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
X is universally catenary,
for every z \in Z the formal fibres of \mathcal{O}_{X, z} are (S_ n).
In this situation the following are equivalent
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,
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 (51.11.1.1). Having said this the lemma follows from Theorem 51.11.6 as elucidated by Remark 51.11.7.
\square
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