The Stacks project

Remark 51.11.7. The astute reader will have realized that we can get away with a slightly weaker condition on the formal fibres of the local rings of $A$. Namely, in the situation of Theorem 51.11.6 assume $A$ is universally catenary but make no assumptions on the formal fibres. Suppose we have an $n$ and we want to prove that $H^ i_ Z(M)$ are finite for $i \leq n$. Then the exact same proof shows that it suffices that $s_{A, I}(M) > n$ and that the formal fibres of local rings of $A$ are $(S_ n)$. On the other hand, if we want to show that $H^ s_ Z(M)$ is not finite where $s = s_{A, I}(M)$, then our arguments prove this if the formal fibres are $(S_{s - 1})$.