By Jean Renault

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**Example text**

If we combine this embedding with the embedding of Q in the reals we obtain an embedding ¯ × {real completion of Q} Q →Q as a discrete subgroup with a compact quotient. g. Q (ξ) = Q (x)/(xp − 1) and GL (n, Z). These Adele groups have natural measures, and the volumes of the corresponding compact quotients have interesting number theoretical signiﬁcance. ) In the number ﬁeld case the Adeles form a ring. The units in this ring are called ideles. The ideles are used to construct Abelian extensions of the number ﬁeld.

N/pα , . . ) has one fewer non-integral component than a. This shows the ﬁnite Adeles Qp are generated by the diagonal Q and Zp . Thus i − j p is onto. 30 As before the proof is completed by observing that a rational number which is also a p-adic integer for every p must actually be an integer. Corollary The ring of integers is the ﬁbre product of the rational numbers and the inﬁnite product of all the various rings of p-adic integers over the ring of ﬁnite Adeles. More generally, for a ﬁnitely generated Abelian group G and a non-void set of primes there is a ﬁbre square G⊗Z ≡G -adic completion GG ≡G⊗Z localization at zero localization at zero formal completionG − G ⊗ Q ≡ G0 (G )0 ≡ (G0 ) ≡ G ⊗ Q ⊗ Z Taking to be “all primes” we see that the group G can be recovered from appropriate maps of its localization at zero G ⊗ Q and its Gp into G⊗ “ﬁnite Adeles”.

The proof of ii) has two points. 8 if two of the homologies H∗ F , H∗ E , H∗ B are local the third is also. g. H∗ (F ) ∼ = H∗ (F ; Z ) . = H∗ (F ) ⊗ Z ∼ But this last point is clear since if two of f , g, h induce isomorphisms on H ∗ ( ; Z ) the third does also by the spectral sequence comparison Theorem. With these remarks in mind it is easy now to see that a map of simple spaces →X X− localizes homotopy iﬀ it localizes homology. Step 1. The case (X − →X) = K(π, 1) − → K(π , 1) . If localizes homology, then it localizes homotopy since π = H1 X, π = H1 X .