The Stacks project

Example 15.69.4. Let $k$ be a field and let $R$ be the ring of dual numbers over $k$, i.e., $R = k[x]/(x^2)$. Denote $\epsilon \in R$ the class of $x$. Let $M = R/(\epsilon )$. Then $M$ is quasi-isomorphic to the complex

\[ \ldots \to R \xrightarrow {\epsilon } R \xrightarrow {\epsilon } R \]

and $R$ is an injective $R$-module. However one usually does not consider $M$ to have finite injective dimension in this situation. This explains why we consider bounded (in both directions) complexes of injective modules in our definition of bounded injective dimension of objects of $D(R)$.

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