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Suppose that H is a finitely generated Abelian group. Denote by ı : H → N(H ) the natural morphism. If P , Q ∈ N(H ) then P |Q ⇐⇒ P |Q(ı(h) − 1), ∀h ∈ H. 29. Suppose H ∼ = Z ⊕ G where G is a finite Abelian group. Denote by t the generator of Z. Then T = It = 1 St = N 1 N g t, N := |G| g∈G The group algebra Q[H ] is isomorphic to the ring of Laurent polynomials Q[H ] ∼ = = Q[G][t, t −1 ] ∼ Then Ki [t, t −1 ]. N[H ] = Z[H ] + SZ[T , T −1 , (1 − T )−1 ]. The correspondence H → N(H ) is functorial.

This is an idempotent with respect to the pointwise multiplication. In particular, the function 1 − Iˆ can be interpreted as the identity element ˆ \ {1} → C. on the algebra of functions f : G We have seen that if φ : G0 → G1 is an epimorphism of finite Abelian groups there is an induced morphism φ : N(G0 ) → N(G1 ). We want to present a description of this morphism using Fourier analysis. ˆ i \ {1}, i = 0, 1. The Fourier transform maps N(Gi ) isomorphically ˆ∗ = G Let G i ˆ i → C consisting of functions vanishˆ i ) of the ring of functions G onto a subring N(G ˆ ∗ → C.

U Then b¯m τ m . SB = S m Denote by αm,u (resp. βm ) the image of am,u (resp. b¯m ) in Q/Z. Observe that νh βm = 0 = νH αm,u . 10) ∀u, v ∈ Tors(H ), ∀m. Denote by αm the common value of αm,u , u ∈ Tors(H ) and by km the integer 0 ≤ km < νH such that km = αm νH in Q/Z. 5 Abelian group algebras Define K := m 27 1 km m τ ∈ SZ[H ]. 10) can be rephrased in the following compact form, A − SK ∈ Z[H ]. On the other hand, the identities (1 − τ )x ∈ Z ⇐⇒ αm+1 − αm + βm+1 = 0, ∀m can now be rewritten S(1 − τ )K + SB ∈ SZ[H ] so that x = (A − SK) + S B + (1 − τ )K (1 − τ )−1 ∈ Z[H ] + SZ[H ](1 − τ )−1 = N1 .