The Stacks project

Example 15.68.3. 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

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

but $M$ does not have finite projective dimension as defined in Algebra, Definition 10.109.2. This explains why we consider bounded (in both directions) complexes of projective modules in our definition of finite projective dimension of objects of $D(R)$.

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View Example 15.68.3 as pdf