Remark 38.40.4. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be a perfect object such that $H^ i(E)$ is a perfect $\mathcal{O}_ X$-module of tor dimension $\leq 1$ for all $i \in \mathbf{Z}$. This property sometimes allows one to reduce questions about $E$ to questions about $H^ i(E)$. For example, suppose

is a bounded complex of finite locally free $\mathcal{O}_ X$-modules representing $E$. Then $\mathop{\mathrm{Im}}(d^ i)$ and $\mathop{\mathrm{Ker}}(d^ i)$ are finite locally free $\mathcal{O}_ X$-modules for all $i$. Namely, suppose by induction we know this for all indices bigger than $i$. Then we can first use the short exact sequence

and the assumption that $H^{i + 1}(E)$ is perfect of tor dimension $\leq 1$ to conclude that $\mathop{\mathrm{Im}}(d^ i)$ is finite locally free. The same argument used again for the short exact sequence

then gives that $\mathop{\mathrm{Ker}}(d^ i)$ is finite locally free. It follows that the distinguished triangles

are represented by the following short exact sequences of bounded complexes of finite locally free modules

Here the complexes are the rows and the “obvious” zeros are omitted from the display.

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