Example 4

Let A be a noetherian local integral ring; let K denote its field of fractions, and suppose that $A \neq K$. It is an injective A-module (A, X, p. 18, example 1), so that the module $H_A(K)$ is zero (example 1). From the exact sequence $0 \to A \to K \to K/A \to 0$, one obtains for every $i$ an isomorphism $H_A^i(K/A) \to H_A^{i+1}(A)$.

More generally, for every torsion-free A-module M and every integer $i$, one deduces from the exact sequence

an isomorphism $H_A^i((K/A) \otimes_A M) \to H_A^{i+1}(M)$.