The Stacks project

Lemma 51.12.1. Let $X$ be a locally Noetherian scheme. Let $j : U \to X$ be the inclusion of an open subscheme with complement $Z$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. Let $n \geq 0$ be an integer. Assume

  1. $X$ is universally catenary,

  2. for every $z \in Z$ the formal fibres of $\mathcal{O}_{X, z}$ are $(S_ n)$.

In this situation the following are equivalent

  1. for $x \in \text{Supp}(\mathcal{F})$ and $z \in Z \cap \overline{\{ x\} }$ we have $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) + \dim (\mathcal{O}_{\overline{\{ x\} }, z}) > n$,

  2. $R^ pj_*\mathcal{F}$ is coherent for $0 \leq p < n$.

Proof. The statement is local on $X$, hence we may assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = V(I)$. Let $M$ be a finite $A$-module whose associated coherent $\mathcal{O}_ X$-module restricts to $\mathcal{F}$ over $U$, see Lemma 51.8.2. This lemma also tells us that $R^ pj_*\mathcal{F}$ is coherent if and only if $H^{p + 1}_ Z(M)$ is a finite $A$-module. Observe that the minimum of the expressions $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) + \dim (\mathcal{O}_{\overline{\{ x\} }, z})$ is the number $s_{A, I}(M)$ of ( Having said this the lemma follows from Theorem 51.11.6 as elucidated by Remark 51.11.7. $\square$

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