The Stacks project

Remark 52.16.12. In Lemma 52.16.11 if $A$ is universally catenary with Cohen-Macaulay formal fibres (for example if $A$ has a dualizing complex), then the condition that $H^1_\mathfrak a(A/fA)$ and $H^2_\mathfrak a(A/fA)$ are finite $A$-modules, is equivalent with

\[ \text{depth}((A/f)_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) > 2 \]

for all $\mathfrak p \in V(f) \setminus V(\mathfrak a)$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$ by Local Cohomology, Theorem 51.11.6.

For example, if $A/fA$ is $(S_2)$ and if every irreducible component of $Z = V(\mathfrak a)$ has codimension $\geq 3$ in $Y = \mathop{\mathrm{Spec}}(A/fA)$, then we get the finiteness of $H^1_\mathfrak a(A/fA)$ and $H^2_\mathfrak a(A/fA)$. This should be contrasted with the slightly weaker conditions found in Lemma 52.20.1 (see also Remark 52.20.2).

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