Lemma 92.5.8. 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 92.5.5 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} \to 0 (Algebra, Lemma 10.138.9) and since L_{B/A} = \Omega _{B/A}[0] (Lemma 92.4.7) we obtain the result. \square
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