots ,g_c$ satisfies $h\mathrm{\notin }\mathfrak{p}$. Since $I$ is finitely generated and $I^{\mathrm{\prime }}=(g_1,\dots ,g_c)R$, there is $s\mathrm{\notin }\mathfrak{p}$ with $sI\subseteq (g_1,\dots ,g_c)$; put $t:=sh\in A$. By [F10] the element $t$ is not in $\mathfrak{q}$ and becomes a unit in $A_t$, and in $A_t$ the ideal $I$ is generated by $g_1,\dots ,g_c$ while $h$ is a unit, so $A_t\cong (P\mathrm{/}(g_1,\dots ,g_c){)}_t$ is a standard smooth $k$-algebra with $n$ variables and $c$ equations.

Openness of the regular locus. In the situation of step 1.2, [F9] applied over the field $k$ shows that every local ring of the standard smooth $k$-algebra $A_t$ is regular; hence the whole distinguished open $D(t)$ consists of regular points and is a neighbourhood of $\mathfrak{q}$. As $\mathfrak{q}$ was an arbitrary regular point, the regular locus is open in $\mathrm{Spec}A$.

Density along a generically reduced component. Let $\mathfrak{p}$ be a minimal prime of $A$ with $A_{\mathfrak{p}}$ reduced. By [F11] the local ring $A_{\mathfrak{p}}$ is Noetherian local of dimension $0$, so all its primes are maximal and it is Artinian by [F12]; its maximal ideal is therefore nilpotent by [F12], and reducedness forces it to be zero, so $A_{\mathfrak{p}}$ is a field, in particular regular. Step 2.1 then provides $t\mathrm{\notin }\mathfrak{p}$ with $D(t)$ contained in the regular locus, and $D(t)\cap V(\mathfrak{p})$ is a nonempty open subset of the irreducible space $V(\mathfrak{p})$, hence dense in $V(\mathfrak{p})$ by [F13]. When $A$ is reduced, every localisation $A_{\mathfrak{p}}$ at a minimal prime is reduced, so this applies to every irreducible component.

The three assertions are steps 1.1, 1.2 and 2.1 with the density statement of step 3.1; all of them use only the perfectness of $k$ through [F3] and [F2], and no later local-presentation theorem.