Example 15.69.3. Let $R$ be a Dedekind domain. Then every nonzero ideal $I$ is a finite projective module, see Lemma 15.22.11. Thus $R/I$ has projective dimension $1$. Hence every $R$-module $M$ has injective dimension $\leq 1$ by Lemma 15.69.2. Thus $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N) = 0$ for $i \geq 2$ and any pair of $R$-modules $M, N$. It follows that any object $K$ in $D^ b(R)$ is isomorphic to the direct sum of its cohomologies: $K \cong \bigoplus H^ i(K)[-i]$, see Derived Categories, Lemma 13.27.10.

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