Submersion criterion for locally standard smooth morphisms

Setup. Since $f$ is of finite type, the point $x$ has an affine open neighbourhood $\mathrm{Spec}C\subseteq X$ and $y$ has an affine open neighbourhood $\mathrm{Spec}D\subseteq Y$ with $f(\mathrm{Spec}C)\subseteq \mathrm{Spec}D$ and $D\to C$ of finite type; shrinking $C$ we may suppose that $k\to C$ has a standard smooth presentation $C\cong (k[x_1,\dots ,x_N]\mathrm{/}(f_1,\dots ,f_c){)}_g$ with leading $c\times c$ minor a unit [F1], and shrinking $D$ that $k\to D$ has a standard smooth presentation $D\cong (k[y_1,\dots ,y_M]\mathrm{/}(G_1,\dots ,G_b){)}_H$ with leading $b\times b$ minor a unit. Let $\mathfrak{q}\subseteq C$ be the prime corresponding to $x$ and $\mathfrak{p}=\mathfrak{q}\cap D$ the prime corresponding to $y$; both are maximal with $C\mathrm{/}\mathfrak{q}=D\mathrm{/}\mathfrak{p}=k$, since $x$ and $y$ are $k$-rational points and the $k$-algebra maps $k[x]\mathrm{/}{Q}^{\mathrm{\prime }}\to \kappa (x)=k$ and $k[y]\mathrm{/}{P}^{\mathrm{\prime }}\to \kappa (y)=k$ have finite-type domains, hence are isomorphisms. Put $A:=D_{\mathfrak{p}}$ and $S:=C_{\mathfrak{q}}$, so that $A\to S$ is a local homomorphism of Noetherian local rings with residue field $k$ [F19], and put $F:=S\mathrm{/}{\mathfrak{m}}_{y}S$.

Converse direction: extending regular parameters. By [F3] applied over $k$ to the two charts of step 1.1, $A$ and $S$ are regular local rings; write $n=\dim A$ and $m=\dim S$. Assume now that the map ${\mathfrak{m}}_y\mathrm{/}{\mathfrak{m}}_y^2\to {\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2$ induced by $f^{\ast }$ is injective. Choose a $k$-basis of ${\mathfrak{m}}_y\mathrm{/}{\mathfrak{m}}_y^2$ and lift it to $y_1,\dots ,y_n\in {\mathfrak{m}}_y$; by [F8] the tuple $(y_1,\dots ,y_n)$ is a regular system of parameters of $A$. Its images form an independent tuple of $n$ elements of the $k$-vector space ${\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2$ of dimension $m$, which therefore extends to a $k$-basis; lifting that basis so that the first $n$ lifts are $y_1,\dots ,y_n$ and the remaining $m-n$ lifts are new elements gives $(x_1,\dots ,x_m)\in {\mathfrak{m}}_x^m$ with $x_i=y_i$ for $i\le n$, and [F8] again makes it a regular system of parameters of $S$.

The local rings and the cotangent identifications. Applying [F3] to the two standard smooth presentations of step 1.1 over the base field $k$ with $\mathfrak{p}=(0)$ and $K=k$, where the corresponding primes $Q^{\mathrm{\prime }}\subseteq k[x_1,\dots ,x_N]$ and $P^{\mathrm{\prime }}\subseteq k[y_1,\dots ,y_M]$ are maximal and hence of heights $N$ and $M$ by [F14], shows that $S$ is a regular local ring of dimension $N-c$ and that $A$ is a regular local ring of dimension $M-b$; by [F12] these integers are $m=\dim S$ and $n=\dim A$, so $N-c=m$ and $M-b=n$. Since the residue fields of $A$ and $S$ are the field $k$, a finite separable extension of $k$, [F4] gives isomorphisms ${\mathfrak{m}}_y\mathrm{/}{\mathfrak{m}}_y^2\cong {\mathrm{\Omega }}_{A\mathrm{/}k}{\otimes }_{A}k$ and ${\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2\cong {\mathrm{\Omega }}_{S\mathrm{/}k}{\otimes }_{S}k$ carrying the class of an element of the maximal ideal to $\mathrm{d}\mathrm{\square }\otimes 1$.

