Existence of certain homomorphism on cochaincomplexes

Question

I found the following problem online. I'm not sure if this is easy or not as I'm not sure how one defines the class of an element in $H^p$.

Let $C=\bigoplus_{p\in\mathbb Z}C^p$, $C^\prime$ ja $C^{\prime\prime}$ be cochaincomplexes and let $$0\to C^\prime\overset u\to C\overset v\to C^{\prime\prime}\to 0$$ be the exact sequence on complexes.

How can I show that there are well defined homomorphisms $$\ldots \to H^p(C^\prime )\overset{u_*}\to H^p(C)\overset{v_*}\to H^p(C^{\prime\prime})\overset\delta\to H^{p+1}(C^\prime)\to\ldots$$ satisfying: $u_*(\overline{z^\prime})=\overline{u(z^\prime)}$ as $z^\prime\in Z^p(C^\prime)$ $(\overline{z^\prime}\in H^p(C^\prime)$ is the class of $z^\prime)$, $v_*(\overline{z})=\overline{v(z)}$ as $z\in Z^p(C)$, and $\delta (\overline{z^{\prime\prime}})=\overline{z^\prime}$, as $z^{\prime\prime}\in Z^p(C^{\prime\prime})$, $z^\prime\in Z^{p+1}(C^\prime)$ and for some $x\in C^p$ satisfies $v(x)=z^{\prime\prime}$, $dx=u(z^\prime)$.

1. How can I show the existence of homomorphisms?

2. How can I show that the pair of homomorphisms $(u_*,v_*)$ is exact?

3. How can I show that the pair $(v_*,\delta)$ is exact?

Answer 1

Well, this is a classical result of homological algebra, the cohomology LES. You get your sequence by applying the Snake lemma to the commutative diagram with exact rows having as columns exact sequences of the form $0\rightarrow H^n(C)\rightarrow C^n/B^n(C)\rightarrow Z^n(C)[1]\rightarrow H^n(C)[1]\rightarrow 0$ (note that $H^n(C)$ is the kernel of the middle map, which is induced by $d$, and $H^n(C)[1]$ the cokernel), and the rows are induced by the short exact sequence $0\rightarrow C'\rightarrow C\rightarrow C''\rightarrow 0$ of complexes. Alternatively you do it by a diagram chase. But you should learn about the snake lemma, it is very useful to establish diagram lemmas, see e.g. here: https://math.stackexchange.com/questions/581698/on-a-commutative-diagram/1085070#1085070 . Details can be found in any homological algebra textbook, e.g. Weibel or Bourbaki