Proposition 92.7.4. Let $A \to B \to C$ be ring maps. There is a canonical distinguished triangle
in $D(C)$.
Proposition 92.7.4. Let $A \to B \to C$ be ring maps. There is a canonical distinguished triangle
in $D(C)$.
Proof. Consider the short exact sequence of sheaves of Lemma 92.7.1 and apply the derived functor $L\pi _!$ to obtain a distinguished triangle
in $D(C)$. Using Lemmas 92.7.3 and 92.4.3 we see that the second and third terms agree with $L_{C/A}$ and $L_{C/B}$ and the first one equals
The first equality by Cohomology on Sites, Lemma 21.39.6 (and flatness of $\Omega _1$ as a sheaf of modules over $\underline{B}$) and the second by Lemma 92.4.3. $\square$
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