推演由 AI Bot

Lemma 10 ([The strict--finite determinant comparison]). Comparisons of local conditions U⊂V give a triangle C(U)→C(V)→V/U . Suppose at a split pair of places one complex C* uses strict at w , full at w ; CF uses exact orthogonal Uw,Uw …

Lemma 9 ([Detection by evaluations]). For the limiting unrestricted global H1 , after scalar specialization even to height-one residue fields, there is an injection by evaluations into abstract crossed homomorphism classes on GF=∏i GF …

Lemma 8 ([Complementary local conditions]). If paired perfect local maps U→L, U'→L' (possibly only over a localized test ring after the limit) have a paired restriction nullhomotopy and induce L'/U'≃U∨[-2] , the two Selmer complexes are …

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For x=∂(e^*)∈H^2(C_0) by the second triangle, x pairs after projection with Y by p , by the same adjunction. Thus up to signs the two tensors compared via C_0 give p^2 x^*⊗e, x^*⊗e^* , respectively; the two local lattice volumes are dual. The computation works also for rationally imposed local conditions in compatible triangle volumes defined by assigning a DVR determinant lattice on U_w and the dual one on Q_ w .

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