Skip to content
Permalink
Browse files

Lemma about excess conormal sheaf; schemes version

  • Loading branch information
aisejohan committed Nov 27, 2019
1 parent 31ec4b8 commit 9155485e21cd233bbcebc32a95e5d6ade2e25949
Showing with 53 additions and 1 deletion.
  1. +53 −1 more-morphisms.tex
@@ -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}

0 comments on commit 9155485

Please sign in to comment.
You can’t perform that action at this time.