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Triangle NLs when the last arrow is lci

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aisejohan committed Nov 27, 2019
1 parent 8b678e5 commit 0f3b2fd73868dbe4b940f67b8462bbae347fc036
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  1. +27 −0 more-morphisms.tex
@@ -3838,6 +3838,33 @@ \section{The naive cotangent complex}
Modules, Lemma \ref{modules-lemma-exact-sequence-NL-ringed-topoi}.

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}.

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
To translate into algebra use Lemma \ref{lemma-NL-affine}.

Let $f : X \to Y$ be a morphism of schemes which factors

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