The Stacks project

Lemma 52.19.4. In Situation 52.16.1 let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module and $d \geq 1$. Assume

  1. $A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$,

  2. the completion $\mathcal{F}^\wedge $ of $\mathcal{F}$ satisfies the strict $(1, 1 + d)$-inequalities.

Let $x \in X$ be a point. Let $W = \overline{\{ x\} }$. If $W \cap Y$ has an irreducible component contained in $Z$ and one which is not, then $\text{depth}(\mathcal{F}_ x) \geq 1$.

Proof. Let $W \cap Y = W_1 \cup \ldots \cup W_ n$ be the decomposition into irreducible components. By assumption, after renumbering, we can find $0 < m < n$ such that $W_1, \ldots , W_ m \subset Z$ and $W_{m + 1}, \ldots , W_ n \not\subset Z$. We conclude that

\[ W \cap Y \setminus \left((W_1 \cup \ldots \cup W_ m) \cap (W_{m + 1} \cup \ldots \cup W_ n)\right) \]

is disconnected. By Lemma 52.14.2 we can find $1 \leq i \leq m < j \leq n$ and $z \in W_ i \cap W_ j$ such that $\dim (\mathcal{O}_{W, z}) \leq d + 1$. Choose an immediate specialization $y \leadsto z$ with $y \in W_ j$, $y \not\in Z$; existence of $y$ follows from Properties, Lemma 28.6.4. Observe that $\delta ^ Y_ Z(y) = 1$ and $\dim (\mathcal{O}_{W, y}) \leq d$. Let $\mathfrak p \subset \mathcal{O}_{X, y}$ be the prime corresponding to $x$. Let $\mathfrak p' \subset \mathcal{O}_{X, y}^\wedge $ be a minimal prime over $\mathfrak p\mathcal{O}_{X, y}^\wedge $. Then we have

\[ \text{depth}(\mathcal{F}_ x) = \text{depth}((\mathcal{F}^\wedge _ y)_{\mathfrak p'}) \quad \text{and}\quad \dim (\mathcal{O}_{W, y}) = \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p') \]

See Algebra, Lemma 10.163.1 and Local Cohomology, Lemma 51.11.3. Now we read off the conclusion from the inequalities given to us. $\square$

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