Lemma 51.8.2. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Set $X = \mathop{\mathrm{Spec}}(A)$, $Z = V(I)$, $U = X \setminus Z$, and $j : U \to X$ the inclusion morphism. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module. Then

there exists a finite $A$-module $M$ such that $\mathcal{F}$ is the restriction of $\widetilde{M}$ to $U$,

given $M$ there is an exact sequence

\[ 0 \to H^0_ Z(M) \to M \to H^0(U, \mathcal{F}) \to H^1_ Z(M) \to 0 \]and isomorphisms $H^ p(U, \mathcal{F}) = H^{p + 1}_ Z(M)$ for $p \geq 1$,

given $M$ and $p \geq 0$ the following are equivalent

$R^ pj_*\mathcal{F}$ is coherent,

$H^ p(U, \mathcal{F})$ is a finite $A$-module,

$H^{p + 1}_ Z(M)$ is a finite $A$-module,

if the equivalent conditions in (3) hold for $p = 0$, we may take $M = \Gamma (U, \mathcal{F})$ in which case we have $H^0_ Z(M) = H^1_ Z(M) = 0$.

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