Arithmetic examples Clause Samples

Arithmetic examples. ‌ In this section, we give arithmetic examples of (strictly) admissible complexes that will be discussed in further detail in this thesis. At the outset we fix a finite abelian extension F/k of number fields and set G := Gal(F/k). We also fix a finite set of places S of k that contains all places that are either archimedean or ramify in F/k and an auxiliary finite set of places T in k that is disjoint from S. We write OF,S for the ring of SF -integers in F . 2.4.1 Weil-´etale cohomology for Gm‌ In [16], ▇▇▇▇▇, ▇▇▇▇▇▇▇▇ and ▇▇▇▇ describe a canonical ‘Weil-´etale cohomology’ complex RΓT ((OF,S)W , Gm) for the multiplicative group Gm over F . Instead of recalling the detailed construction of this complex, we will record its relevant properties here. To do this we write OF×,S,T for the (finite index) subgroup of OF×,S defined by OF×,S,T := {a ∈ OF×,S : a ≡ 1 (mod w) for all w ∈ TF } and SelT (F )tr for the transpose integral Selmer module defined in [16, Def. 2.6] (where it was denoted Str (Gm/F )). We recall that it is shown in [16, Rem. 2.7] that the module SelT (F )tr lies in a canonical short exact sequence 0 → ClT (F ) → SelT (F )tr → XF,S → 0 (2.10) L where ClT (F ) is the ray class group of OF,S modulo the product of all places in TF (to be more specific, it is the quotient of the group of fractional ideals of F whose supports are coprime to all places in SF ∪ TF by the subgroup of principal ideals with a generator congruent to 1 modulo all places in TF ) and XF,S is the kernel of the natural morphism w∈SF Z → Z. Note that in the case when T is empty, we will suppress it from any notation. Q Σ Remark 2.4.1. Recall from [16] that the ‘S-relative T -trivialized Selmer group’ SelT (F ) for Gm over F is the cokernel of the canonical homomorphism w Z −→ HomZ(FT×, Z). Here w runs over all places of F that are not in SF ∪ TF , FT× is the subgroup of F × comprising elements a with ordw(a − 1) > 0 for all places w of TF and the homomorphism sends each element (xw)w to the map a '→ w ordw(a)xw. This group is a natural analogue for Gm of the ‘integral Selmer groups’ that are defined for abelian varieties by ▇▇▇▇▇ and ▇▇▇▇ in [52]. In particular, it lies in an exact sequence [16, Prop. 2.2] 0 → HomZ(ClT (F ), Q/Z) → SelT (F ) → HomZ(O× , Z) → 0. (2.11) We also note that the module SelT (F )tr can be regarded as the canonical transpose of SelT (F ) in the sense of ▇▇▇▇▇▇▇’▇ homotopy theory of modules [39]. Proposition 2.4.2. Let CF,S,T := RΓT ((OF,S)W , Gm)[−1]. If t...