5 J. TATE unique to an inner automorphism of WF by an element of the connected component W% = Ker (p in the R-cases. To prove (1.5.1) in the leases, we note first that, since p is surjective, we are reduced immediately to the case of the corollary: We must show that if (1, a)e Aut(F, WF)9 then a = Inn(vv) for some w e W$. Going back to the proof of (1.3.1) with WF = WF we find that the group of these a's is given by proj lim (CE/CF) = proj lim (CE/CF) (1 for norm 1) E E = proj lim C^/proj lim CF (by compacity) E E = w°'i/(z n w°*) = w°/z (existence theorem 0 for con- nected component) as claimed, where Z is the center of W. Suppose now we are in a Z-case. Since p is injective, i.e., WF c GF, it is clear that Aut(F, WF) consists only of the pairs (7, aa). The center of GF is 1, because GF/GE « Gal(F/F) acts faithfully on CE c Gf for each finite Galois E/F. Hence, since WF is dense in GFt aa is not an inner automorphism of WF unless a WF. However, aa does induce an inner automorphism of WE/F for finite E/F. Since WF is dense in GF it suffices to prove this last statement for o close to 1, say o e GE. Then aff induces an isomorphism of the group extension 0 -+ CE -• WE/F -• Gal(F/F) -• 0 which is identity on the extremities, and hence is an inner automor- phism by an element of CEy since Hl (Gal(F/F), CE) = 0. (1.6) The local-global relationship. Suppose now Fis global. Let v be a place of F and Fv the completion of Fat v. Let F(resp. Fv) be a separable algebraic closure of F(resp. Fv) and let WF (resp. WF) be a Weil group for F/F(resp. for FJFV). (1.6.1) PROPOSITION. Let iv : F Fv be an F-homomorphism. For each finite ex- tension E of F in F, let Ev = i(E)F0 be the induced completion of E. There exists a continuous homomorphism 0P: WFv -+ WF such that the following diagrams are com- mutative W Fv induced by iv Wt induced by iv WP w$ where nv maps a e E* to the class of the idele whose v-component is a and whose other components are 1. If Fis a function field, then^ is unique. In the number field case, dv is unique up to composition with an inner automorphism of WF defined by an element of the connected component IV$ = Ker p. The proof of this is analogous to the proof of (1.3.1) and (1.5.1), using the stand-
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