ASPECTS OF IWASAWA THEORY OVER FUNCTION FIELDS 1 ...

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IWASAWA THEORY OVER FUNCTION FIELDS 16be the exact sequence coming from the structure theorem for Λ d -modules (see section 3) whereE(F d ) := ⊕ iΛ d /(f i,d )is an elementary module and N(F d ), R(F d ) are pseudo-null. Let Ch Λd (A(F d )) = (Π i f i,d ) bethe characteristic ideal of A(F d ): we want to compare Ch Λd−1 (A(F d−1 )) with π F dF d−1(Ch Λd (A(F d )))for some F d−1 ⊂ F d and show that these characteristic ideals form an inverse system (in Λ).Consider the module B(F d ) := N(F d ) ⊕ R(F d ). We need the following hypothesis.Assumption 5.3. There is a choice of the pseudo-isomorphism ι in (5.7) and a splitting ofthe projection Γ d ↠ Γ d−1 so thati) Γ d = 〈γ d 〉 ⊕ Γ d−1 ;ii) B(F d ) is a finitely generated torsion Z p [[Γ d−1 ]]-module.As explained in [20] (see the remarks just before Lemma 3), for any F d and ι one can finda subfield F d−1 so that Assumption 5.3 holds.In order to ease notations, we put γ = γ d , so that Γ d d−1 = 〈γ〉.Lemma 5.4. With the above notations, one hasCh Λd−1 (A(F d )/I d d−1 A(F d)) = π d d−1(ChΛd (A(F d )) ) · Ch Λd−1 (A(F d ) Γd d−1 ) .Proof. We split the previous sequence in two byN(F d ) ↩→ A(F d ) ↠ C(F d ) , C(F d ) ↩→ E(F d ) ↠ R(F d )and consider the snake lemma sequences coming from the following diagrams(5.8) N(F d ) A(F d ) C(F d ) C(F d ) E(F d ) R(F d )γ−1N(F d ) γ−1γ−1γ−1A(F d ) C(F d ) C(F d ) i.e.,(5.9) N(F d ) Γd d−1 A(F d ) Γd d−1γ−1γ−1 E(F d ) R(F d ) , C(F d ) Γd d−1C(F d )/Id−1 d C(F d) A(F d )/Id−1 d A(F d) N(F d )/Id−1 d N(F d)and(5.10) C(F d ) Γd d−1 E(F d ) Γd d−1 R(F d ) Γd d−1R(F d )/Id−1 d R(F d) E(F d )/Id−1 d E(F d) C(F d )/Id−1 d C(F d) .From (5.7) we get an exact sequenceA(F d )/I d d−1 A(F d) −→ ⊕ iΛ d /(γ − 1, f i,d ) −→ R(F d )/I d d−1 R(F d)where the last term is a torsion(Λ d−1 -module. So is A(F d−1 ) for d ≥ 3 and, by (5.6),Ch Λd−1 (A(F d−1 )) = Ch Λd−1 A(Fd )/Id−1 d A(F d) ) . It follows that none of the f i,d ’s belongto Id−1 d . Therefore:
IWASAWA THEORY OVER FUNCTION FIELDS 171. the map γ − 1 : E(F d ) −→ E(F d ) has trivial kernel, i.e., E(F d ) Γd d−1 = 0 so thatC(F d ) Γd d−1 = 0 as well;2. the characteristic ideal of the Λ d−1 -module E(F d )/I d d−1 E(F d) is generated by theproduct of the f i,d ’s modulo I d d−1 , hence it is obviously equal to πd d−1 (Ch Λ d(A(F d ))).Moreover, from the fact that N(F d ) and R(F d ) are finitely generated torsion Λ d−1 -modules3 and the multiplicativity of characteristic ideals, looking at the left (resp. right) verticalsequence of the first (resp. second) diagram in (5.8), one findsCh Λd−1 (N(F d ) Γd d−1 ) = ChΛd−1 (N(F d )/I d d−1 N(F d))andCh Λd−1 (R(F d ) Γd d−1 ) = ChΛd−1 (R(F d )/Id−1 d R(F d)) .Hence from (5.9) one hasCh Λd−1 (A(F d )/I d d−1 A(F d)) = Ch Λd−1(C(Fd )/I d d−1 C(F d) ) · Ch Λd−1 (N(F d ) Γd d−1 )= Ch Λd−1(C(Fd )/I d d−1 C(F d) ) · Ch Λd−1 (A(F d ) Γd d−1 )(where the last line comes from the isomorphism A(F d ) Γd d−1 ≃ N(Fd ) Γd d−1 ). The sequence(5.10) provides the equalityCh Λd−1 (C(F d )/I d d−1 C(F d)) = Ch Λd−1 (E(F d )/I d d−1 E(F d))= π d d−1 (Ch Λ d(A(F d ))) .Therefore one concludes that(5.11) Ch Λd−1 (A(F d )/Id−1 d A(F d)) = πd−1d (ChΛd (A(F d )) ) · Ch Λd−1 (A(F d ) Γd d−1 ) .□Our next step is to prove that A(F d ) Γd d−1 = 0 (note that it would be enough to prove thatit is pseudo-null as a Λ d−1 -module). For this we need first a few lemmas.Lemma 5.5. Let G be a finite group and endow Z p with the trivial G-action. Then for anyG-module M we haveH i (G, M ⊗ Z p ) = H i (G, M) ⊗ Z pfor all i ≥ 0.This result should be well-known. Since we were not able to find a suitable reference, hereis a sketch of the proof.Proof. Let X be a G-module which has no torsion as an abelian group and put Y := X ⊗ Q.It is not hard to prove that Y G ⊗ Z p = (Y ⊗ Z p ) G and it follows that the same holds for X,since X G ⊗ Z p is a saturated submodule of X ⊗ Z p . Applying this to the standard complex bymeans of which the H i (G, M) are defined, one can prove the equality in the case M has notorsion as an abelian group. The general case follows because any G-module is the quotientof two such modules.□Up to now we have mainly considered M(L) as an Iwasawa module (for various L), nowwe focus on its interpretation as a group of divisor classes. Let L be a finite extensions of Fand recall that we defined M(L) = (Div(L)/P L ) ⊗ Z p . From the exact sequenceF ∗ L ↩→ L ∗ ↠ P Land the fact that |F ∗ L | is prime with p, one finds an isomorphism between L∗ ⊗Z p and P L ⊗Z p .Hence we can (and will) identify the two.holds.3 It might be worth to notice that this is the only point where we use the hypothesis that Assumption 5.3
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