Example 2

If $H_A^0(M) = M$ (for example if M is artinian), $H_A^i(M)$ is zero for $i > 0$. Let in fact (I, e) be an injective envelope of M. The submodule $H_A^0(I)$ of I is injective (example 1) and contains $e(M)$, hence is equal to I. Put $N = \mathrm{Coker}\,e$ and consider the exact sequence $0 \to M \xrightarrow{e} I \xrightarrow{p} N \to 0$. Since $I = H_A^0(I)$, one has $N = H_A^0(N)$ and the homomorphism $H_A^0(p)$ is surjective. Since $H_A^i(I)$ is zero for $i > 0$ (example 1), $H_A^1(M)$ is zero and $H_A^i(M)$ is isomorphic to $H_A^{i-1}(N)$ for $i > 1$; one concludes by reasoning by induction on the integer $i$.