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