Example 5

Let us retain the hypotheses of the preceding example and suppose moreover that $\dim(A) = 1$. Let N be a torsion A-module; since every nonzero ideal of A distinct from A is an ideal of definition (VIII, § 1, No. 3, prop. 6, e)), one has $H_A^0(N) = N$, and consequently $H_A^i(N) = 0$ for $i > 0$ (example 2).

Let M be an A-module; let $T(M)$ denote its torsion submodule. Consider the long exact local cohomology sequence associated with the exact sequence

in view of the foregoing, one deduces from it canonical isomorphisms $T(M) \to H_A^0(M)$ and $H_A^1(M) \to H_A^1(M/T(M))$. Since the canonical homomorphism $(K/A) \otimes_A M \to (K/A) \otimes_A (M/T(M))$ is bijective, one finally obtains canonical isomorphisms