Example 48.3.4. If $f : X \to Y$ is a proper or even finite morphism of Noetherian schemes, then the right adjoint of $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ does not map $D_\mathit{QCoh}^-(\mathcal{O}_ Y)$ into $D_\mathit{QCoh}^-(\mathcal{O}_ X)$. Namely, let $k$ be a field, let $k[\epsilon ]$ be the dual numbers over $k$, let $X = \mathop{\mathrm{Spec}}(k)$, and let $Y = \mathop{\mathrm{Spec}}(k[\epsilon ])$. Then $\mathop{\mathrm{Ext}}\nolimits ^ i_{k[\epsilon ]}(k, k)$ is nonzero for all $i \geq 0$. Hence $a(\mathcal{O}_ Y)$ is not bounded above by Example 48.3.2.

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