We continue our quest to climb Mount Takagi-Artin.

In class field theory, it will be important to compute and keep track of the orders of groups such as ${(K^*:NL^*)}$, where ${L/K}$ is a Galois extension of local fields. A convenient piece of machinery for doing this is the Herbrand quotient, which we discuss today. I only sketch the proofs though, and a little familiarity with the Tate cohomology groups will be useful (but is not strictly necessary if one accepts the essentially combinatorial results without proof or proves them directly).

1. Definition

Let ${G}$ be a cyclic group generated by ${\sigma}$ and ${A}$ a ${G}$-module. It is well-known that the Tate cohomology groups ${H^i_T(G, A)}$ are periodic with period two and thus determined by ${H^0}$ and ${H^{-1}}$. By definition,

$\displaystyle H^0(G,A) = A^G/ NA ,$

where ${A^G}$ consists of the elements of ${A}$ fixed by ${G}$, and ${N: A \rightarrow A}$ is the norm map, ${a \rightarrow \sum_g ga}$. Moreover,

$\displaystyle H^{-1}(G, A) = \mathrm{ker} N/ (\sigma -1) A.$

(Normally, for ${G}$ only assumed finite, we would quotient by the sum of ${(\sigma - 1)A}$ for ${\sigma \in G}$ arbitrary, but here it is enough to do it for a generator—easy exercise.)

If both cohomology groups are finite, define the Herbrand quotient ${Q(A)}$ as

$\displaystyle Q(A) = \frac{ |H_T^0(G,A)|}{|H_T^{-1}(G,A)|}.$ (more…)

Group cohomology is a useful language for expressing the results of class field theory, among (many) other things. There are a few ways I could introduce this. I could define them as derived functors (i.e. as a special case of ${\mathrm{Tor},\mathrm{Ext}}$) or satellites, which would be the most general, but I try to keep my posts somewhat self-contained. I could define them additionally as cochains or coboundaries. I’ve decided to give an axiomatic definition, which will include the previous ones.  (more…)