The Stacks project

Lemma 37.13.3. Let $f : X \to Y$ be a morphism of schemes. 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, Section 17.30 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 $\Omega _{X/Y}$. The sheaf of differentials is quasi-coherent (by Morphisms, Lemma 29.32.7). To finish the proof it suffices to show that $H^{-1}(\mathop{N\! L}\nolimits _{X/Y})$ is quasi-coherent. This follows by checking over affines using Lemma 37.13.2. $\square$


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