The group-algebra of an abelian group is commutative, so we can consider the spectrum of this algebra. Are there any information about the abelian group that we can obtain from such considerations? That is to say, could we study abelian groups by considering the spectrum and the scheme of its group-algebra?  

Since I know nothing about the subject, any reference is mostly welcomed. Thanks in advance.  

P.S. I also posted in mathmatics stack exchange
[Here.](https://math.stackexchange.com/questions/308939/commutative-ring-and-its-group-algebra-and-abelian-group-algebra-as-a-commutati)



Spectrum and scheme of the commutative group-algebra of an abelian group.
The spectrum of the group algebra of a commutative group is called a diagonalizable group scheme. This is defined in [SGA 3](http://www.math.jussieu.fr/~polo/SGA3/) Exposé VIII Section 1. Several geometric characterizations of group-theoretic properties are given in Proposition 2.1. A lot more is written in later sections, such as material on principal homogeneous spaces, quotients of affine schemes by diagonalizable group schemes, and representability of restriction of scalars.


If that isn't enough for you, Exposés 9-11 are about group schemes that are locally-on-the-base isomorphic to diagonalizable group schemes.


