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