# stacks/stacks-project

Lemma about excess conormal sheaf; schemes version

 @@ -3807,6 +3807,19 @@ \section{The naive cotangent complex} (Algebra, Lemma \ref{algebra-lemma-finite-projective}). \end{proof} \begin{lemma} \label{lemma-NL-immersion} Let $i : Z \to X$ be an immersion of schemes. Then $\NL_{Z/X}$ is isomorphic to $\mathcal{C}_{Z/X}[1]$ in $D(\mathcal{O}_Z)$ where $\mathcal{C}_{Z/X}$ is the conormal sheaf of $Z$ in $X$. \end{lemma} \begin{proof} This follows from Algebra, Lemma \ref{algebra-lemma-NL-surjection}, Morphisms, Lemma \ref{morphisms-lemma-affine-conormal}, and Lemma \ref{lemma-NL-affine}. \end{proof} \begin{lemma} \label{lemma-exact-sequence-NL} Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of schemes. @@ -3821,7 +3834,46 @@ \section{The naive cotangent complex} \end{lemma} \begin{proof} Special case of Modules, Lemma \ref{modules-lemma-exact-sequence-NL}. Special case of Modules, Lemma \ref{modules-lemma-exact-sequence-NL-ringed-topoi}. \end{proof} \begin{lemma} \label{lemma-get-NL} Let $f : X \to Y$ be a morphism of schemes which factors as $f = g \circ i$ with $i$ an immersion and $g : P \to Y$ formally smooth (for example smooth). Then there is a canonical isomorphism $$\NL_{X/Y} \cong \left(\mathcal{C}_{X/P} \to i^*\Omega_{P/Y}\right)$$ in $D(\mathcal{O}_X)$ where the conormal sheaf $\mathcal{C}_{X/P}$ is placed in degree $-1$. \end{lemma} \begin{proof} (For the parenthetical statement see Lemma \ref{lemma-smooth-formally-smooth}.) By Lemmas \ref{lemma-NL-immersion} and \ref{lemma-NL-formally-smooth} we have $\NL_{X/P} = \mathcal{C}_{X/P}[1]$ and $\NL_{P/Y} = \Omega_{P/Y}$ with $\Omega_{P/Y}$ locally projective. This implies that $i^*\NL_{P/Y} \to i^*\Omega_{P/Y}$ is a quasi-isomorphism too (small detail omitted; the reason is that $i^*\NL_{P/Y}$ is the same thing as $\tau_{\geq -1}Li^*\NL_{P/Y}$, see More on Algebra, Lemma \ref{more-algebra-lemma-tensor-NL}). Thus the canonical map $$i^*\NL_{P/Y} \to \text{Cone}(\NL_{X/Y} \to \NL_{X/P})[-1]$$ of Modules, Lemma \ref{modules-lemma-exact-sequence-NL-ringed-topoi} is an isomorphism in $D(\mathcal{O}_X)$ because the cohomology group $H^{-1}(i^*\NL_{P/Y})$ is zero by what we said above. In other words, we have a distinguished triangle $$i^*\NL_{P/Y} \to \NL_{X/Y} \to \NL_{X/P} \to i^*\NL_{P/Y}[1]$$ Clearly, this means that $\NL_{X/Y}$ is the cone on the map $\NL_{X/P}[-1] \to i^*\NL_{P/Y}$ which is equivalent to the statement of the lemma by our computation of the cohomology sheaves of these objects in the derived category given above. \end{proof} \begin{lemma}