# stacks/stacks-project

Triangle NLs when the last arrow is lci

 @@ -3838,6 +3838,33 @@ \section{The naive cotangent complex} Modules, Lemma \ref{modules-lemma-exact-sequence-NL-ringed-topoi}. \end{proof} \begin{lemma} \label{lemma-get-triangle-NL} Let $f : X \to Y$ and $Y \to Z$ be morphisms of schemes. Assume $X \to Y$ is a complete intersection morphism. Then there is a canonical distinguished triangle $$f^*\NL_{Y/Z} \to \NL_{X/Z} \to \NL_{X/Y} \to f^*\NL_{Y/Z}[1]$$ in $D(\mathcal{O}_X)$ which recovers the $6$-term exact sequence of Lemma \ref{lemma-exact-sequence-NL}. \end{lemma} \begin{proof} It suffices to show the canonical map $$f^*\NL_{Y/Z} \to \text{Cone}(\NL_{X/Y} \to \NL_{X/Z})[-1]$$ of Modules, Lemma \ref{modules-lemma-exact-sequence-NL-ringed-topoi} is an isomorphism in $D(\mathcal{O}_X)$. In order to show this, it suffices to show that the $6$-term sequence has a zero on the left, i.e., that $H^{-1}(f^*\NL_{Y/Z}) \to H^{-1}(\NL_{X/Z})$ is injective. Affine locally this follows from the corresponding algebra result in More on Algebra, Lemma \ref{more-algebra-lemma-transitive-lci-at-end}. To translate into algebra use Lemma \ref{lemma-NL-affine}. \end{proof} \begin{lemma} \label{lemma-get-NL} Let $f : X \to Y$ be a morphism of schemes which factors