Dense relative-dimension strata in flat finitely presented fibres

Reduction to affine charts. The claims of the Statement are local on $X$. Choose, using the definition of local finite presentation, an affine open cover of $X$ by charts $U=\mathrm{Spec}B$ with $f(U)\subseteq V=\mathrm{Spec}A$ and $A\to B$ finitely presented; such charts exist at every point by [F1]. For a point $x\in U$ with prime $\mathfrak{q}\subseteq B$ over the prime $\mathfrak{p}\subseteq A$ of $s$, the identification of [F1] exhibits the chart fibre $\mathrm{Spec}(B{\otimes }_A\kappa (\mathfrak{p}))$ as the open subscheme $U\cap X_s$ of $X_s$, gives ${\mathcal{O}}_{X_s,x}=B_{\mathfrak{q}}\mathrm{/}\mathfrak{p}{B}_{\mathfrak{q}}$, and gives the same local dimension ${\dim }_x{X}_s$ whether computed in $U\cap X_s$ or in $X_s$ by [F2]. A subset of a topological space is open (respectively dense) exactly when its trace in every member of an open cover is so, and a union of pairwise disjoint open subsets is determined chart by chart. Hence it suffices to prove the following affine statement: for a flat finitely presented ring map $R\to B$ and a prime $\mathfrak{q}\subseteq B$ over $\mathfrak{p}$, writing $W_B=\{{\mathfrak{q}}^{\mathrm{\prime }}\in \mathrm{Spec}B:B_{{\mathfrak{q}}^{\mathrm{\prime }}}\mathrm{/}{\mathfrak{p}}^{\mathrm{\prime }}{B}_{{\mathfrak{q}}^{\mathrm{\prime }}}$ is Cohen--Macaulay$\}$ and $W_{B,d}=\{{\mathfrak{q}}^{\mathrm{\prime }}\in W_B:{\dim }_{{\mathfrak{q}}^{\mathrm{\prime }}}(B\mathrm{/}R)=d\}$, the set $W_B$ is open, each $W_{B,d}$ is open, the $W_{B,d}$ partition $W_B$, $W_B$ is dense in every nonempty fibre $\mathrm{Spec}(B{\otimes }_R\kappa (\mathfrak{p}))$, and the supremum of the labels $d$ met by a nonempty fibre is its dimension.

The polynomial chart at a point of $W$. Let $\mathfrak{q}\in W_B$ and put $n:={\dim }_{\mathfrak{q}}(B\mathrm{/}R)$, the local dimension of the fibre at $\mathfrak{q}$. Applying [F4] (AC) to the finite type map $R\to B$ at $\mathfrak{q}$ gives $g\in B\setminus \mathfrak{q}$ and an $R$-algebra map $\phi \text{\hspace{0.17em}⁣}:P:=R[T_1,\dots ,T_n]\to B_g$ that is quasi-finite. Since $B$ is finitely presented over $R$, the localisation $B_g$ is finitely presented over $P$ as well (composition of finitely presented maps, and localisation is finitely presented). The fibre ring at $\mathfrak{q}$ is unchanged by the localisation $B\to B_g$, and $g\mathrm{\notin }\mathfrak{q}$.

Fibres are locally Noetherian. In the chart of step 1.1 the fibre ring $B{\otimes }_A\kappa (\mathfrak{p})$ is generated over the field $\kappa (\mathfrak{p})$ by the images of a finite algebra generating family of $B$ over $A$ [F1]; it is therefore a finite type algebra over a field, hence Noetherian by [F8]. As the charts $U\cap X_s$ cover $X_s$, the fibre $X_s$ is locally Noetherian. In particular each local ring ${\mathcal{O}}_{X_s,x}$ is a nonzero Noetherian local ring, so the Cohen--Macaulay condition defining $W$ is meaningful by [F8].

