Local fibre-dimension bound from polynomial quasi-finiteness

Write $\kappa =\kappa (\mathfrak{p})$ and $\bar{S}=B{\otimes }_A\kappa$, a finite-type $\kappa$-algebra with a prime $\bar{\mathfrak{q}}$ corresponding to $\mathfrak{q}$, whose spectrum has the subspace topology from $\mathrm{Spec}B$ by [F14]; by the local-dimension convention of [F5] the point $\bar{\mathfrak{q}}$ has an open neighbourhood $U$ of dimension exactly $n$ (the infimum of the dimensions of the open neighbourhoods is attained because these dimensions are natural numbers: the fibre is finite type over the field $\kappa$, so each of its points has an affine neighbourhood of finite dimension by [F6], [F7] and [F8]), every open neighbourhood of $\bar{\mathfrak{q}}$ has dimension at least $n$, and since basic opens form a base for the induced topology there is $g_1\in B\setminus \mathfrak{q}$ with $\bar{\mathfrak{q}}\in D(g_1)\cap \mathrm{Spec}\bar{S}\subseteq U$.

The set $D(g_1)\cap \mathrm{Spec}\bar{S}=\mathrm{Spec}({\bar{S}}_{g_1})$ is an open neighbourhood of $\bar{\mathfrak{q}}$, so it has dimension at least $n$ by the convention of [F5] and at most $\dim U=n$; replacing $B$ by $B_{g_1}$ (whose fibre over $\mathfrak{p}$ is $\mathrm{Spec}{\bar{S}}_{g_1}$, localisation commuting with the tensor product) we may assume for the rest of the proof that the fibre $\bar{S}$ has dimension exactly $n$.

Apply Noether normalisation [F6] to the finite-type $\kappa$-algebra $\bar{S}$ of dimension $n$: there are algebraically independent $z_1,\dots ,z_d\in \bar{S}$ with $\bar{S}$ module-finite over $\kappa [z_1,\dots ,z_d]$, the extension $\kappa [z_1,\dots ,z_d]\subseteq \bar{S}$ is injective and integral, so $\dim \bar{S}=\dim \kappa [z_1,\dots ,z_d]=d$ by [F7] and [F8], and $d=n$.

By [F14], every $z_i\in \bar{S}=(B\mathrm{/}\mathfrak{p}B{)}_{A\setminus \mathfrak{p}}$ can be written $z_i={\bar{y}}_i\mathrm{/}{\bar{a}}_i$ for $y_i\in B$ and $a_i\in A\setminus \mathfrak{p}$. Put $w_i={\bar{a}}_i{z}_i={\bar{y}}_i$. The scalars ${\bar{a}}_i$ are nonzero in $\kappa$, so $\kappa [w_1,\dots ,w_n]=\kappa [z_1,\dots ,z_n]$ and the $w_i$ are algebraically independent. Thus the map $\phi \text{\hspace{0.17em}⁣}:A[T_1,\dots ,T_n]\to B$, $T_i\mapsto y_i$, becomes finite after tensoring with $\kappa$. It is itself of finite type because any finite list of $A$-algebra generators of $B$ also generates it over $A[T_1,\dots ,T_n]$. If $\mathfrak{r}={\phi }^{-1}(\mathfrak{q})$, then $\mathfrak{r}\cap A=\mathfrak{p}$, and the fibre of $\phi$ at $\mathfrak{r}$ is the corresponding fibre of this finite $\kappa [T_1,\dots ,T_n]$-algebra. It is finite-dimensional over $\kappa (\mathfrak{r})$, as is its localization at the point of $\mathfrak{q}$, so [F1] proves quasi-finiteness at $\mathfrak{q}$. For $n=0$ the lists are empty and the same argument applies.

By [F9] the quasi-finite locus of $\phi$ is open in $\mathrm{Spec}B$ and contains $\mathfrak{q}$, so there is $g_2\in B\setminus \mathfrak{q}$ such that $\phi$ is quasi-finite at every prime of $B_{g_2}$; write $g_2=b\mathrm{/}{g}_1^N$ in the original ring localized at $g_1$, with $b\mathrm{\notin }\mathfrak{q}$, and put $g=g_{1}b$, still not in $\mathfrak{q}$. Then the induced $A$-algebra map ${\phi }_g\text{\hspace{0.17em}⁣}:A[T_1,\dots ,T_n]\to B_g$ is quasi-finite at every prime, which is assertion 1.

Now let ${\mathfrak{q}}^{\mathrm{\prime }}\in \mathrm{Spec}{B}_g$, put ${\mathfrak{p}}^{\mathrm{\prime }}={\mathfrak{q}}^{\mathrm{\prime }}\cap A$, ${\kappa }^{\mathrm{\prime }}=\kappa ({\mathfrak{p}}^{\mathrm{\prime }})$ and $S^{\mathrm{\prime }}=B_g{\otimes }_A{\kappa }^{\mathrm{\prime }}\cong B_g{\otimes }_{A[T_1,\dots ,T_n]}{\kappa }^{\mathrm{\prime }}[T_1,\dots ,T_n]$, where $A[T_1,\dots ,T_n]\to {\kappa }^{\mathrm{\prime }}[T_1,\dots ,T_n]$ extends $A\to {\kappa }^{\mathrm{\prime }}$ and preserves the variables. Applying [F15] to the base change of ${\phi }_g$ along this polynomial-ring map shows that $\psi \text{\hspace{0.17em}⁣}:{\kappa }^{\mathrm{\prime }}[T_1,\dots ,T_n]\to S^{\mathrm{\prime }}$ is of finite type and quasi-finite at the prime corresponding to ${\mathfrak{q}}^{\mathrm{\prime }}$.

