Exercise 6, § 10

Let $A$ be a local noetherian ring admitting a dualizing module, $M$ a finitely generated $A$-module, $n$ an integer.

a) For the $A$-module $H^n_A(M)$ to be of finite length, it is necessary and sufficient that for every prime ideal $\mathfrak{p}$ of $A$ distinct from $\mathfrak{m}_A$, one has $H^{n-\dim(A/\mathfrak{p})}_{A_{\mathfrak{p}}}(M_{\mathfrak{p}}) = 0$ (use Grothendieck duality).

b) For $H^i_A(M)$ to be of finite length for every $i \leq n$, it is necessary and sufficient that one has $\operatorname{prof}(M_{\mathfrak{p}}) + \dim(A/\mathfrak{p}) > n$ for every prime ideal $\mathfrak{p}$ distinct from $\mathfrak{m}_A$