Lemma 21.26.1. In the situation above, choose a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}$-modules representing $K$. Using $-1$ times the canonical map for one of the four arrows we get maps of complexes

\[ \mathcal{I}^\bullet (X) \xrightarrow {\alpha } \mathcal{I}^\bullet (Z) \oplus \mathcal{I}^\bullet (Y) \xrightarrow {\beta } \mathcal{I}^\bullet (E) \]

with $\beta \circ \alpha = 0$. Thus a canonical map

\[ c^ K_{X, Z, Y, E} : \mathcal{I}^\bullet (X) \longrightarrow C(\beta )^\bullet [-1] \]

This map is canonical in the sense that a different choice of K-injective complex representing $K$ determines an isomorphic arrow in the derived category of abelian groups. If $c^ K_{X, Z, Y, E}$ is an isomorphism, then using its inverse we obtain a canonical distinguished triangle

\[ R\Gamma (X, K) \to R\Gamma (Z, K) \oplus R\Gamma (Y, K) \to R\Gamma (E, K) \to R\Gamma (X, K)[1] \]

All of these constructions are functorial in $K$.

**Proof.**
This lemma proves itself. For example, if $\mathcal{J}^\bullet $ is a second K-injective complex representing $K$, then we can choose a quasi-isomorphism $\mathcal{I}^\bullet \to \mathcal{J}^\bullet $ which determines quasi-isomorphisms between all the complexes in sight. Details omitted. For the construction of cones and the relationship with distinguished triangles see Derived Categories, Sections 13.9 and 13.10.
$\square$

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