The Stacks project

Lemma 90.5.8. Let $A \to B \to C$ be ring maps. If $B$ is a polynomial algebra over $A$, then there is a distinguished triangle $L_{B/A} \otimes _ B^\mathbf {L} C \to L_{C/A} \to L_{C/B} \to L_{B/A} \otimes _ B^\mathbf {L} C[1]$ in $D(C)$.

Proof. We will use the observations of Remark 90.5.5 without further mention. Choose a resolution $\epsilon : P_\bullet \to C$ of $C$ over $B$ (for example the standard resolution). Since $B$ is a polynomial algebra over $A$ we see that $P_\bullet $ is also a resolution of $C$ over $A$. Hence $L_{C/A}$ is computed by $\Omega _{P_\bullet /A} \otimes _{P_\bullet , \epsilon } C$ and $L_{C/B}$ is computed by $\Omega _{P_\bullet /B} \otimes _{P_\bullet , \epsilon } C$. Since for each $n$ we have the short exact sequence $0 \to \Omega _{B/A} \otimes _ B P_ n \to \Omega _{P_ n/A} \to \Omega _{P_ n/B}$ (Algebra, Lemma 10.138.9) and since $L_{B/A} = \Omega _{B/A}[0]$ (Lemma 90.4.7) we obtain the result. $\square$


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