symmetric monoidal (∞,1)-category of spectra
The group algebra of a group $G$ over a ring $R$ is the associative algebra whose elements are formal linear combinations over $R$ of the elements of $G$ and whose multiplication is given on these basis elements by the group operation in $G$.
Let $G$ be a discrete group. Let $R$ be a commutative ring.
The group $R$-algebra $R[G]$ is the associative algebra over $R$
whose underlying $R$-module is the the free module over $R$ on the underlying set of $G$;
whose multiplication is given on basis elements by the group operation.
By the discussion at free module an element $r$ in $R[G]$ is a formal linear combination of basis elements in $G$ with coefficients in $R$, hence a formal sum
with $\forall_{g \in G} (r_g \in R)$ and only finitely many of the coefficients different from $0 \in R$.
The addition of algebra elements is given by the componentwise addition of coefficients
and the multiplication is given by
The formal linear combinations over $R$ of element in $G$ may equivalently be thought of as functions
from the underlying set of $G$ to the underlying set of $R$ which have finite support. Accordingly, oftn the underlying set of the group $R$-algebra is written as
and for the basis elements one writes
the characteristic function of an element $g \in G$, defined by
In terms of this the product in the group algebra is called the convolution product on functions.
The notion of group algebra is a special case of that of a groupoid algebra, hence of category algebra.
The completed group ring of a profinite group is a pseudocompact ring. Let $\hat{\mathbb{Z}}$ be the profinite completion of the ring of integers, $\mathbb{Z}$, then $\hat{\mathbb{Z}}$ is itself a pseudocompact ring as it is the inverse limit of its finite quotients. Now let $G$ be a profinite group.
The completed group algebra, $\hat{\mathbb{Z}}[\![G]\!]$, of $G$ over $\hat{\mathbb{Z}}$ is the inverse limit of the ordinary group algebras, $\hat{\mathbb{Z}}[G/U]$, of the finite quotients, $G/U$ (for $U$ in the directed set, $\Omega(G)$, of open normal subgroups of $G$), over $\hat{\mathbb{Z}}$;
For $R$ a pseudocompact ring, it is then easy to construct the corresponding pseudo-compact group algebra of $G$ over $R$; see the paper by Brumer.
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A group algebra is in particular a Hopf algebra and a $G$-graded algebra.
The following states a universal property of the construction of the group algebra.
There is an adjunction
between the category of associative algebras over $R$ and that of groups, where $R[-]$ forms group rings and $(-)^\times$ assigns to an $R$-algebra its group of units.
Let $V$ be an abelian group. A homomorphism of rings $R[G]\to End(V)$ of the group ring to the endomorphism ring of $V$ is equivalently a $R[G]$-module structure on $V$. And any homomorphism of groups $p:G\to Aut(V)$ to the automorphism group of $V$ extends to to a morphism of rings. This observation is used extensively in the theory of group representations. See also at module – Abelian groups with G-action as modules over a ring.
Let $G$ be a finite group, let $R = k$ be a field.
Then $k[G]$ is a semi-simple algebra precisely if the order of $G$ is not divisible by the characteristic of k.
Lecture notes include
The universal localization of group rings (see also at Snaith's theorem) is discussed in
M. Farber, P. Vogel, The Cohn localization if the free group ring, Math. Proc. Camb. Phil. Soc. (1992) 111, 433 (pdf)
Davidson, Nicholas, Modules Over Localized Group Rings for Groups Mapping Onto Free Groups (2011). Boise State University Theses and Dissertations. Paper 170. (web)
For the case of profinite groups, see