The Stacks project

Lemma 76.21.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The cohomology sheaves of the complex $\mathop{N\! L}\nolimits _{X/Y}$ are quasi-coherent, zero outside degrees $-1$, $0$ and equal to $\Omega _{X/Y}$ in degree $0$.

Proof. By construction of the naive cotangent complex in Modules on Sites, Section 18.35 we have that $\mathop{N\! L}\nolimits _{X/Y}$ is a complex sitting in degrees $-1$, $0$ and that its cohomology in degree $0$ is $\Omega _{X/Y}$ (by our construction of $\Omega _{X/Y}$ in Section 76.7). The sheaf of differentials is quasi-coherent (by Lemma 76.7.4). To finish the proof it suffices to show that $H^{-1}(\mathop{N\! L}\nolimits _{X/Y})$ is quasi-coherent. This follows by checking ├ętale locally (allowed by Lemma 76.21.2 and Properties of Spaces, Lemma 66.29.6) reducing to the case of schemes (Lemma 76.21.3) and finally using the result in the case of schemes (More on Morphisms, Lemma 37.13.3). $\square$

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