Lemma about excess conormal sheaf; algebra version

more-algebra.tex

@@ -20094,6 +20094,42 @@ \section{The naive cotangent complex}
 in $[-1, 0]$.
 \end{proof}
 
+\begin{lemma}
+\label{lemma-base-change-lci-bis}
+Consider a cocartesian diagram of rings
+$$
+\xymatrix{
+B \ar[r] & B' \\
+A \ar[r] \ar[u] & A' \ar[u]
+}
+$$
+If $A \to B$ and $A' \to B'$ are local complete intersections as in
+Definition \ref{definition-local-complete-intersection}, then
+the kernel of $H^{-1}(\NL_{B/A} \otimes_B B') \to H^{-1}(\NL_{B'/A'})$
+is a finite projective $B'$-module.
+\end{lemma}
+
+\begin{proof}
+By Lemma \ref{lemma-lci-NL} the complexes $\NL_{B/A}$ and $\NL_{B'/A'}$
+are perfect of tor-amplitude in $[-1, 0]$.
+Combining Lemmas \ref{lemma-tensor-NL}, \ref{lemma-pull-perfect}, and
+\ref{lemma-pull-tor-amplitude} we have
+$\NL_{B/A} \otimes_B B' = \NL_{B/A} \otimes_B^\mathbf{L} B'$
+and this complex is also perfect of tor-amplitude in $[-1, 0]$.
+Choose a distinguished triangle
+$$
+C \to \NL_{B/A} \otimes_B B'  \to \NL_{B'/A'} \to C[1]
+$$
+in $D(B')$. By Lemmas \ref{lemma-two-out-of-three-perfect} and
+\ref{lemma-cone-tor-amplitude} we conclude that $C$ is perfect
+with tor-amplitude in $[-1, 1]$. By Lemma \ref{lemma-base-change-NL}
+the complex $C$ has only one nonzero cohomology module, namely the module
+of the lemma sitting in degree $-1$. This module is of finite presentation
+(Lemma \ref{lemma-n-pseudo-module}) and flat
+(Lemma \ref{lemma-tor-dimension}). Hence it is finite projective by
+Algebra, Lemma \ref{algebra-lemma-finite-projective}.
+\end{proof}
+