Basic Iwasawa Theory Question I'm looking at a paper that introduces some terms and intends to use concepts from Iwasawa Theory. I instantly find myself stuck at the second sentence and even after much searching on the internet, I still can't quite understand it.
The paragraph that I'm trying to understand is taken from page 19 of http://wstein.org/papers/shark/shark.pdf
This is the bit that I do not understand:
Let $\Lambda$ be the completed group algebra $\mathbb{Z}_p[[\Gamma]]$. We use a fixed topological generator $\gamma$ of $\Gamma$ to identify $\Lambda$ with $\mathbb{Z}_p[[T]]$ by sending $\gamma$ to $1+T$. Any finitely-generated $\Lambda$-module admits a decomposition up to quasi-isomorphism as a direct sum of elementary $\Lambda$-modules.


*

*What is a completed group algebra $\mathbb{Z}_p[[\Gamma]]$? I've seen the wikipedia page http://en.wikipedia.org/wiki/Group_algebra but it doesn't seem to give me anything I can understand. An elaboration or an example might be good.

*How is $\mathbb{Z}_p[[\Gamma]]$ different from $\mathbb{Z}_p[[T]]$?

*What are elementary modules?
Thanks for any help.
 A: 1) Think of the completed group algebra as "convergent power series in elements of $\Gamma$ with coefficients in $\mathbb{Z}_p$". Alternatively, since as it turns out this is isomorphic to $\mathbb{Z}_p[[T]]$, think of it just as power series in a variable $T$ with coefficients in $\mathbb{Z}_p$. 
In general, a group algebra $R[G]$ is the set of polynomials in the "variables" which are just the elements of the group $G$ whose coefficients are elements of some ring $R$. The addition of two such polynomials is the obvious thing, while multiplication is given by the standard "FOIL" you learn in elementary school, only now we are multiplying elements of the group $G$ rather than indeterminates like $x,y,z$ etc. So to "complete" such a thing is literally a completion, in the sense that we can put a topology on the group algebra and then complete it so that all its Cauchy sequences converge. As it turns out, in this case the "completion" of polynomials gives what you expect, which are power series!
2) The difference is a technical one, which I explained in the first paragraph. But you should think of them as behaving the same, since they are isomorphic! The particular choice of isomorphism is important, though.
3) An elementary $\Lambda$ module is the analogue of a finitely generated module over a PID. In the latter case, the structure theorem tells us all f.g. modules over a PID $R$ are isomorphic to $R^n\times torsion$, where the torsion piece is just $R$ modulo various proper (principal!) ideals satisfying some divisibility criterion. In this case, an elementary $\Lambda$ module is similar: It is just a free part $\Lambda^n$ plus a direct sum of things that look like $\Lambda/(f^{\alpha})$ where $f$ is either the prime $p$ or else an irreducible distinguished polynomial, and $\alpha$ is just some power.