Forward direction: charts give freeness, flatness and the fibre dimension. Assume $f$ is locally standard smooth at $x$; after shrinking the charts of step 1.1 we may suppose that $C$ carries a standard smooth presentation over $D$ of relative dimension $d:=N^{\mathrm{\prime }}-c^{\mathrm{\prime }}$, say $C\cong (D[x_1,\dots ,x_{N^{\mathrm{\prime }}}]\mathrm{/}(f_1^{\mathrm{\prime }},\dots ,f_{c^{\mathrm{\prime }}}^{\mathrm{\prime }}){)}_{g^{\mathrm{\prime }}}$ with an invertible $c^{\mathrm{\prime }}\times c^{\mathrm{\prime }}$ Jacobian minor [F1]. By [F7] the $S$-module ${\mathrm{\Omega }}_{S\mathrm{/}A}\cong S{\otimes }_C{\mathrm{\Omega }}_{C\mathrm{/}D}$ [F6] is the cokernel of the Jacobian matrix $S^{c^{\mathrm{\prime }}}\to S^{N^{\mathrm{\prime }}}$, hence is free of rank $d$ because the minor is a unit of $S$. Base changing this presentation along $D\to A$ exhibits $S$ as a localisation of the standard smooth $A$-algebra $(A[x_1,\dots ,x_{N^{\mathrm{\prime }}}]\mathrm{/}(f_1^{\mathrm{\prime }},\dots ,f_{c^{\mathrm{\prime }}}^{\mathrm{\prime }}){)}_{g^{\mathrm{\prime }}}$ [F13], which is flat over $A$ by [F2]; localisation is flat and flatness is transitive [F17], so $S$ is flat over $A$. Finally, $F=S\mathrm{/}{\mathfrak{m}}_{y}S$ is the local ring of the fibre algebra $(k[x_1,\dots ,x_{N^{\mathrm{\prime }}}]\mathrm{/}({\bar{f}}_1^{\mathrm{\prime }},\dots ,{\bar{f}}_{c^{\mathrm{\prime }}}^{\mathrm{\prime }}){)}_{{\bar{g}}^{\mathrm{\prime }}}$ at the prime ${\mathfrak{Q}}^{\mathrm{\prime }}$ corresponding to $x$, which is maximal because its residue field is $\kappa (x)=k$; so [F3] and [F14] give $\dim F=\mathrm{ht}({\mathfrak{Q}}^{\mathrm{\prime }})-c^{\mathrm{\prime }}=N^{\mathrm{\prime }}-c^{\mathrm{\prime }}=d$.

Converse direction: flatness and a regular local fibre. The images in $S$ of the regular system of parameters $y_1,\dots ,y_n$ of $A$ are the initial segment of the regular system of parameters $(x_1,\dots ,x_m)$ of $S$ from step 2.1, so the second assertion of [F11] shows that $S$ is flat over $A$. By [F9] the tuple $(x_1,\dots ,x_n)$ is $S$-regular and $S\mathrm{/}(x_1,\dots ,x_n)$ is a regular local ring of dimension $m-n$; since $(x_1,\dots ,x_n)=(y_1,\dots ,y_n)S={\mathfrak{m}}_{y}S$, this quotient is $F=S\mathrm{/}{\mathfrak{m}}_{y}S$, the local ring of the fibre $X_y$ at $x$ [F16].

Forward direction: relative dimension $m-n$ and injectivity of the cotangent map. Composing the standard smooth presentation of $C$ over $D$ from step 2.3 with the standard smooth presentation of $D$ over $k$ from step 1.1 presents the finite-type $k$-algebra $C$ as standard smooth over $k$ with relative dimension $n+d$ [F13]; its localisation at the $k$-rational prime $\mathfrak{q}$ is $S$, so [F12] identifies that relative dimension with $\dim S=m$, whence $d=m-n$ and, by step 2.3, $\dim F=m-n$. The transitivity sequence ${\mathrm{\Omega }}_{A\mathrm{/}k}{\otimes }_{A}S\to {\mathrm{\Omega }}_{S\mathrm{/}k}\to {\mathrm{\Omega }}_{S\mathrm{/}A}\to 0$ of [F5] is right exact, and tensoring it with $S\to k$ yields, using [F4] and [F18], the exact sequence ${\mathfrak{m}}_y\mathrm{/}{\mathfrak{m}}_y^2\to {\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2\to {\mathrm{\Omega }}_{S\mathrm{/}A}{\otimes }_{S}k\to 0$, in which the first arrow is the map induced by $f^{\ast }$; since ${\mathrm{\Omega }}_{S\mathrm{/}A}\cong S^d$ the last term is a $k$-vector space of dimension $d$, so the image has dimension $m-d=n$, which equals ${\dim }_k({\mathfrak{m}}_y\mathrm{/}{\mathfrak{m}}_y^2)=n$ by step 2.2 and forces the map to be injective.

