I'm learning homological algebra using Weibel's book. And I have trouble in exercise 5.4.4:

Let $f : B \to C$ be a map of filtered chain complexes. For each $r \geq 0$, define a filtration on the mapping cone $cone(f)$ by $$F_p cone(f)_n = F_{p-r} B_{n-1} \oplus F_p C_n.$$ Show that $E^r_p (cone f)$ is the mapping cone of $f^r : E^r_p (B) \to E^r_p (C)$.

I'm not used to deal with mapping cone and spectral sequence, and I'm curious about the approach to this problem.

At first, I tried to represent $E^r_p (conef)$ explicitly by the construction process of $E^r_p$, but the indexing and calculation were so complicated that I doubted if it was the right way.

Is there a hint about this exercise?

  • $\begingroup$ Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please edit the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. $\endgroup$ Nov 25 '21 at 8:17
  • $\begingroup$ What edition of the book are you reading? The numbering changes from one to another. $\endgroup$
    – Pedro Tamaroff
    Nov 26 '21 at 12:27

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