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I want to construct an example of Herbrand quotient's hexagon diagram.

Let $0 \to p\mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} \to 0$ be a short exact sequence of $\mathbb{Z}$-modules and we get induced long exact sequence

$$H^0(\mathbb{Z}, p\mathbb{Z}) \to H^0(\mathbb{Z}, \mathbb{Z}) \to H^0(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z}) \to H^1(\mathbb{Z}, p\mathbb{Z}) \to H^1(\mathbb{Z}, \mathbb{Z}) \to$$ $$H^1(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z}) \to H^2(\mathbb{Z}, p\mathbb{Z})\to \dots $$

and $H^0(\mathbb{Z}, p\mathbb{Z}) \cong H^2(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z})$, so we can get Herbrand's hexagon diagram of $\mathbb{Z}$-modules.

$$H^0(\mathbb{Z}, p\mathbb{Z}) \to H^0(\mathbb{Z}, \mathbb{Z}) \to H^0(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z}) \to H^1(\mathbb{Z}, p\mathbb{Z}) \to H^1(\mathbb{Z}, \mathbb{Z}) \to$$ $$H^1(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z}) \to H^2(\mathbb{Z},p\mathbb{Z})$$

But I have stuck with calculating each component of the sequence above. I would be appreciated if you could give me help completing the exact sequence. If you know more good example of Herbrand quotient hexagon, such an example is also appreciated, thank you.

P.S.

　Thanks to a comment, I noticed I cannot make an example of Herbrand quotient hexagon by $\mathbb{Z}$-modules.

I would be appreciated if you could construct example using $\mathbb{Z}/p\mathbb{Z}$-modules.

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    I believe the last term in both displayed sequences should be $H^2(\mathbb{Z}, p\mathbb{Z})$ not $H^2(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z})$. – Michael Albanese Dec 28 '19 at 14:30
  • Thank you very much I edited. –  Dec 28 '19 at 14:36
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    Note that $p\mathbb{Z} \cong \mathbb{Z}$, so all you need to know is $H^i(\mathbb{Z}, \mathbb{Z})$ for $i = 0, 1, 2$ and $H^j(\mathbb{Z}, \mathbb{Z}/p\mathbb{Z})$ for $j = 0, 1$. Do you know these groups? – Michael Albanese Dec 28 '19 at 14:44
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    I thought the Herbrand hexagon was about functors $\widehat H^i(C_n,-)$ where $C_n$ is finite cyclic, not functors $H^i(\Bbb Z,-)$. – Angina Seng Dec 28 '19 at 14:47
  • Then an example cannot be constructed by Z-modules? We should take, for example, Z/pZ-modules? I would be appreciated if you could make an example in Z/pZ-modules sequence..Thank you very much for your advice. –  Dec 28 '19 at 15:16

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