Arithmetic application: Complete group ring and group ring for infinite group

Question

Let $G$ be a profinite(infinite) group, $\Lambda$ be the complete group ring(Iwasawa algerbra) of $G$ over a unity ring $R$. My first question is that do we know something about the relation with $\Lambda$ and the group ring $R[G]$. I have some feeling that $\Lambda$ contains $R[G]$, but I'm not sure.

My second question is that does $R[G]$ have any arithmetic application, like group rings for finite groups?

The motiviation for this problem is that when I read the book, Selmer complex, JAN NEKOVÁR, I find that in the settings, like on page 95 and 96, there always occur many profinite $G$ and $R[G]$-modules. I'm a little confused about it and not sure whether it is just a quirk of the author to use $R[G]$ to denote $R[[G]]$. Maybe it caused by my ignorance.

But later I find in other article there exists the same expression:

Let $G$ be a profinite group, $R$ a topological ring and let $M$ be a topological $R$-module with a continuous $R$-action and we additionally assume that $M$ is a continuous $R[G]$-module, i.e. the group homomorphism $G \rightarrow Aut_{R,cts}(M)$ is continuous.

So I want to know if it makes sense for using group ring for infinite group to study number theory, what will happen differently with group ring for finite group and complete group ring. Thanks very much.

Answer 1

About the first question, There exists a map $R[G]\rightarrow R[[G]]$ given by $x\mapsto (x\mod H)_{H}$ which is only injective if $\cap H=0$ (maybe you have to assume $G$ is commutative for that, I'm too lazy to check).

Does $R[G]$ have arithmetic applications? Much of arithmetic involves understanding of continuous $\mathbb{Z}_{p}$-Galois representations which are $\mathbb{Z}_{ p}[G_{\mathbb{Q}}]$-modules, so $\mathbb{Z}_{p}[G_{\mathbb{Q}}]$ (and in general $R [G_{\mathbb{ Q}}]$) have many applications.

So why $R[[G]]$? This is the idea of ​​Iwasawa's theory. For example, in Iwasawa's cyclotomic theory, if $M$ is a (finite-dimensional) $\mathbb{Z}_{p}[G_{\mathbb{Q}}]$-module then $\displaystyle \lim_{ \leftarrow_ {n}}M/\chi_{\text{cyc}}^{n}M$ is a $\mathbb{Z}_{p}[[G(\mathbb{Q}(\zeta_{p ^{ \infty}}),\mathbb{Q})]]$-module which is easier to understand.

You can do the same for cohmology groups: instead of $H(G_{\mathbb{Q}},M)$ you consider $\lim H(G_{\mathbb{Q}(\zeta_{p ^{ n}})},M)$.

What about Nekovar's book? The reason Nekovar considers $R[[G]]$ is not because it makes more sense in arithmetic, but because he wants to expand a duality theory for the limit of the group cohomology as above (more precisely for what we call Selmer complexes).

I hope this will help

Answer 2

I am not sure precisely what your question is asking, but I have found that the old paper of Brumer, Pseudocompact algebras, profinite groups and class formations, J. Alg, 4, (1966), 442–470, link, is very useful for the general area of pseudocompact algebras and their use as analogues of the usual group algebra construction especially from a homological point of view.

I know there is a lot more recent work than that in the area, but that paper from 1966 does seem to answer your question as to whether the construction is useful in number theoretic contexts and also is fairly accessible.

Much more recently, the algebraic side of work on Knots and Primes, can be viewed from a perspective of the modules over various topological algebras as you seem to be considering, and the analogies between group rings (as used in parts of elementary Knot theory in, say, the classic text by Crowell and Fox) and the completed group algebras are central to those considerations.