## reference request – A long exact sequence involving the mapping cone (Exercise 1.5.9 in Weibel)

Let $$f:Bto C$$ be a morphism of chain complexes. Exercise 1.5.9 from Weibel’s “An introduction to homological algebra”, page 24, states that the natural maps $$ker f(-1)xrightarrowalpha{rm cone}(f)xrightarrowbeta{rm coker} f$$ give rise to a long exact sequence:
$$cdotsxrightarrowdelta H_{n-1}(ker f)xrightarrowalpha H_n({rm cone}(f))xrightarrowbeta H_n({rm coker} f)xrightarrowdelta H_{n-2}(ker f)xrightarrowalphacdots$$
(In the book there is a small mistake. The map $$ker f(-1)to{rm cone}(f)$$ was denoted by $$delta$$ instead of $$alpha$$.)

My question is, is there any quotable source for this result? I cannot quote an exercise, especially when it is incomplete. It doesn’t state what the map $$delta$$ is and in my paper I need that $$delta$$.

I figured out that $$delta=delta’delta”$$, where $$delta’$$ is the connecting map $$H_{n-1}({rm Im}f)to H_{n-2}(ker f)$$, coming from the exact sequence $$0toker fto Bxrightarrow f{rm Im}fto 0$$, and $$delta”$$ is the connecting map $$H_n({rm coker}f)to H_{n-1}({rm Im}f)$$, coming from the exact sequence $$0to{rm Im}fto Cto{rm coker} fto 0.$$

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