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 étale 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 étale 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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