30.11 Depth
In this section we talk a little bit about depth and property $(S_ k)$ for coherent modules on locally Noetherian schemes. Note that we have already discussed this notion for locally Noetherian schemes in Properties, Section 28.12.
Definition 30.11.1. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $k \geq 0$ be an integer.
We say $\mathcal{F}$ has depth $k$ at a point $x$ of $X$ if $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) = k$.
We say $X$ has depth $k$ at a point $x$ of $X$ if $\text{depth}(\mathcal{O}_{X, x}) = k$.
We say $\mathcal{F}$ has property $(S_ k)$ if
\[ \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) \geq \min (k, \dim (\text{Supp}(\mathcal{F}_ x))) \]
for all $x \in X$.
We say $X$ has property $(S_ k)$ if $\mathcal{O}_ X$ has property $(S_ k)$.
Any coherent sheaf satisfies condition $(S_0)$. Condition $(S_1)$ is equivalent to having no embedded associated points, see Divisors, Lemma 31.4.3.
Lemma 30.11.2. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules and $x \in X$.
If $\mathcal{G}_ x$ has depth $\geq 1$, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})_ x$ has depth $\geq 1$.
If $\mathcal{G}_ x$ has depth $\geq 2$, then $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})_ x$ has depth $\geq 2$.
Proof.
Observe that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is a coherent $\mathcal{O}_ X$-module by Lemma 30.9.4. Coherent modules are of finite presentation (Lemma 30.9.1) hence taking stalks commutes with taking $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ and $\mathop{\mathrm{Hom}}\nolimits $, see Modules, Lemma 17.22.4. Thus we reduce to the case of finite modules over local rings which is More on Algebra, Lemma 15.23.10.
$\square$
Lemma 30.11.3. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules.
If $\mathcal{G}$ has property $(S_1)$, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ has property $(S_1)$.
If $\mathcal{G}$ has property $(S_2)$, then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ has property $(S_2)$.
Proof.
Follows immediately from Lemma 30.11.2 and the definitions.
$\square$
We have seen in Properties, Lemma 28.12.3 that a locally Noetherian scheme is Cohen-Macaulay if and only if $(S_ k)$ holds for all $k$. Thus it makes sense to introduce the following definition, which is equivalent to the condition that all stalks are Cohen-Macaulay modules.
Definition 30.11.4. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. We say $\mathcal{F}$ is Cohen-Macaulay if and only if $(S_ k)$ holds for all $k \geq 0$.
Lemma 30.11.5. Let $X$ be a regular scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. The following are equivalent
$\mathcal{F}$ is Cohen-Macaulay and $\text{Supp}(\mathcal{F}) = X$,
$\mathcal{F}$ is finite locally free of rank $> 0$.
Proof.
Let $x \in X$. If (2) holds, then $\mathcal{F}_ x$ is a free $\mathcal{O}_{X, x}$-module of rank $> 0$. Hence $\text{depth}(\mathcal{F}_ x) = \dim (\mathcal{O}_{X, x})$ because a regular local ring is Cohen-Macaulay (Algebra, Lemma 10.106.3). Conversely, if (1) holds, then $\mathcal{F}_ x$ is a maximal Cohen-Macaulay module over $\mathcal{O}_{X, x}$ (Algebra, Definition 10.103.8). Hence $\mathcal{F}_ x$ is free by Algebra, Lemma 10.106.6.
$\square$
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