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 ; C_F uses exact orthogonal U_w,U_w …

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 .