Converse direction: the fibre is geometrically regular. Write $Q^{\mathrm{\prime }}\subseteq k[x_1,\dots ,x_N]$ for the maximal ideal corresponding to $x$ in the presentation of step 1.1, and put $R^{\mathrm{\prime }}:=k[x_1,\dots ,x_N{]}_{Q^{\mathrm{\prime }}}$. The parameters $y_1,\dots ,y_n\in A=D_{\mathfrak{p}}$ generate $\mathfrak{p}{D}_{\mathfrak{p}}$. Since $D$ is Noetherian, after shrinking $\mathrm{Spec}D$ around $y$ and its inverse-image chart around $x$, we may represent every $y_i$ by an element of $D$ and arrange that $\mathfrak{p}D=(y_1,\dots ,y_n)D$ on these charts: first clear their denominators outside $\mathfrak{p}$, then invert an element outside $\mathfrak{p}$ annihilating the finite module $\mathfrak{p}D\mathrm{/}(y_1,\dots ,y_n)D$. Absorb the corresponding principal localisations into the polynomial chart of $C$. Write each image of $y_i$ in $C$ as $P_i\mathrm{/}{g}^{e_i}$ with $P_i\in k[x_1,\dots ,x_N]$ and $e_i\ge 0$. Since $g$ is a unit of $C$, the images of the numerators $P_i$ generate the same ideal as those of the $y_i$, and each $P_i$ vanishes at $x$. The finite-type fibre algebra $B:=C{\otimes }_{D}k=C\mathrm{/}\mathfrak{p}C$ is then presented on this chart by $B\cong (k[x_1,\dots ,x_N]\mathrm{/}(f_1,\dots ,f_c,P_1,\dots ,P_n){)}_g$, and its local ring at $x$ is $F\cong R^{\mathrm{\prime }}\mathrm{/}I$ for $I:=(f_1,\dots ,f_c,P_1,\dots ,P_n)R^{\mathrm{\prime }}$ [F16]. The ring $R^{\mathrm{\prime }}$ is regular local of dimension $N$ by [F3] with $c=0$ and [F14], and $\dim F=m-n$ by step 3.1, so [F10] gives ${\dim }_k((I+Q^{\mathrm{\prime }2})\mathrm{/}{Q}^{\mathrm{\prime }2})=N-\dim F=N-m+n=c+n$. The classes of the $c+n$ generators $f_1,\dots ,f_c,P_1,\dots ,P_n$ span that space, hence form a basis. By [F4] and [F7], their classes in $Q^{\mathrm{\prime }}\mathrm{/}{Q}^{\mathrm{\prime }2}$ are the $c+n$ rows of the Jacobian matrix evaluated at $Q^{\mathrm{\prime }}$, so some $(c+n)\times (c+n)$ minor $h$ does not lie in $Q^{\mathrm{\prime }}$. Localising the finite-type algebra $B$ at the image of $h$ gives a standard smooth $k$-presentation with the displayed $c+n$ equations [F1]; this open chart contains $x$. By [F3], after every field extension $K\mathrm{/}k$ every local ring of $(B_h){\otimes }_{k}K$ is regular. Every prime of the extended fibre lying over $x$ belongs to this chart because $h\mathrm{\notin }{Q}^{\mathrm{\prime }}$, so the fibre $C{\otimes }_{D}k$ is geometrically regular at $x$ in the sense of [F15].

Converse direction: concluding local standard smoothness. The $k$-algebra map $D\to C$ of step 1.1 is of finite type, hence finitely presented because $D$ is a localisation of a finite-type $k$-algebra and therefore Noetherian [F19]. Its localisation $A\to S$ is flat by step 3.1, and step 4.1 proves that the finite-type fibre $C{\otimes }_D\kappa (\mathfrak{p})$ is geometrically regular at the point induced by $\mathfrak{q}$ (its local ring there is $F=S\mathrm{/}{\mathfrak{m}}_{y}S$), so clause 1 of [F12] shows that $D\to C$ is standard smooth at $\mathfrak{q}$; that is exactly the assertion that $f$ is locally standard smooth at $x$. Together with step 3.2 this proves the equivalence of clause 1, step 2.3 and step 3.1 give flatness and the regularity and dimension $m-n$ of the fibre local ring in both directions, and steps 3.2 and 2.3 show that a witnessing chart has relative dimension $m-n$.