The Stacks project

Lemma 37.13.11. 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

\[ \mathop{N\! L}\nolimits _{X/Y} \cong \left(\mathcal{C}_{X/P} \to i^*\Omega _{P/Y}\right) \]

in $D(\mathcal{O}_ X)$ where the conormal sheaf $\mathcal{C}_{X/P}$ is placed in degree $-1$.

Proof. (For the parenthetical statement see Lemma 37.11.7.) By Lemmas 37.13.8 and 37.13.5 we have $\mathop{N\! L}\nolimits _{X/P} = \mathcal{C}_{X/P}[1]$ and $\mathop{N\! L}\nolimits _{P/Y} = \Omega _{P/Y}$ with $\Omega _{P/Y}$ locally projective. This implies that $i^*\mathop{N\! L}\nolimits _{P/Y} \to i^*\Omega _{P/Y}$ is a quasi-isomorphism too (small detail omitted; the reason is that $i^*\mathop{N\! L}\nolimits _{P/Y}$ is the same thing as $\tau _{\geq -1}Li^*\mathop{N\! L}\nolimits _{P/Y}$, see More on Algebra, Lemma 15.78.1). Thus the canonical map

\[ i^*\mathop{N\! L}\nolimits _{P/Y} \to \text{Cone}(\mathop{N\! L}\nolimits _{X/Y} \to \mathop{N\! L}\nolimits _{X/P})[-1] \]

of Modules, Lemma 17.28.7 is an isomorphism in $D(\mathcal{O}_ X)$ because the cohomology group $H^{-1}(i^*\mathop{N\! L}\nolimits _{P/Y})$ is zero by what we said above. In other words, we have a distinguished triangle

\[ i^*\mathop{N\! L}\nolimits _{P/Y} \to \mathop{N\! L}\nolimits _{X/Y} \to \mathop{N\! L}\nolimits _{X/P} \to i^*\mathop{N\! L}\nolimits _{P/Y}[1] \]

Clearly, this means that $\mathop{N\! L}\nolimits _{X/Y}$ is the cone on the map $\mathop{N\! L}\nolimits _{X/P}[-1] \to i^*\mathop{N\! L}\nolimits _{P/Y}$ 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. $\square$


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