# stacks/stacks-project

Lemma about excess conormal sheaf; algebra version

 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}