The Stacks project

Lemma 17.28.7. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Let $C$ be the cone of the map $\mathop{N\! L}\nolimits _{X/Z} \to \mathop{N\! L}\nolimits _{X/Y}$ of complexes of $\mathcal{O}_ X$-modules. There is a canonical map

\[ f^*\mathop{N\! L}\nolimits _{Y/Z} \to C[-1] \]

which produces a canonical six term exact sequence

\[ \xymatrix{ H^0(f^*\mathop{N\! L}\nolimits _{Y/Z}) \ar[r] & H^0(\mathop{N\! L}\nolimits _{X/Z}) \ar[r] & H^0(\mathop{N\! L}\nolimits _{X/Y}) \ar[r] & 0 \\ H^{-1}(f^*\mathop{N\! L}\nolimits _{Y/Z}) \ar[r] & H^{-1}(\mathop{N\! L}\nolimits _{X/Z}) \ar[r] & H^{-1}(\mathop{N\! L}\nolimits _{X/Y}) \ar[llu] } \]

of cohomology sheaves.

Proof. Consider the maps of sheaves rings

\[ (g \circ f)^{-1}\mathcal{O}_ Z \to f^{-1}\mathcal{O}_ Y \to \mathcal{O}_ X \]

and apply Lemma 17.28.5. $\square$


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