The local dimension calculation on the closed fibre. Base change the chart of step 1.2 to $k=\kappa (\mathfrak{p})$ and write $S=B_g{\otimes }_{R}k$, $R_0=k[T_1,\dots ,T_n]$, ${\mathfrak{q}}_0$ for the selected prime of $S$, and ${\mathfrak{p}}_0$ for its contraction to $R_0$. Quasi-finiteness at ${\mathfrak{q}}_0$ gives a finite residue extension $\kappa ({\mathfrak{q}}_0)\mathrm{/}\kappa ({\mathfrak{p}}_0)$ [F3]. Let $d=\dim S_{{\mathfrak{q}}_0}$. For each minimal prime ${\mathfrak{m}}_i$ of the Cohen--Macaulay local ring $S_{{\mathfrak{q}}_0}$, [F5] makes it associated and gives $\dim (S_{{\mathfrak{q}}_0}\mathrm{/}{\mathfrak{m}}_i)=d$. Let $P_i$ be the corresponding minimal prime of $S$ contained in ${\mathfrak{q}}_0$. Apply the affine-domain dimension formula [F5] to the finite-type domain $S\mathrm{/}{P}_i$ at ${\mathfrak{q}}_0\mathrm{/}{P}_i$: its component has dimension $d+{\mathrm{trdeg}}_k\kappa ({\mathfrak{q}}_0)$. The local topological dimension at ${\mathfrak{q}}_0$ is the maximum of the dimensions of these components by [F2], and equals $n$ by the choice in step 1.2; thus $d=n-{\mathrm{trdeg}}_k\kappa ({\mathfrak{q}}_0)$. The residue extension is finite, so this transcendence degree equals that of $\kappa ({\mathfrak{p}}_0)$; applying the same affine-domain formula to the polynomial domain $R_0$ gives $n-{\mathrm{trdeg}}_k\kappa ({\mathfrak{p}}_0)=\mathrm{ht}({\mathfrak{p}}_0)=\dim (R_0{)}_{{\mathfrak{p}}_0}$. Hence $\dim S_{{\mathfrak{q}}_0}=\dim (R_0{)}_{{\mathfrak{p}}_0}=d$, which may be strictly less than the local topological dimension $n$ at a nonclosed point.

Flatness from regular parameters. The local ring $(R_0{)}_{{\mathfrak{p}}_0}$ is regular of dimension $d$; choose a regular parameter system $u_1,\dots ,u_d$. Since $S_{{\mathfrak{q}}_0}\mathrm{/}{\mathfrak{p}}_0{S}_{{\mathfrak{q}}_0}$ is a finite-dimensional local $\kappa ({\mathfrak{p}}_0)$-algebra by quasi-finiteness [F3], it is Artinian, so ${\mathfrak{p}}_0{S}_{{\mathfrak{q}}_0}$ is ${\mathfrak{q}}_0{S}_{{\mathfrak{q}}_0}$-primary. The images of the $u_i$ therefore form a system of parameters of the $d$-dimensional Cohen--Macaulay local ring $S_{{\mathfrak{q}}_0}$ by step 2.2, hence a regular sequence by [F5]. Tensoring the Koszul resolution of $\kappa ({\mathfrak{p}}_0)$ over $(R_0{)}_{{\mathfrak{p}}_0}$ with $S_{{\mathfrak{q}}_0}$ gives its Koszul complex on this regular sequence, so ${\mathrm{Tor}}_1^{(R_0{)}_{{\mathfrak{p}}_0}}(\kappa ({\mathfrak{p}}_0),S_{{\mathfrak{q}}_0})=0$ by [F5]. The published local Tor-flatness criterion [F5] applies to the finite $S_{{\mathfrak{q}}_0}$-module $S_{{\mathfrak{q}}_0}$ and yields flatness over $(R_0{)}_{{\mathfrak{p}}_0}$. For $d=0$ the parameter list is empty, the source local ring is a field and the same conclusion is immediate.

The generic point of every irreducible component of a fibre lies in $W$. Let $s\in S$ and let $C$ be an irreducible component of $X_s$ with generic point $\eta$ [F9]. By step 2.1 the fibre $X_s$ is locally Noetherian, so ${\mathcal{O}}_{X_s,\eta }$ is a Noetherian local ring. We claim its Krull dimension is $0$, i.e. that its maximal ideal is its only prime. Primes of ${\mathcal{O}}_{X_s,\eta }$ correspond to the points $\xi \in X_s$ with $\eta \in \overline{\{\xi \}}$, that is, to the generisations of $\eta$ [F9]; if $\xi$ is such a point, then $\overline{\{\xi \}}$ is a closed irreducible subset of $X_s$ containing $\eta$, hence contains $\overline{\{\eta \}}=C$, and since $C$ is an irreducible component and $\overline{\{\xi \}}$ is irreducible we get $\overline{\{\xi \}}=C$; thus $\xi$ is a generic point of the component $C$, so $\xi =\eta$. Hence $\dim {\mathcal{O}}_{X_s,\eta }=0$, and by [F8] this nonzero Noetherian local ring is Cohen--Macaulay. Therefore $\eta \in W$.

