The Stacks project

Lemma 76.21.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If $f$ is locally of finite presentation, then $\mathop{N\! L}\nolimits _{X/Y}$ is ├ętale locally on $X$ quasi-isomorphic to a complex

\[ \ldots \to 0 \to \mathcal{F}^{-1} \to \mathcal{F}^0 \to 0 \to \ldots \]

of quasi-coherent $\mathcal{O}_ X$-modules with $\mathcal{F}^0$ of finite presentation and $\mathcal{F}^{-1}$ of finite type.

Proof. Formation of the naive cotangent complex commutes with ├ętale localization by Lemma 76.21.2. This reduces us to the case of schemes by Lemma 76.21.3. The result in the case of schemes is More on Morphisms, Lemma 37.13.4. $\square$


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