The Stacks project

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

\[ \Psi : \mathop{\mathrm{colim}}\nolimits _{f \in A, f \not\in \mathfrak p} H^ i_{V((I, f))}(M) \longrightarrow H^ i_{V(I)}(M) \]


  1. $\mathop{\mathrm{Im}}(\Psi )$ is the set of elements which map to zero in $H^ i_{V(I)}(M)_\mathfrak p$,

  2. if $H^{i - 1}_{V(I)}(M)_\mathfrak p = 0$, then $\Psi $ is injective,

  3. if $H^{i - 1}_{V(I)}(M)_\mathfrak p = H^ i_{V(I)}(M)_\mathfrak p = 0$, then $\Psi $ is an isomorphism.

Proof. For $f \in A$, $f \not\in \mathfrak p$ the spectral sequence of Dualizing Complexes, Lemma 47.9.6 degenerates to give short exact sequences

\[ 0 \to H^1_{V(f)}(H^{i - 1}_{V(I)}(M)) \to H^ i_{V((I, f))}(M) \to H^0_{V(f)}(H^ i_{V(I)}(M)) \to 0 \]

This proves (1) and part (2) follows from this and Lemma 51.2.4. Part (3) is a formal consequence. $\square$

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