Flatness over the polynomial chart. The polynomial ring $P=R[T_1,\dots ,T_n]$ is free, hence flat, over $R$; the ring $B_g$ is finitely presented over $P$; the localisation $B_{\mathfrak{q}}$ is flat over $R_{\mathfrak{p}}$ because $B$ is flat over $R$ and localisation is exact; and step 3.1 exhibits the closed-fibre local algebra as flat over the polynomial fibre local ring. By the critère de platitude par fibres in the polynomial-chart form [F6], the local ring $B_{\mathfrak{q}}$ is flat over $P_{\mathfrak{Q}}$, where $\mathfrak{Q}=P\cap \mathfrak{q}$. This is the exact use of [F6].

The supremum of the relative dimensions over a fibre is its dimension. Let $s\in S$ with $X_s\mathrm{\ne }\mathrm{\varnothing }$. If $W_d\cap X_s\mathrm{\ne }\mathrm{\varnothing }$ and $x$ lies in it, then $d={\dim }_x{X}_s\le \dim X_s$ by [F2]; this gives one inequality, including when $\dim X_s=\mathrm{\infty }$. For the reverse inequality, cover $X_s$ by the finite-type affine fibre charts $U=\mathrm{Spec}A$ of step 2.1. Every finite chain of irreducible closed subsets $Z_0⊊\cdots ⊊Z_m$ of $X_s$ remains a strict chain after intersection with an affine chart $U$ containing a point of $Z_0$: each intersection is nonempty and irreducible, and equality of two consecutive intersections would put a nonempty open dense subset of $Z_{i+1}$ inside its proper closed subset $Z_i$. Consequently $\dim X_s={\sup }_U\dim U$, even if the fibre is not quasi-compact and this supremum is infinite. Each affine chart $U$ is finite type over $\kappa (s)$, so it has finitely many irreducible components [F9], each of finite dimension by Affine-domain dimension equals transcendence degree; choose one component $C_U$ of dimension $\dim U$ and its generic point ${\eta }_U$. Its local ring has dimension zero by step 3.2, so ${\eta }_U\in W$, and [F2] applied inside $U$ gives ${\dim }_{{\eta }_U}{X}_s={\dim }_{{\eta }_U}U=\dim C_U=\dim U$. Thus $W_{\dim U}\cap X_s\mathrm{\ne }\mathrm{\varnothing }$ for every affine chart $U$, proving $\sup \{d:W_d\cap X_s\mathrm{\ne }\mathrm{\varnothing }\}\ge {\sup }_U\dim U=\dim X_s$. If $\dim X_s<\mathrm{\infty }$, a nonempty subset of $\mathbb{N}$ with finite supremum attains it, so the supremum is a largest value in that case.

Spreading flatness and quasi-finiteness out. The map $P\to B_g$ is finitely presented and $B_{\mathfrak{q}}$ is flat over $P_{\mathfrak{Q}}$ by step 4.1, so the openness of the flat locus [F7] gives $h_1\in B_g\setminus \mathfrak{q}$ with $(B_g{)}_{h_1}$ flat over $P$; this is the exact use of [F7]. The map $P\to B_g$ is of finite type and quasi-finite at $\mathfrak{q}$ by step 1.2, so openness of the quasi-finite locus [F3] gives $h_2\in B_g\setminus \mathfrak{q}$ such that $P\to (B_g{)}_{h_2}$ is quasi-finite at every prime, and this remains true after base change to each residue field by [F3]. Replacing $B$ by $B_{g{h}_1{h}_2}$, which by [F3] leaves the fibre ring and the local fibre dimension at every prime of $B$ not containing $g{h}_1{h}_2$ unchanged, we may assume for the remainder of the affine proof: $R\to B$ is flat and finitely presented, $P=R[T_1,\dots ,T_n]\to B$ arises from a quasi-finite map and is flat at every prime of $B$, and $P\to B$ is quasi-finite at every prime of $B$.

