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Cassels-adeles - J W S CASSELS 62 Now suppose that a E...

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62 J. W. S. CASSELS Now suppose that a E R.H.S. and that ,!I=\$b,,w,ER.H.S. (b,Ek,). 1 (12.2) Then for any m, 1 I m I N we have [email protected], ,..., w,,-~,P,w,,+~ ,..., d=b:W’, ,..., R>, and so dbi E o, (1 I m 5 N) where d=D(o,,...,w,)Ek. But (Appendix B) we have d # 0, and so ldlv = 1 for almost all u. For almost all u the condition (12.2) thus implies b, E o, (15 m 5 N), i.e. R.H.S. c L.H.S. This proves the lemma. [COROLLARY. Almost all v are unramijied in the extension K/k. For by the results of Chapter I a necessary and sutlicient condition for v to be un- ramified is that there are yl,. . ., yN E R.H.S. with ID(y,,. . ., yN)lv = 1. And for almost all v we can put yn = ~f”-~.] 13. Restricted Topological Product We describe here a topological tool which will be needed later: DEFINITION. Let sZA (A E A) be a family of topological spaces and for almost alIt L let On c Q, be an open subsetof a,. Consider the space n whose points are sets u = h&h where aA E a, for every A and cr, E On for almost all 1. We give Q a topology by taking as a basis of open sets the sets l-m where rl c G12, is open for all rZ and rA = O1 for almost all 1. With this topology R is the restricted topologicalproduct of the R, with respect to the O1. COROLLARY. Let S be a finite subset of A and let Rs be the set of a E fi with aAE O1 (2 q! S), i.e. (13.1) Then G!, is open in Sz and the topology induced in as as a subset of R is the sameas the product topology. Beweis. Klar. The restricted topological product depends on the totality of the 0, but not on the individual 0,: t i.e. all except possibly finitely many.

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GLOBAL FIELDS 63 LEMMA. Let 0; c Jz, be open sets defined for almost all II and suppose that OA = 0: for almost all 2. Then the restricted product of the f2, with respect to the 0; is the same ast the restrictedproduct with respect to the 0,. Beweis. Klar. LEMMA. Suppose that the RA are locally compact and that the 0, are compact. Then R is locally compact. Proof. The as are locally compact by (13.1) since S is finite. Since n = u C& and the S& are open in 0, the result follows. DEFINITION. Suppose that measures ,uI are defined on the RA with ~~(0,) = 1 when OA is defined. We define the product measure p on R to be that for which a basis of measurable sets is the where MA c Sz, has finite PA-measure and MA = On for aImost all 1 and where COROLLARY. The restriction of ,u to Q, is just the ordinary product measure. 14. Adele Ring (or Ring of Valuation Vectors) Let k be a global field. For each normalized valuation 1 1” of k denote by k, the completion of k. If 1 1” is non-archimedean denote by D, the ring of integers of k,. The adele ring V, of k is the topological ring whose under- lying topological space is the restricted product of the k, with respect to the D, and where addition and multiplication are defined componentwise: W% = aA (a+/% = av+/L a,BE 6. (14.1) It is readily verified (i) that this definition makes sense, i.e. if a, fi E V, then an, a+ fl whose components are given by (14.1) are also in V, and (ii) that addition and multiplication are continuous in the V,-topology, so V, is a topological ring, as asserted.
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