The Stacks project

Definition 52.19.1. In Situation 52.16.1 let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(U, I\mathcal{O}_ U)$. Let $a, b$ be integers. Let $\delta ^ Y_ Z$ be as in (52.18.0.1). We say $(\mathcal{F}_ n)$ satisfies the $(a, b)$-inequalities if for $y \in U \cap Y$ and a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $\mathfrak p \not\in V(I\mathcal{O}_{X, y}^\wedge )$

1. if $V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge ) \not= \{ \mathfrak m_ y^\wedge \} $, then

   \[ \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \delta ^ Y_ Z(y) \geq a \quad \text{or}\quad \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) + \delta ^ Y_ Z(y) > b \]

2. if $V(\mathfrak p) \cap V(I\mathcal{O}_{X, y}^\wedge ) = \{ \mathfrak m_ y^\wedge \} $, then

   \[ \text{depth}((\mathcal{F}^\wedge _ y)_\mathfrak p) + \delta ^ Y_ Z(y) > a \]

We say $(\mathcal{F}_ n)$ satisfies the strict $(a, b)$-inequalities if for $y \in U \cap Y$ and a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $\mathfrak p \not\in V(I\mathcal{O}_{X, y}^\wedge )$ we have