Lemma 17.31.7. 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 17.31.5.
\square
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