It is usual in studying questions involving group rings to pay attention to their structure as rings and algebras, and to investigate modules over them as well as their groups of units. It is well known that from this point of view they have been and are being intensively studied (see [8], [9], [11], [12]). But if the group is commutative, these rings are commutative rings and as such have their naturally associated affine schemes. This was already in some sense studied before: the prime ideals of the spectrum of the representationring of G was considered in [2], when G is finite, and in [10], when G is a compact Lie group. On the other hand, such problems as the determination of the singular points, ramification points and Ka¨hler differentials, seemingly have not been treated yet. Let G be a finite abelian group, A := Z[G] its integral group ring, and

Spec A its spectrum, that is, the set of all prime ideals in A, considered 183

as a topological space with the Zariski (spectral) topology (also denoted by Spec A) and endowed with the structural sheaf. Then A is a noetherian commutative ring and Spec A is a noetherian affine scheme. In Classical Algebraic Geometry the points of a variety correspond to the maximal ideals of its coordinate ring and the tangent space at a point x is the dual of the Ox/mx-vector space mx/mx2, where Ox is the local ring of x and mx is its maximal ideal. The point x is regular if and only if the ring Ox is regular. These concepts extend to arbitrary locally noetherian schemes, in particular to the spectrum of a commutative noetherian ring. We remind that in the spectral topology the closed points are exactly the maximal ideals. In the case of groups of prime order Cp the underlying topological space Spec Z[Cp], consists of two irreducible components, one of which is homeomorphic to Spec (Z) and the other to Spec OK , where OK is the ring of integers of the cyclotomic field Q(ζp) of p-th roots of unity. As it is well known, OK = Z[ζp]. The components intersect at a single point, which is the only singular point of Spec Z[Cp]. In this paper we describe Spec Z[G], for finite abelian G; in particular we determine its components and singularities. We shall use the following notations. For an ideal I in A we denote by

V (I) the set of prime ideals in A containing I. These are the closed sets of the spectral topology. By dim A we mean the Krull dimension of A.