Lemma 15.28.11. Let $R$ be a ring. Let $f_1, \ldots , f_{r - 1}$ be a sequence of elements of $R$. Let $f, g \in R$. The complex $K_\bullet (f_1, \ldots , f_{r - 1}, fg)$ is homotopy equivalent to the cone of a map of complexes

\[ K_\bullet (f_1, \ldots , f_{r - 1}, f)[1] \longrightarrow K_\bullet (f_1, \ldots , f_{r - 1}, g) \]

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