The Stacks project

42.7 Setup

We will throughout work over a locally Noetherian universally catenary base $S$ endowed with a dimension function $\delta $. Although it is likely possible to generalize (parts of) the discussion in the chapter, it seems that this is a good first approximation. It is exactly the generality discussed in [Thorup]. We usually do not assume our schemes are separated or quasi-compact. Many interesting algebraic stacks are non-separated and/or non-quasi-compact and this is a good case study to see how to develop a reasonable theory for those as well. In order to reference these hypotheses we give it a number.

Situation 42.7.1. Here $S$ is a locally Noetherian, and universally catenary scheme. Moreover, we assume $S$ is endowed with a dimension function $\delta : S \longrightarrow \mathbf{Z}$.

See Morphisms, Definition 29.17.1 for the notion of a universally catenary scheme, and see Topology, Definition 5.20.1 for the notion of a dimension function. Recall that any locally Noetherian catenary scheme locally has a dimension function, see Properties, Lemma 28.11.3. Moreover, there are lots of schemes which are universally catenary, see Morphisms, Lemma 29.17.5.

Let $(S, \delta )$ be as in Situation 42.7.1. Any scheme $X$ locally of finite type over $S$ is locally Noetherian and catenary. In fact, $X$ has a canonical dimension function

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

associated to $(f : X \to S, \delta )$ given by the rule $\delta _{X/S}(x) = \delta (f(x)) + \text{trdeg}_{\kappa (f(x))}\kappa (x)$. See Morphisms, Lemma 29.52.3. Moreover, if $h : X \to Y$ is a morphism of schemes locally of finite type over $S$, and $x \in X$, $y = h(x)$, then obviously $\delta _{X/S}(x) = \delta _{Y/S}(y) + \text{trdeg}_{\kappa (y)}\kappa (x)$. We will freely use this function and its properties in the following.

Here are the basic examples of setups as above. In fact, the main interest lies in the case where the base is the spectrum of a field, or the case where the base is the spectrum of a Dedekind ring (e.g. $\mathbf{Z}$, or a discrete valuation ring).

Example 42.7.2. Here $S = \mathop{\mathrm{Spec}}(k)$ and $k$ is a field. We set $\delta (pt) = 0$ where $pt$ indicates the unique point of $S$. The pair $(S, \delta )$ is an example of a situation as in Situation 42.7.1 by Morphisms, Lemma 29.17.5.

Example 42.7.3. Here $S = \mathop{\mathrm{Spec}}(A)$, where $A$ is a Noetherian domain of dimension $1$. For example we could consider $A = \mathbf{Z}$. We set $\delta (\mathfrak p) = 0$ if $\mathfrak p$ is a maximal ideal and $\delta (\mathfrak p) = 1$ if $\mathfrak p = (0)$ corresponds to the generic point. This is an example of Situation 42.7.1 by Morphisms, Lemma 29.17.5.

Example 42.7.4. Here $S$ is a Cohen-Macaulay scheme. Then $S$ is universally catenary by Morphisms, Lemma 29.17.5. We set $\delta (s) = -\dim (\mathcal{O}_{S, s})$. If $s' \leadsto s$ is a nontrivial specialization of points of $S$, then $\mathcal{O}_{S, s'}$ is the localization of $\mathcal{O}_{S, s}$ at a nonmaximal prime ideal $\mathfrak p \subset \mathcal{O}_{S, s}$, see Schemes, Lemma 26.13.2. Thus $\dim (\mathcal{O}_{S, s}) = \dim (\mathcal{O}_{S, s'}) + \dim (\mathcal{O}_{S, s}/\mathfrak p) > \dim (\mathcal{O}_{S, s'})$ by Algebra, Lemma 10.104.4. Hence $\delta (s') > \delta (s)$. If $s' \leadsto s$ is an immediate specialization, then there is no prime ideal strictly between $\mathfrak p$ and $\mathfrak m_ s$ and we find $\delta (s') = \delta (s) + 1$. Thus $\delta $ is a dimension function. In other words, the pair $(S, \delta )$ is an example of Situation 42.7.1.

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

Lemma 42.7.5. Let $(S, \delta )$ be as in Situation 42.7.1. Assume in addition $S$ is a Jacobson scheme, and $\delta (s) = 0$ for every closed point $s$ of $S$. Let $X$ be locally of finite type over $S$. Let $Z \subset X$ be an integral closed subscheme and let $\xi \in Z$ be its generic point. The following integers are the same:

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

  2. $\dim (Z)$, and

  3. $\dim (\mathcal{O}_{Z, z})$ where $z$ is a closed point of $Z$.

Proof. Let $X \to S$, $\xi \in Z \subset X$ be as in the lemma. Since $X$ is locally of finite type over $S$ we see that $X$ is Jacobson, see Morphisms, Lemma 29.16.9. Hence closed points of $X$ are dense in every closed subset of $Z$ and map to closed points of $S$. Hence given any chain of irreducible closed subsets of $Z$ we can end it with a closed point of $Z$. It follows that $\dim (Z) = \sup _ z(\dim (\mathcal{O}_{Z, z})$ (see Properties, Lemma 28.10.3) where $z \in Z$ runs over the closed points of $Z$. Note that $\dim (\mathcal{O}_{Z, z}) = \delta (\xi ) - \delta (z)$ by the properties of a dimension function. For each closed $z \in Z$ the field extension $\kappa (z)/\kappa (f(z))$ is finite, see Morphisms, Lemma 29.16.8. Hence $\delta _{X/S}(z) = \delta (f(z)) = 0$ for $z \in Z$ closed. It follows that all three integers are equal. $\square$

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. However, in general we cannot expect the equality to hold. For example if $S = \mathop{\mathrm{Spec}}(\mathbf{C}[[t]])$ and $X = \mathop{\mathrm{Spec}}(\mathbf{C}((t)))$ then we would get $\delta (x) = 1$ for the unique point of $X$, but $\dim (X) = 0$. Still we want to think of $\delta _{X/S}$ as giving the dimension of the irreducible closed subschemes. Thus we introduce the following terminology.

Definition 42.7.6. Let $(S, \delta )$ as in Situation 42.7.1. For any scheme $X$ locally of finite type over $S$ and any irreducible closed subset $Z \subset X$ we define

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

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

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