The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

10.112 The dimension formula

Recall the definitions of catenary (Definition 10.104.1) and universally catenary (Definition 10.104.3).

Lemma 10.112.1. Let $R \to S$ be a ring map. Let $\mathfrak q$ be a prime of $S$ lying over the prime $\mathfrak p$ of $R$. Assume that

  1. $R$ is Noetherian,

  2. $R \to S$ is of finite type,

  3. $R$, $S$ are domains, and

  4. $R \subset S$.

Then we have

\[ \text{height}(\mathfrak q) \leq \text{height}(\mathfrak p) + \text{trdeg}_ R(S) - \text{trdeg}_{\kappa (\mathfrak p)} \kappa (\mathfrak q) \]

with equality if $R$ is universally catenary.

Proof. Suppose that $R \subset S' \subset S$ is a finitely generated $R$-subalgebra of $S$. In this case set $\mathfrak q' = S' \cap \mathfrak q$. The lemma for the ring maps $R \to S'$ and $S' \to S$ implies the lemma for $R \to S$ by additivity of transcendence degree in towers of fields (Fields, Lemma 9.26.5). Hence we can use induction on the number of generators of $S$ over $R$ and reduce to the case where $S$ is generated by one element over $R$.

Case I: $S = R[x]$ is a polynomial algebra over $R$. In this case we have $\text{trdeg}_ R(S) = 1$. Also $R \to S$ is flat and hence

\[ \dim (S_{\mathfrak q}) = \dim (R_{\mathfrak p}) + \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) \]

see Lemma 10.111.7. Let $\mathfrak r = \mathfrak pS$. Then $\text{trdeg}_{\kappa (\mathfrak p)} \kappa (\mathfrak q) = 1$ is equivalent to $\mathfrak q = \mathfrak r$, and implies that $\dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) = 0$. In the same vein $\text{trdeg}_{\kappa (\mathfrak p)} \kappa (\mathfrak q) = 0$ is equivalent to having a strict inclusion $\mathfrak r \subset \mathfrak q$, which implies that $\dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}) = 1$. Thus we are done with case I with equality in every instance.

Case II: $S = R[x]/\mathfrak n$ with $\mathfrak n \not= 0$. In this case we have $\text{trdeg}_ R(S) = 0$. Denote $\mathfrak q' \subset R[x]$ the prime corresponding to $\mathfrak q$. Thus we have

\[ S_{\mathfrak q} = (R[x])_{\mathfrak q'}/\mathfrak n(R[x])_{\mathfrak q'} \]

By the previous case we have $\dim ((R[x])_{\mathfrak q'}) = \dim (R_{\mathfrak p}) + 1 - \text{trdeg}_{\kappa (\mathfrak p)} \kappa (\mathfrak q)$. Since $\mathfrak n \not= 0$ we see that the dimension of $S_{\mathfrak q}$ decreases by at least one, see Lemma 10.59.12, which proves the inequality of the lemma. To see the equality in case $R$ is universally catenary note that $\mathfrak n \subset R[x]$ is a height one prime as it corresponds to a nonzero prime in $F[x]$ where $F$ is the fraction field of $R$. Hence any maximal chain of primes in $S_\mathfrak q = R[x]_{\mathfrak q'}/\mathfrak nR[x]_{\mathfrak q'}$ corresponds to a maximal chain of primes with length 1 greater between $\mathfrak q'$ and $(0)$ in $R[x]$. If $R$ is universally catenary these all have the same length equal to the height of $\mathfrak q'$. This proves that $\dim (S_\mathfrak q) = \dim (R[x]_{\mathfrak q'}) - 1$ and this implies equality holds as desired. $\square$

The following lemma says that generically finite maps tend to be quasi-finite in codimension $1$.

Lemma 10.112.2. Let $A \to B$ be a ring map. Assume

  1. $A \subset B$ is an extension of domains,

  2. the induced extension of fraction fields is finite,

  3. $A$ is Noetherian, and

  4. $A \to B$ is of finite type.

Let $\mathfrak p \subset A$ be a prime of height $1$. Then there are at most finitely many primes of $B$ lying over $\mathfrak p$ and they all have height $1$.

Proof. By the dimension formula (Lemma 10.112.1) for any prime $\mathfrak q$ lying over $\mathfrak p$ we have

\[ \dim (B_{\mathfrak q}) \leq \dim (A_{\mathfrak p}) - \text{trdeg}_{\kappa (\mathfrak p)} \kappa (\mathfrak q). \]

As the domain $B_\mathfrak q$ has at least $2$ prime ideals we see that $\dim (B_{\mathfrak q}) \geq 1$. We conclude that $\dim (B_{\mathfrak q}) = 1$ and that the extension $\kappa (\mathfrak p) \subset \kappa (\mathfrak q)$ is algebraic. Hence $\mathfrak q$ defines a closed point of its fibre $\mathop{\mathrm{Spec}}(B \otimes _ A \kappa (\mathfrak p))$, see Lemma 10.34.9. Since $B \otimes _ A \kappa (\mathfrak p)$ is a Noetherian ring the fibre $\mathop{\mathrm{Spec}}(B \otimes _ A \kappa (\mathfrak p))$ is a Noetherian topological space, see Lemma 10.30.5. A Noetherian topological space consisting of closed points is finite, see for example Topology, Lemma 5.9.2. $\square$


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