## 52.18 A distance function

Let $Y$ be a Noetherian scheme and let $Z \subset Y$ be a closed subset. We define a function

52.18.0.1
\begin{equation} \label{algebraization-equation-delta-Z} \delta ^ Y_ Z = \delta _ Z : Y \longrightarrow \mathbf{Z}_{\geq 0} \cup \{ \infty \} \end{equation}

which measures the “distance” of a point of $Y$ from $Z$. For an informal discussion, please see Remark 52.18.3. Let $y \in Y$. We set $\delta _ Z(y) = \infty$ if $y$ is contained in a connected component of $Y$ which does not meet $Z$. If $y$ is contained in a connected component of $Y$ which meets $Z$, then we can find $k \geq 0$ and a system

$V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \supset V_ k \subset W_ k$

of integral closed subschemes of $Y$ such that $V_0 \subset Z$ and $y \in V_ k$ is the generic point. Set $c_ i = \text{codim}(V_ i, W_ i)$ for $i = 0, \ldots , k$ and $b_ i = \text{codim}(V_{i + 1}, W_ i)$ for $i = 0, \ldots , k - 1$. For such a system we set

$\delta (V_0, W_0, V_1, \ldots , W_ k) = k + \max _{i = 0, 1, \ldots , k} (c_ i + c_{i + 1} + \ldots + c_ k - b_ i - b_{i + 1} - \ldots - b_{k - 1})$

This is $\geq k$ as we can take $i = k$ and we have $c_ k \geq 0$. Finally, we set

$\delta _ Z(y) = \min \delta (V_0, W_0, V_1, \ldots , W_ k)$

where the minimum is over all systems of integral closed subschemes of $Y$ as above.

Lemma 52.18.1. Let $Y$ be a Noetherian scheme and let $Z \subset Y$ be a closed subset.

1. For $y \in Y$ we have $\delta _ Z(y) = 0 \Leftrightarrow y \in Z$.

2. The subsets $\{ y \in Y \mid \delta _ Z(y) \leq k\}$ are stable under specialization.

3. For $y \in Y$ and $z \in \overline{\{ y\} } \cap Z$ we have $\dim (\mathcal{O}_{\overline{\{ y\} }, z}) \geq \delta _ Z(y)$.

4. If $\delta$ is a dimension function on $Y$, then $\delta (y) \leq \delta _ Z(y) + \delta _{max}$ where $\delta _{max}$ is the maximum value of $\delta$ on $Z$.

5. If $Y = \mathop{\mathrm{Spec}}(A)$ is the spectrum of a catenary Noetherian local ring with maximal ideal $\mathfrak m$ and $Z = \{ \mathfrak m\}$, then $\delta _ Z(y) = \dim (\overline{\{ y\} })$.

6. Given a pattern of specializations

$\xymatrix{ & y'_0 \ar@{~>}[ld] \ar@{~>}[rd] & & y'_1 \ar@{~>}[ld] & \ldots & y'_{k - 1} \ar@{~>}[rd] & \\ y_0 & & y_1 & & \ldots & & y_ k = y }$

between points of $Y$ with $y_0 \in Z$ and $y_ i' \leadsto y_ i$ an immediate specialization, then $\delta _ Z(y_ k) \leq k$.

7. If $Y' \subset Y$ is an open subscheme, then $\delta ^{Y'}_{Y' \cap Z}(y') \geq \delta ^ Y_ Z(y')$ for $y' \in Y'$.

Proof. Part (1) is essentially true by definition. Namely, if $y \in Z$, then we can take $k = 0$ and $V_0 = W_0 = \overline{\{ y\} }$.

Proof of (2). Let $y \leadsto y'$ be a nontrivial specialization and let $V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \subset W_ k$ is a system for $y$. Here there are two cases. Case I: $V_ k = W_ k$, i.e., $c_ k = 0$. In this case we can set $V'_ k = W'_ k = \overline{\{ y'\} }$. An easy computation shows that $\delta (V_0, W_0, \ldots , V'_ k, W'_ k) \leq \delta (V_0, W_0, \ldots , V_ k, W_ k)$ because only $b_{k - 1}$ is changed into a bigger integer. Case II: $V_ k \not= W_ k$, i.e., $c_ k > 0$. Observe that in this case $\max _{i = 0, 1, \ldots , k} (c_ i + c_{i + 1} + \ldots + c_ k - b_ i - b_{i + 1} - \ldots - b_{k - 1}) > 0$. Hence if we set $V'_{k + 1} = W_{k + 1} = \overline{\{ y'\} }$, then although $k$ is replaced by $k + 1$, the maximum now looks like

$\max _{i = 0, 1, \ldots , k + 1} (c_ i + c_{i + 1} + \ldots + c_ k + c_{k + 1} - b_ i - b_{i + 1} - \ldots - b_{k - 1} - b_ k)$

with $c_{k + 1} = 0$ and $b_ k = \text{codim}(V_{k + 1}, W_ k) > 0$. This is strictly smaller than $\max _{i = 0, 1, \ldots , k} (c_ i + c_{i + 1} + \ldots + c_ k - b_ i - b_{i + 1} - \ldots - b_{k - 1})$ and hence $\delta (V_0, W_0, \ldots , V'_{k + 1}, W'_{k + 1}) \leq \delta (V_0, W_0, \ldots , V_ k, W_ k)$ as desired.

