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Lemma about excess conormal sheaf; algebra version

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aisejohan committed Nov 27, 2019
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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}




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