Remark 37.59.9. It is not true that a morphism between schemes $X, Y$ perfect over a base $S$ is perfect. An example is $S = \mathop{\mathrm{Spec}}(k)$, $X = \mathop{\mathrm{Spec}}(k)$, $Y = \mathop{\mathrm{Spec}}(k[x]/(x^2)$ and $X \to Y$ the unique $S$-morphism.

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