symmetric monoidal (∞,1)-category of spectra
The group algebra of a group over a ring is the associative algebra whose elements are formal linear combinations over of the elements of and whose multiplication is given on these basis elements by the group operation in .
The group -algebra is the associative algebra over
with and only finitely many of the coefficients different from .
The addition of algebra elements is given by the componentwise addition of coefficients
and the multiplication is given by
from the underlying set of to the underlying set of which have finite support. Accordingly, oftn the underlying set of the group -algebra is written as
and for the basis elements one writes
the characteristic function of an element , defined by
In terms of this the product in the group algebra is called the convolution product on functions.
The completed group ring of a profinite group is a pseudocompact ring. Let be the profinite completion of the ring of integers, , then is itself a pseudocompact ring as it is the inverse limit of its finite quotients. Now let be a profinite group.
The completed group algebra, , of over is the inverse limit of the ordinary group algebras, , of the finite quotients, (for in the directed set, , of open normal subgroups of ), over ;
For a pseudocompact ring, it is then easy to construct the corresponding pseudo-compact group algebra of over ; see the paper by Brumer.
The following states a universal property of the construction of the group algebra.
There is an adjunction
Let be an abelian group. A homomorphism of rings of the group ring to the endomorphism ring of is equivalently a -module structure on . And any homomorphism of groups to the automorphism group of 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 be a finite group, let be a field.
Lecture notes include
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