Lemma 15.85.5 (0FV1)—The Stacks project

Lemma 15.85.5. 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 15.33.2, 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 15.85.4 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 15.85.1, 15.74.9, and 15.66.13 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 15.74.4 and 15.66.5 we conclude that $C$ is perfect with tor-amplitude in $[-1, 1]$ . By Lemma 15.85.2 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 15.64.4) and flat (Lemma 15.66.6). Hence it is finite projective by Algebra, Lemma 10.78.2. $\square$

The Stacks project

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