Since ${\kappa }^{\mathrm{\prime }}$ is a field, $S^{\mathrm{\prime }}$ is a finite-type ${\kappa }^{\mathrm{\prime }}$-algebra, and $\psi$ is quasi-finite at every prime because the prime of step 6.1 was arbitrary; applying [F10] to $\psi$ and the prime ${\mathfrak{q}}^{\mathrm{\prime }\mathrm{\prime }}$ of $S^{\mathrm{\prime }}$ corresponding to ${\mathfrak{q}}^{\mathrm{\prime }}$ gives a finite ${\kappa }^{\mathrm{\prime }}[T_1,\dots ,T_n]$-subalgebra $T$ of the relative integral closure and an element $g^{\mathrm{\prime }\mathrm{\prime }}\in T$, $g^{\mathrm{\prime }\mathrm{\prime }}\mathrm{\notin }{\mathfrak{q}}^{\mathrm{\prime }\mathrm{\prime }}\cap T$, with $T_{g^{\mathrm{\prime }\mathrm{\prime }}}\cong (S^{\mathrm{\prime }}{)}_{g^{\mathrm{\prime }\mathrm{\prime }}}$, so that $S_{{\mathfrak{q}}^{\mathrm{\prime }\mathrm{\prime }}}^{\mathrm{\prime }}\cong (S_{g^{\mathrm{\prime }\mathrm{\prime }}}^{\mathrm{\prime }}{)}_{{\mathfrak{q}}^{\mathrm{\prime }\mathrm{\prime }}}\cong (T_{g^{\mathrm{\prime }\mathrm{\prime }}}{)}_{{\mathfrak{q}}^{\mathrm{\prime }\mathrm{\prime }}}$ is a localisation of $T$.

Let $R^{\mathrm{\prime }\mathrm{\prime }}\subseteq T$ be the image of ${\kappa }^{\mathrm{\prime }}[T_1,\dots ,T_n]$; the extension $R^{\mathrm{\prime }\mathrm{\prime }}\subseteq T$ is injective and integral, so $\dim T=\dim R^{\mathrm{\prime }\mathrm{\prime }}$ by [F7], while $R^{\mathrm{\prime }\mathrm{\prime }}$ is a quotient of the polynomial ring ${\kappa }^{\mathrm{\prime }}[T_1,\dots ,T_n]$, so $\dim R^{\mathrm{\prime }\mathrm{\prime }}\le \dim {\kappa }^{\mathrm{\prime }}[T_1,\dots ,T_n]=n$ by [F12] and [F8]; hence $\dim T\le n$ and $\dim S_{{\mathfrak{q}}^{\mathrm{\prime }\mathrm{\prime }}}^{\mathrm{\prime }}\le \dim T\le n$ by [F11], a localisation of $T$ not increasing dimension.

Every strict chain of primes of $S^{\mathrm{\prime }}$ ends at some prime, and a chain ending at a prime $\mathfrak{r}$ has length at most $\mathrm{ht}(\mathfrak{r})=\dim (S_{\mathfrak{r}}^{\mathrm{\prime }})\le n$ by [F2] and [F3]; taking the supremum over chains gives $\dim S^{\mathrm{\prime }}\le n$.

The ring $S^{\mathrm{\prime }}$ is finite type over the field ${\kappa }^{\mathrm{\prime }}$, hence Noetherian by [F16] and [F17], so $\mathrm{Spec}{S}^{\mathrm{\prime }}$ is a Noetherian topological space by [F18] and its local dimension at the point corresponding to ${\mathfrak{q}}^{\mathrm{\prime }}$ is the infimum of the dimensions of the open neighbourhoods of that point [F5]; the whole space $\mathrm{Spec}{S}^{\mathrm{\prime }}$ is one of these neighbourhoods, so that local dimension is at most $\dim S^{\mathrm{\prime }}\le n$.

For ${\mathfrak{q}}^{\mathrm{\prime }}\in D(g)$ the fibre of $\mathrm{Spec}B\to \mathrm{Spec}A$ over ${\mathfrak{p}}^{\mathrm{\prime }}$ is $\mathrm{Spec}(B{\otimes }_A\kappa ({\mathfrak{p}}^{\mathrm{\prime }}))$ by [F14], and the fibre of $\mathrm{Spec}{B}_g$ over ${\mathfrak{p}}^{\mathrm{\prime }}$ is its open subscheme $D(g)$ with the same local dimension at ${\mathfrak{q}}^{\mathrm{\prime }}$, because $\mathrm{Spec}(B_g{\otimes }_A\kappa ({\mathfrak{p}}^{\mathrm{\prime }}))$ is the open subscheme $D(g)\cap \mathrm{Spec}(B{\otimes }_A\kappa ({\mathfrak{p}}^{\mathrm{\prime }}))$ of the fibre and the local dimension at a point is unchanged on passing to an open neighbourhood (open neighbourhoods inside the open piece give the same infimum [F5]); by step 10.1 this local dimension is at most $n$, and since $D(g)$ is an open neighbourhood of $\mathfrak{q}$ in $\mathrm{Spec}B$, assertion 2 holds with $V=D(g)$. The Axiom of Choice [F13] licenses the cited results, in particular [F7], [F9], [F10] and [F18]; the proof makes finitely many choices of preimages and localising elements. [F4, F5, F13, F14, step 5.1, step 10.1] $\mathrm{\square }$