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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