Lemma 51.2.5. Let $I \subset A$ be a finitely generated ideal of a ring $A$. Let $\mathfrak p$ be a prime ideal. Let $M$ be an $A$-module. Let $i \geq 0$ be an integer and consider the map
\[ \Psi : \mathop{\mathrm{colim}}\nolimits _{f \in A, f \not\in \mathfrak p} H^ i_{V((I, f))}(M) \longrightarrow H^ i_{V(I)}(M) \]
Then
$\mathop{\mathrm{Im}}(\Psi )$ is the set of elements which map to zero in $H^ i_{V(I)}(M)_\mathfrak p$,
if $H^{i - 1}_{V(I)}(M)_\mathfrak p = 0$, then $\Psi $ is injective,
if $H^{i - 1}_{V(I)}(M)_\mathfrak p = H^ i_{V(I)}(M)_\mathfrak p = 0$, then $\Psi $ is an isomorphism.
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