Lemma 90.20.3. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Then there is a canonical distinguished triangle

in $D(\mathcal{O}_ X)$.

Lemma 90.20.3. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Then there is a canonical distinguished triangle

\[ Lf^* L_{Y/Z} \to L_{X/Z} \to L_{X/Y} \to Lf^*L_{Y/Z}[1] \]

in $D(\mathcal{O}_ X)$.

**Proof.**
Set $h = g \circ f$ so that $h^{-1}\mathcal{O}_ Z = f^{-1}g^{-1}\mathcal{O}_ Z$. By Lemma 90.18.3 we have $f^{-1}L_{Y/Z} = L_{f^{-1}\mathcal{O}_ Y/h^{-1}\mathcal{O}_ Z}$ and this is a complex of flat $f^{-1}\mathcal{O}_ Y$-modules. Hence the distinguished triangle above is an example of the distinguished triangle of Lemma 90.18.8 with $\mathcal{A} = h^{-1}\mathcal{O}_ Z$, $\mathcal{B} = f^{-1}\mathcal{O}_ Y$, and $\mathcal{C} = \mathcal{O}_ X$.
$\square$

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