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edited Jul 25, 2021 at 22:44

I don't know the answer to your question in all generality. However, if one continues with $M=\mathbb{Z}$ and $C=C_K$ then one can ask what happens as we vary $i$.

for $i=-3$ we identify $H^{-3}(C, \mathbb{Z})$ with elements of $C_K$ of norm $1$ modulo ``principal elements'' (those of the form $I_GC_K$ where $I_G$ is the augmentation ideal). This isomorphism is used in section 2 of Peter Roquette's article in Cassels and Frohlich,
for $i=-1$ the groups are trivial,
for $i=0$ we identify $H^2(G, C_k)$ with $\mathbb{Z}/n\mathbb{Z}$,
for $i=1$ the groups are trivial,
the $i=2$ case is described in the exercises on page 157 of

Artin, Emil; Tate, John, Class field theory, Providence, RI: AMS Chelsea Publishing (ISBN 978-0-8218-4426-7/hbk). vii, 192 p. (2009). ZBL1179.11040.

For some applications of these isomorphisms where $M$ is not $\mathbb{Z}$, you can look at the following paper by Kottwitz (in particular see section 4): https://arxiv.org/abs/1401.5728

I don't know the answer to your question in all generality. However, if one continues with $M=\mathbb{Z}$ and $C=C_K$ then one can ask what happens as we vary $i$.

for $i=-3$ we identify $H^{-3}(C, \mathbb{Z})$ with elements of $C_K$ of norm $1$ modulo ``principal elements'' (those of the form $I_GC_K$ where $I_G$ is the augmentation ideal). This isomorphism is used in section 2 of Peter Roquette's article in Cassels and Frohlich,
for $i=-1$ the groups are trivial,
for $i=0$ we identify $H^2(G, C_k)$ with $\mathbb{Z}/n\mathbb{Z}$,
for $i=1$ the groups are trivial,
the $i=2$ case is described in the exercises on page 157 of

Artin, Emil; Tate, John, Class field theory, Providence, RI: AMS Chelsea Publishing (ISBN 978-0-8218-4426-7/hbk). vii, 192 p. (2009). ZBL1179.11040.

I don't know the answer to your question in all generality. However, if one continues with $M=\mathbb{Z}$ and $C=C_K$ then one can ask what happens as we vary $i$.

for $i=-3$ we identify $H^{-3}(C, \mathbb{Z})$ with elements of $C_K$ of norm $1$ modulo ``principal elements'' (those of the form $I_GC_K$ where $I_G$ is the augmentation ideal). This isomorphism is used in section 2 of Peter Roquette's article in Cassels and Frohlich,
for $i=-1$ the groups are trivial,
for $i=0$ we identify $H^2(G, C_k)$ with $\mathbb{Z}/n\mathbb{Z}$,
for $i=1$ the groups are trivial,
the $i=2$ case is described in the exercises on page 157 of

Artin, Emil; Tate, John, Class field theory, Providence, RI: AMS Chelsea Publishing (ISBN 978-0-8218-4426-7/hbk). vii, 192 p. (2009). ZBL1179.11040.

For some applications of these isomorphisms where $M$ is not $\mathbb{Z}$, you can look at the following paper by Kottwitz (in particular see section 4): https://arxiv.org/abs/1401.5728

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created Jul 25, 2021 at 22:37

Alexander

I don't know the answer to your question in all generality. However, if one continues with $M=\mathbb{Z}$ and $C=C_K$ then one can ask what happens as we vary $i$.

for $i=-3$ we identify $H^{-3}(C, \mathbb{Z})$ with elements of $C_K$ of norm $1$ modulo ``principal elements'' (those of the form $I_GC_K$ where $I_G$ is the augmentation ideal). This isomorphism is used in section 2 of Peter Roquette's article in Cassels and Frohlich,
for $i=-1$ the groups are trivial,
for $i=0$ we identify $H^2(G, C_k)$ with $\mathbb{Z}/n\mathbb{Z}$,
for $i=1$ the groups are trivial,
the $i=2$ case is described in the exercises on page 157 of

Artin, Emil; Tate, John, Class field theory, Providence, RI: AMS Chelsea Publishing (ISBN 978-0-8218-4426-7/hbk). vii, 192 p. (2009). ZBL1179.11040.