The Stacks project

Proposition 89.13.4. Let $A \to B$ be a local complete intersection map. Then $L_{B/A}$ is a perfect complex with tor amplitude in $[-1, 0]$.

Proof. Choose a surjection $P = A[x_1, \ldots , x_ n] \to B$ with kernel $J$. By Lemma 89.10.3 we see that $J/J^2 \to \bigoplus B\text{d}x_ i$ is quasi-isomorphic to $\tau _{\geq -1}L_{B/A}$. Note that $J/J^2$ is finite projective (More on Algebra, Lemma 15.31.3), hence $\tau _{\geq -1}L_{B/A}$ is a perfect complex with tor amplitude in $[-1, 0]$. Thus it suffices to show that $H^ i(L_{B/A}) = 0$ for $i \not\in [-1, 0]$. This follows from (89.7.0.1)

\[ L_{P/A} \otimes _ P^\mathbf {L} B \to L_{B/A} \to L_{B/P} \to L_{P/A} \otimes _ P^\mathbf {L} B[1] \]

and Lemma 89.13.3 to see that $H^ i(L_{B/P})$ is zero unless $i \in \{ -1, 0\} $. (We also use Lemma 89.4.7 for the term on the left.) $\square$


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