Lemma 51.16.3. Let $A$ be a Noetherian local ring of dimension $d$. Let $f \in A$ be an element which is not contained in any minimal prime of dimension $d$. Then $f : H^ d_{V(I)}(M) \to H^ d_{V(I)}(M)$ is surjective for any finite $A$-module $M$ and any ideal $I \subset A$.

Proof. The support of $M/fM$ has dimension $< d$ by our assumption on $f$. Thus $H^ d_{V(I)}(M/fM) = 0$ by Lemma 51.4.7. Thus $H^ d_{V(I)}(fM) \to H^ d_{V(I)}(M)$ is surjective. Since by Lemma 51.4.7 we know $\text{cd}(A, I) \leq d$ we also see that the surjection $M \to fM$, $x \mapsto fx$ induces a surjection $H^ d_{V(I)}(M) \to H^ d_{V(I)}(fM)$. $\square$

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