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81.2 Setup

We first fix the category of algebraic spaces we will be working with. Please keep in mind throughout this chapter that “decent $+$ locally Noetherian” is the same as “quasi-separated $+$ locally Noetherian” according to Decent Spaces, Lemma 67.14.1.

Situation 81.2.1. Here $S$ is a scheme and $B$ is an algebraic space over $S$. We assume $B$ is quasi-separated, locally Noetherian, and universally catenary (Decent Spaces, Definition 67.25.4). Moreover, we assume given a dimension function $\delta : |B| \longrightarrow \mathbf{Z}$. We say $X/B$ is good if $X$ is an algebraic space over $B$ whose structure morphism $f : X \to B$ is quasi-separated and locally of finite type. In this case we define

\[ \delta = \delta _{X/B} : |X| \longrightarrow \mathbf{Z} \]

as the map sending $x$ to $\delta (f(x))$ plus the transcendence degree of $x/f(x)$ (Morphisms of Spaces, Definition 66.33.1). This is a dimension function by More on Morphisms of Spaces, Lemma 75.32.2.

A special case is when $S = B$ is a scheme and $(S, \delta )$ is as in Chow Homology, Situation 42.7.1. Thus $B$ might be the spectrum of a field (Chow Homology, Example 42.7.2) or $B = \mathop{\mathrm{Spec}}(\mathbf{Z})$ (Chow Homology, Example 42.7.3).

Many lemma, proposition, theorems, definitions on algebraic spaces are easier in the setting of Situation 81.2.1 because the algebraic spaces we are working with are quasi-separated (and thus a fortiori decent) and locally Noetherian. We will sprinkle this chapter with remarks such as the following to point this out.

Remark 81.2.2. In Situation 81.2.1 if $X/B$ is good, then $|X|$ is a sober topological space. See Properties of Spaces, Lemma 65.15.1 or Decent Spaces, Proposition 67.12.4. We will use this without further mention to choose generic points of irreducible closed subsets of $|X|$.

Remark 81.2.3. In Situation 81.2.1 if $X/B$ is good, then $X$ is integral (Spaces over Fields, Definition 71.4.1) if and only if $X$ is reduced and $|X|$ is irreducible. Moreover, for any point $\xi \in |X|$ there is a unique integral closed subspace $Z \subset X$ such that $\xi $ is the generic point of the closed subset $|Z| \subset |X|$, see Spaces over Fields, Lemma 71.4.7.

If $B$ is Jacobson and $\delta $ sends closed points to zero, then $\delta $ is the function sending a point to the dimension of its closure.

Lemma 81.2.4. In Situation 81.2.1 assume $B$ is Jacobson and that $\delta (b) = 0$ for every closed point $b$ of $|B|$. Let $X/B$ be good. If $Z \subset X$ is an integral closed subspace with generic point $\xi \in |Z|$, then the following integers are the same:

  1. $\delta (\xi ) = \delta _{X/B}(\xi )$,

  2. $\dim (|Z|)$,

  3. $\text{codim}(\{ z\} , |Z|)$ for $z \in |Z|$ closed,

  4. the dimension of the local ring of $Z$ at $z$ for $z \in |Z|$ closed, and

  5. $\dim (\mathcal{O}_{Z, \overline{z}})$ for $z \in |Z|$ closed.

Proof. Let $X$, $Z$, $\xi $ be as in the lemma. Since $X$ is locally of finite type over $B$ we see that $X$ is Jacobson, see Decent Spaces, Lemma 67.23.1. Hence $X_{\text{ft-pts}} \subset |X|$ is the set of closed points by Decent Spaces, Lemma 67.23.3. Given a chain $T_0 \supset \ldots \supset T_ e$ of irreducible closed subsets of $|Z|$ we have $T_ e \cap X_{\text{ft-pts}}$ nonempty by Morphisms of Spaces, Lemma 66.25.6. Thus we can always assume such a chain ends with $T_ e = \{ z\} $ for some $z \in |Z|$ closed. It follows that $\dim (Z) = \sup _ z \text{codim}(\{ z\} , |Z|)$ where $z$ runs over the closed points of $|Z|$. We have $\text{codim}(\{ z\} , Z) = \delta (\xi ) - \delta (z)$ by Topology, Lemma 5.20.2. By Morphisms of Spaces, Lemma 66.25.4 the image of $z$ is a finite type point of $B$, i.e., a closed point of $|B|$. By Morphisms of Spaces, Lemma 66.33.4 the transcendence degree of $z/b$ is $0$. We conclude that $\delta (z) = \delta (b) = 0$ by assumption. Thus we obtain equality

\[ \dim (|Z|) = \text{codim}(\{ z\} , Z) = \delta (\xi ) \]

for all $z \in |Z|$ closed. Finally, we have that $\text{codim}(\{ z\} , Z)$ is equal to the dimension of the local ring of $Z$ at $z$ by Decent Spaces, Lemma 67.20.2 which in turn is equal to $\dim (\mathcal{O}_{Z, \overline{z}})$ by Properties of Spaces, Lemma 65.22.4. $\square$

In the situation of the lemma above the value of $\delta $ at the generic point of a closed irreducible subset is the dimension of the irreducible closed subset. This motivates the following definition.

Definition 81.2.5. In Situation 81.2.1 for any good $X/B$ and any irreducible closed subset $T \subset |X|$ we define

\[ \dim _\delta (T) = \delta (\xi ) \]

where $\xi \in T$ is the generic point of $T$. We will call this the $\delta $-dimension of $T$. If $T \subset |X|$ is any closed subset, then we define $\dim _\delta (T)$ as the supremum of the $\delta $-dimensions of the irreducible components of $T$. If $Z$ is a closed subspace of $X$, then we set $\dim _\delta (Z) = \dim _\delta (|Z|)$.

Of course this just means that $\dim _\delta (T) = \sup \{ \delta (t) \mid t \in T\} $.

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