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The Stacks project

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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