The Stacks project

Lemma 15.78.5. 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 15.32.2, then the kernel of $H^{-1}(\mathop{N\! L}\nolimits _{B/A} \otimes _ B B') \to H^{-1}(\mathop{N\! L}\nolimits _{B'/A'})$ is a finite projective $B'$-module.

Proof. By Lemma 15.78.4 the complexes $\mathop{N\! L}\nolimits _{B/A}$ and $\mathop{N\! L}\nolimits _{B'/A'}$ are perfect of tor-amplitude in $[-1, 0]$. Combining Lemmas 15.78.1, 15.70.9, and 15.63.13 we have $\mathop{N\! L}\nolimits _{B/A} \otimes _ B B' = \mathop{N\! L}\nolimits _{B/A} \otimes _ B^\mathbf {L} B'$ and this complex is also perfect of tor-amplitude in $[-1, 0]$. Choose a distinguished triangle

\[ C \to \mathop{N\! L}\nolimits _{B/A} \otimes _ B B' \to \mathop{N\! L}\nolimits _{B'/A'} \to C[1] \]

in $D(B')$. By Lemmas 15.70.4 and 15.63.5 we conclude that $C$ is perfect with tor-amplitude in $[-1, 1]$. By Lemma 15.78.2 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 15.62.4) and flat (Lemma 15.63.6). Hence it is finite projective by Algebra, Lemma 10.77.2. $\square$


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