Proof of (3). Given $y \in Y$ and $z \in \overline{\{ y\} } \cap Z$ we get the system

$V_0 = \overline{\{ z\} } \subset W_0 = \overline{\{ y\} }$

and $c_0 = \text{codim}(V_0, W_0) = \dim (\mathcal{O}_{\overline{\{ y\} }, z})$ by Properties, Lemma 28.10.3. Thus we see that $\delta (V_0, W_0) = 0 + c_0 = c_0$ which proves what we want.

Proof of (4). Let $\delta$ be a dimension function on $Y$. Let $V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \subset W_ k$ be a system for $y$. Let $y'_ i \in W_ i$ and $y_ i \in V_ i$ be the generic points, so $y_0 \in Z$ and $y_ k = y$. Then we see that

$\delta (y_ i) - \delta (y_{i - 1}) = \delta (y'_{i - 1}) - \delta (y_{i - 1}) - \delta (y'_{i - 1}) + \delta (y_ i) = c_{i - 1} - b_{i - 1}$

Finally, we have $\delta (y'_ k) - \delta (y_{k - 1}) = c_ k$. Thus we see that

$\delta (y) - \delta (y_0) = c_0 + \ldots + c_ k - b_0 - \ldots - b_{k - 1}$

We conclude $\delta (V_0, W_0, \ldots , W_ k) \geq k + \delta (y) - \delta (y_0)$ which proves what we want.

Proof of (5). The function $\delta (y) = \dim (\overline{\{ y\} })$ is a dimension function. Hence $\delta (y) \leq \delta _ Z(y)$ by part (4). By part (3) we have $\delta _ Z(y) \leq \delta (y)$ and we are done.

Proof of (6). Given such a sequence of points, we may assume all the specializations $y'_ i \leadsto y_{i + 1}$ are nontrivial (otherwise we can shorten the chain of specializations). Then we set $V_ i = \overline{\{ y_ i\} }$ and $W_ i = \overline{\{ y'_ i\} }$ and we compute $\delta (V_0, W_1, V_1, \ldots , W_{k - 1}) = k$ because all the codimensions $c_ i$ of $V_ i \subset W_ i$ are $1$ and all $b_ i > 0$. This implies $\delta _ Z(y'_{k - 1}) \leq k$ as $y'_{k - 1}$ is the generic point of $W_ k$. Then $\delta _ Z(y) \leq k$ by part (2) as $y$ is a specialization of $y_{k - 1}$.

Proof of (7). This is clear as their are fewer systems to consider in the computation of $\delta ^{Y'}_{Y' \cap Z}$. $\square$

Lemma 52.18.2. Let $Y$ be a universally catenary Noetherian scheme. Let $Z \subset Y$ be a closed subscheme. Let $f : Y' \to Y$ be a finite type morphism all of whose fibres have dimension $\leq e$. Set $Z' = f^{-1}(Z)$. Then

$\delta _ Z(y) \leq \delta _{Z'}(y') + e - \text{trdeg}_{\kappa (y)}(\kappa (y'))$

for $y' \in Y'$ with image $y \in Y$.

Proof. If $\delta _{Z'}(y') = \infty$, then there is nothing to prove. If $\delta _{Z'}(y') < \infty$, then we choose a system of integral closed subschemes

$V'_0 \subset W'_0 \supset V'_1 \subset W'_1 \supset \ldots \subset W'_ k$

of $Y'$ with $V'_0 \subset Z'$ and $y'$ the generic point of $W'_ k$ such that $\delta _{Z'}(y') = \delta (V'_0, W'_0, \ldots , W'_ k)$. Denote

$V_0 \subset W_0 \supset V_1 \subset W_1 \supset \ldots \subset W_ k$

the scheme theoretic images of the above schemes in $Y$. Observe that $y$ is the generic point of $W_ k$ and that $V_0 \subset Z$. For each $i$ we look at the diagram

$\xymatrix{ V'_ i \ar[r] \ar[d] & W'_ i \ar[d] & V'_{i + 1} \ar[l] \ar[d] \\ V_ i \ar[r] & W_ i & V_{i + 1} \ar[l] }$

