inverse limits of group algebras and profinite groups - MathOverflow

inverse limits of group algebras and profinite groups

2011-12-05T06:03:58Z

For an inverse system {$G_i$} of finite groups, and a fixed field $\mathbb{k}$, one can consider the corresponding group algebras $\mathbb{k}[G_i]$. The latter form an inverse system of $\mathbb{k}$-algebras {$\mathbb{k}[G_i]$} (unless I miss something obvious). Is it true that the inverse limit of {$\mathbb{k}[G_i]$} is the group algebra $\mathbb{k}[G]$, for $G=\lim\limits_\leftarrow${$G_i$} ?

In my case $\mathbb{k}=\mathbb{Z}/p\mathbb{Z}$, and $G_i$ are abelian $p$-groups, if this helps.

Added: I see that the answer is much less trivial than I expected. What about the simplest case, perhaps, when $G_i=\mathbb{Z}/p^i\mathbb{Z}$, for $i\geq 1$, and thus $G$ is the additive group of $\mathbb{Z}_p$? Are there fields for which the complete group algebra $[[\mathbb{k}G]]$ is easy to describe (particularly interesting for me would be the case $\mathbb{k}=\mathbb{Z}/p\mathbb{Z}$.) ? [This is answered by Simon Wadsley in the comment below.]

P.S. This is a spill-over of an innocently looking final year project on invertible circulant matrices of an undergraduate student of mine---I am unfamiliar with profinite things...

Answer by Gjergji Zaimi for inverse limits of group algebras and profinite groups

2011-12-05T07:28:05Z

See section 5.3 in "Profinite Groups" by Ribes and Zalesskii. The inverse limit of group algebras you are referring to is called the complete group algebra and it is the completion of the ordinary group algebra $\mathbb k[G]$ with its natural profinite topology. In other words $\mathbb k[G]$ is densely embedded in this algebra.

This works for any profinite ring $\mathbb k$. What I mean by natural profinite topology on $\mathbb k[G]$ is the one given by the fundamental system of neighbourhoods of $0$ consisting of the ideals $$\text{Ker}(\mathbb k[G]\to (\mathbb k/I)[G/U])$$ where $I$ ranges over the open ideals of $\mathbb k$ and $U$ over the open normal subgroups of $G$.

Answer by Tim Porter for inverse limits of group algebras and profinite groups

2011-12-05T09:43:56Z

Although the reference to Ribes and Zalesski (given in another answer) is excellent, another very good starting point for this area is:

A. Brumer, Pseudocompact algebras, profinite groups and class formations , J. Alg, 4, (1966), 442–470.

The key point is the pseudocompactness of the result. Brumer goes into the homological algebra of these algebras. Of course, in the 45 years since that was published there have been a lot of advances, but the basic theory is very well explained there.