The Stacks project

Lemma 52.19.2. 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.

  1. If $(\mathcal{F}_ n)$ is annihilated by a power of $I$, then $(\mathcal{F}_ n)$ satisfies the $(a, b)$-inequalities for any $a, b$.

  2. If $(\mathcal{F}_ n)$ satisfies the $(a + 1, b)$-inequalities, then $(\mathcal{F}_ n)$ satisfies the strict $(a, b)$-inequalities.

If $\text{cd}(A, I) \leq d$ and $A$ has a dualizing complex, then

  1. $(\mathcal{F}_ n)$ satisfies the $(s, s + d)$-inequalities if and only if for all $y \in U \cap Y$ the tuple $\mathcal{O}_{X, y}^\wedge , I\mathcal{O}_{X, y}^\wedge , \{ \mathfrak m_ y^\wedge \} , \mathcal{F}_ y^\wedge , s - \delta ^ Y_ Z(y), d$ is as in Situation 52.10.1.

  2. If $(\mathcal{F}_ n)$ satisfies the strict $(s, s + d)$-inequalities, then $(\mathcal{F}_ n)$ satisfies the $(s, s + d)$-inequalities.

Proof. Immediate except for part (4) which is a consequence of Lemma 52.10.5 and the translation in (3). $\square$

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