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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