Lemma 52.19.3. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. If $\text{cd}(A, I) = 1$, then $\mathcal{F}$ satisfies the $(2, 3)$-inequalities if and only if

$\text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) + \delta ^ Y_ Z(y) > 3$

for all $y \in U \cap Y$ and $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$ with $\mathfrak p \not\in V(I\mathcal{O}_{X, y}^\wedge )$.

Proof. Observe that for a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge$, $\mathfrak p \not\in V(I\mathcal{O}_{X, y}^\wedge )$ we have $V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge ) = \{ \mathfrak m_ y^\wedge \} \Leftrightarrow \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) = 1$ as $\text{cd}(A, I) = 1$. See Local Cohomology, Lemmas 51.4.5 and 51.4.10. OK, consider the three numbers $\alpha = \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) \geq 0$, $\beta = \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) \geq 1$, and $\gamma = \delta ^ Y_ Z(y) \geq 1$. Then we see Definition 52.19.1 requires

1. if $\beta > 1$, then $\alpha + \gamma \geq 2$ or $\alpha + \beta + \gamma > 3$, and

2. if $\beta = 1$, then $\alpha + \gamma > 2$.

It is trivial to see that this is equivalent to $\alpha + \beta + \gamma > 3$. $\square$

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