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9) implies ¯ = ±T(C, [c])(−1) ε T(D, [d]) = T(D¯ , [d]) ε n+1 , where d = εT −1 (c− ). 4 Some generalizations The notion of torsion can be defined in a much more general context than the one discussed above. We refer the reader to [19, 72] for a more in depth study. We will need only a mild generalization of the ideas developed so far. Often, instead of complexes of vector spaces over a field K one encounters complexes C of free modules over an integral domain R. Denote by K the quotient field of R.
B. rank(G) = 1, Tors(G) = 0. Set H := Tors(G), F := G/H and S= h ∈ Z[G]. 6 Abelian harmonic analysis 41 ˜ = Hom(G, C∗ ) is an union of one-dimensional Fix an orientation o on G⊗R. Then G ˆ and thus identifies the complex tori, and the orientation o defines an orientation on G, ∗ ˜ identity component of G with C . A function f ∈ No (G) has noncompact support, but has temperate growth, and ˆ ⊂ D (G) ˆ thus it has a Fourier transform as a temperate distribution. Denote by No (G) the Fourier transform of No (G).
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 .