Openness of $W_B$ and of every $W_{B,d}$. Work in the situation of step 5.1 and let ${\mathfrak{q}}^{\mathrm{\prime }}\in \mathrm{Spec}B$, with images ${\mathfrak{p}}^{\mathrm{\prime }}=R\cap {\mathfrak{q}}^{\mathrm{\prime }}$ and ${\mathfrak{Q}}^{\mathrm{\prime }}=P\cap {\mathfrak{q}}^{\mathrm{\prime }}$. After base change to $k^{\mathrm{\prime }}=\kappa ({\mathfrak{p}}^{\mathrm{\prime }})$, the local map $k^{\mathrm{\prime }}[T_1,\dots ,T_n{]}_{{\mathfrak{Q}}^{\mathrm{\prime }}}\to B_{{\mathfrak{q}}^{\mathrm{\prime }}}\mathrm{/}{\mathfrak{p}}^{\mathrm{\prime }}{B}_{{\mathfrak{q}}^{\mathrm{\prime }}}$ is flat and quasi-finite by step 5.1 and [F3]. The polynomial source is a regular local ring and the closed fibre of this quasi-finite map is zero-dimensional and Cohen--Macaulay, so the flat-local Cohen--Macaulay criterion [F5] makes the target fibre local ring Cohen--Macaulay. Put $h=\mathrm{ht}({\mathfrak{Q}}^{\mathrm{\prime }})$. Going down for the flat local map gives target local-ring dimension at least $h$. For the reverse bound apply the finite integral factorization in [F5] to the quasi-finite $k^{\mathrm{\prime }}[T_1,\dots ,T_n]$-algebra $S^{\mathrm{\prime }}=B{\otimes }_R{k}^{\mathrm{\prime }}$ after the localization of step 5.1. It gives a finite $k^{\mathrm{\prime }}[T_1,\dots ,T_n]$-algebra $T$ and a principal open $D_T(a)$ containing the contraction $\mathfrak{r}$ of the selected prime ${\mathfrak{q}}^{\mathrm{\prime }\mathrm{\prime }}$ of $S^{\mathrm{\prime }}$, with $T_a\cong S_a^{\mathrm{\prime }}$. Hence $S_{{\mathfrak{q}}^{\mathrm{\prime }\mathrm{\prime }}}^{\mathrm{\prime }}\cong T_{\mathfrak{r}}$. Since $T$ is integral over the image of the polynomial ring, distinct comparable primes of $T$ have distinct contractions (incomparability); every strict prime chain in $T_{\mathfrak{r}}$ therefore contracts to a strict chain ending at ${\mathfrak{Q}}^{\mathrm{\prime }}$, so its length is at most $h$. Thus its dimension is $h$. The residue extension is finite, and the CM component and affine-domain dimension calculation of step 2.2, now read in reverse, gives local topological dimension $h+{\mathrm{trdeg}}_{k^{\mathrm{\prime }}}\kappa ({\mathfrak{Q}}^{\mathrm{\prime }})=n$. Therefore every point of this chart lies in $W_B\cap W_{B,n}$. Taking unions over the polynomial charts centered at points of $W_B$ proves that $W_B$ and every $W_{B,d}$ are open.

Density of $W$ in every fibre. Let $s\in S$ and let $C$ be an irreducible component of $X_s$. By step 3.2 its generic point $\eta$ belongs to $W\cap X_s$. Since the closure of $\{\eta \}$ in $X_s$ is $C$, the closure of $W\cap X_s$ contains $C$. Every point of the locally Noetherian fibre $X_s$ lies in an irreducible component [F9], so the closure of $W\cap X_s$ is all of $X_s$. This proves density without using the openness assertion of step 6.1.

The partition $W={⨆}_d{W}_d$. By definition each $x\in W$ lies in exactly one $W_d$, namely the one with $d={\dim }_x{X}_{f(x)}$, because the local dimension is a well-defined nonnegative integer. Hence the $W_d$ are pairwise disjoint and their union is $W$; each is open by step 6.1.

Choice accounting and conclusion. The Axiom of Choice is declared in the Statement and is used exactly through the openness of the quasi-finite locus [F3], the polynomial-chart lemma [F4], the flatness-by-fibres criterion [F6], and the flat-locus openness theorem [F7]; the finitely many chart, element and component selections made in the proof use no further choice, and the reductions of steps 1.1 and 5.1 are finite. Steps 2.1 and 6.1 establish clause 1 of the Statement, steps 3.2 and 7.1 establish clause 2, and steps 6.1, 7.2 and 4.2 establish clause 3; all three clauses are proved under the stated hypotheses; steps 2.1, 3.2, 4.2 and 7.1 do not require the flat-locus openness input, while steps 5.1–6.1 establish the openness claims using [F7]. [F3, F4, F5, F6, F7, F10, step 2.1, step 6.1, step 7.1, step 7.2, step 4.2] $\mathrm{\square }$