Denote $n_ i$ the relative dimension of $V'_ i/V_ i$ and $m_ i$ the relative dimension of $W'_ i/W_ i$; more precisely these are the transcendence degrees of the corresponding extensions of the function fields. Set $c_ i = \text{codim}(V_ i, W_ i)$, $c'_ i = \text{codim}(V'_ i, W'_ i)$, $b_ i = \text{codim}(V_{i + 1}, W_ i)$, and $b'_ i = \text{codim}(V'_{i + 1}, W'_ i)$. By the dimension formula we have

$c_ i = c'_ i + n_ i - m_ i \quad \text{and}\quad b_ i = b'_ i + n_{i + 1} - m_ i$

See Morphisms, Lemma 29.52.1. Hence $c_ i - b_ i = c'_ i - b'_ i + n_ i - n_{i + 1}$. Thus we see that

\begin{align*} & c_ i + c_{i + 1} + \ldots + c_ k - b_ i - b_{i + 1} - \ldots - b_{k - 1} \\ & = c'_ i + c'_{i + 1} + \ldots + c'_ k - b'_ i - b'_{i + 1} - \ldots - b'_{k - 1} + n_ i - n_ k + c_ k - c'_ k \\ & = c'_ i + c'_{i + 1} + \ldots + c'_ k - b'_ i - b'_{i + 1} - \ldots - b'_{k - 1} + n_ i - m_ k \end{align*}

Thus we see that

\begin{align*} \max _{i = 0, \ldots , k} & (c_ i + c_{i + 1} + \ldots + c_ k - b_ i - b_{i + 1} - \ldots - b_{k - 1}) \\ & = \max _{i = 0, \ldots , k} (c'_ i + c'_{i + 1} + \ldots + c'_ k - b'_ i - b'_{i + 1} - \ldots - b'_{k - 1} + n_ i - m_ k) \\ & = \max _{i = 0, \ldots , k} (c'_ i + c'_{i + 1} + \ldots + c'_ k - b'_ i - b'_{i + 1} - \ldots - b'_{k - 1} + n_ i) - m_ k \\ & \leq \max _{i = 0, \ldots , k} (c'_ i + c'_{i + 1} + \ldots + c'_ k - b'_ i - b'_{i + 1} - \ldots - b'_{k - 1}) + e - m_ k \end{align*}

Since $m_ k = \text{trdeg}_{\kappa (y)}(\kappa (y'))$ we conclude that

$\delta (V_0, W_0, \ldots , W_ k) \leq \delta (V'_0, W'_0, \ldots , W'_ k) + e - \text{trdeg}_{\kappa (y)}(\kappa (y'))$

as desired. $\square$

Remark 52.18.3. Let $Y$ be a Noetherian scheme and let $Z \subset Y$ be a closed subset. By Lemma 52.18.1 we have

$\delta _ Z(y) \leq \min \left\{ k \middle | \begin{matrix} \text{ there exist specializations in }Y \\ y_0 \leftarrow y'_0 \rightarrow y_1 \leftarrow y'_1 \rightarrow \ldots \leftarrow y'_{k - 1} \rightarrow y_ k = y \\ \text{ with }y_0 \in Z\text{ and }y_ i' \leadsto y_ i \text{ immediate} \end{matrix} \right\}$

We claim that if $Y$ is of finite type over a field, then equality holds. If we ever need this result we will formulate a precise result and prove it here. However, in general if we define $\delta _ Z$ by the right hand side of this inequality, then we don't know if Lemma 52.18.2 remains true.

Example 52.18.4. Let $k$ be a field and $Y = \mathbf{A}^ n_ k$. Denote $\delta : Y \to \mathbf{Z}_{\geq 0}$ the usual dimension function.

1. If $Z = \{ z\}$ for some closed point $z$, then

1. $\delta _ Z(y) = \delta (y)$ if $y \leadsto z$ and

2. $\delta _ Z(y) = \delta (y) + 1$ if $y \not\leadsto z$.

2. If $Z$ is a closed subvariety and $W = \overline{\{ y\} }$, then

1. $\delta _ Z(y) = 0$ if $W \subset Z$,

2. $\delta _ Z(y) = \dim (W) - \dim (Z)$ if $Z$ is contained in $W$,

3. $\delta _ Z(y) = 1$ if $\dim (W) \leq \dim (Z)$ and $W \not\subset Z$,

4. $\delta _ Z(y) = \dim (W) - \dim (Z) + 1$ if $\dim (W) > \dim (Z)$ and $Z \not\subset W$.

A generalization of case (1) is if $Y$ is of finite type over a field and $Z = \{ z\}$ is a closed point. Then $\delta _ Z(y) = \delta (y) + t$ where $t$ is the minimum length of a chain of curves connecting $z$ to a closed point of $\overline{\{ y\} }$.

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