The Stacks project

Lemma 37.13.4. Let $f : X \to Y$ be a morphism of schemes. If $f$ is locally of finite presentation, then $\mathop{N\! L}\nolimits _{X/Y}$ is 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. By Lemma 37.13.2 it suffices to show that $\mathop{N\! L}\nolimits _{A/R}$ has this shape if $R \to A$ is a finitely presented ring map. Write $A = R[x_1, \ldots , x_ n]/I$ with $I$ finitely generated. Then $I/I^2$ is a finite $A$-module and $\mathop{N\! L}\nolimits _{A/R}$ is quasi-isomorphic to

\[ \ldots \to 0 \to I/I^2 \to \bigoplus \nolimits _{i = 1, \ldots , n} A\text{d}x_ i \to 0 \to \ldots \]

by Algebra, Section 10.134 and in particular Algebra, Lemma 10.134.2. $\square$

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