Characteristic ideals and Iwasawa theory
aa r X i v : . [ m a t h . N T ] J u l CHARACTERISTIC IDEALS AND IWASAWA THEORY
ANDREA BANDINI, FRANCESC BARS, AND IGNAZIO LONGHI
Abstract.
Let Λ be a non-noetherian Krull domain which is the inverse limit of noetherianKrull domains Λ d and let M be a finitely generated Λ-module which is the inverse limit of Λ d -modules M d . Under certain hypotheses on the rings Λ d and on the modules M d , we definea pro-characteristic ideal for M in Λ, which should play the role of the usual characteristicideals for finitely generated modules over noetherian Krull domains. We apply this to thestudy of Iwasawa modules (in particular of class groups) in a non-noetherian Iwasawa algebra Z p [[Gal( F /F )]], where F is a function field of characteristic p and Gal( F /F ) ≃ Z ∞ p . Contents
1. Introduction 12. Pseudo-null modules and characteristic ideals 32.1. Krull domains 32.2. Pseudo-null B -modules 52.3. Pro-characteristic ideals 93. Class groups in global fields 103.1. Iwasawa theory for class groups in function fields 11References 141. Introduction
Let A be a noetherian Krull domain and M a finitely generated torsion A -module. Thestructure theorem for such modules provides an exact sequence(1.1) 0 −→ P −→ M −→ n M i =1 A/ p e i i A −→ Q −→ p i ’s are height 1 prime ideals of A and P and Q are pseudo-null A -modules (for moredetails and precise definitions of all the objects appearing in this Introduction see Section 2).This sequence defines an important invariant of the module M , namely its characteristic ideal Ch A ( M ) := n Y i =1 p e i i . Characteristic ideals play a major role in (commutative) Iwasawa theory for global fields:they provide the algebraic counterpart for the p -adic L -functions associated to Iwasawa mod-ules (such as class groups or duals of Selmer groups). Here the Krull domain one works withis the Iwasawa algebra Z p [[Γ]], where Γ is a commutative p -adic Lie group occurring as Galoisgroup (we shall deal mainly with the case Γ ≃ Z dp for some d ∈ N ). Even if pseudo-nullmodules do not contribute to characteristic ideals, they appear in the descent problem when Mathematics Subject Classification.
Key words and phrases.
Characteristic ideals; Iwasawa theory; Krull rings; class groups.F. Bars supported by MTM2013-40680-P.I. Longhi supported by National Science Council of Taiwan, grant NSC100-2811-M-002-079. one wants to compare the characteristic ideal of an Iwasawa module of a Z dp -extension withthe one of a Z d − p -extension contained in it. The last topic is particularly important when theglobal field has characteristic p , because, in this case, extensions F /F with Gal( F /F ) ≃ Z ∞ p occur quite naturally: in this situation the Iwasawa algebra is non-noetherian and there is noguarantee one can find a sequence such as (1.1). One strategy to overcome this difficulty isto consider a filtration of Z dp -extensions for F , define the characteristic ideals at the Z dp -levelfor all d and then pass to the limit.To deal with the technical complications of inverse limits and projections of pseudo-nullmodules, in Section 2 we prove the following (see Propositions 2.7 and 2.10) Proposition 1.1.
Let A be a noetherian Krull domain and put B := A [[ t ]] . If M is a pseudo-null B -module, then M t (the kernel of multiplication by t ) and M/tM are finitely generatedtorsion A -modules and (1.2) Ch A ( M t ) = Ch A ( M/tM ) . Moreover, for any finitely generated torsion B -module N , we have (1.3) Ch A ( N t ) π ( Ch B ( N )) = Ch A ( N/tN ) (where π : B → A is the canonical projection). This immediately provides a criterion for an B -module to be pseudo-null (Corollary 2.11), butour main application is the definition of an analogue of characteristic ideals in a non-noetherianIwasawa algebra (Theorem 2.13). Theorem 1.2.
Let { Λ d } d > be an inverse system of noetherian Krull domains such that Λ d ≃ Λ d +1 / p d +1 and Λ d +1 ≃ lim ←− n Λ d +1 / p nd +1 f or any d > ( p d +1 a principal prime ideal of Λ d +1 of height 1). Let Λ := lim ←− d Λ d and consider a finitelygenerated Λ -module M = lim ←− d M d (where each M d is a Λ d -module). If, for any d ≫ , the p d -torsion submodule of M d is a pseudo-null Λ d − -module, i.e., Ch Λ d − ( M d [ p d ]) =(1) ; Ch Λ d − ( M d / p d ) ⊆ Ch Λ d − ( M d − ) ,then we can define a pro-characteristic ideal for M as f Ch Λ ( M ) := lim ←− d ( π ΛΛ d ) − ( Ch Λ d ( M d )) (where π ΛΛ d : Λ → Λ d is the natural map provided by the inverse limit defining Λ ). The Iwasawa algebra associated with a Z dp -extension of a global field F is (noncanonically)isomorphic to the Krull ring Z p [[ t , . . . , t d ]], hence descending to a Z d − p -extension correspondsto the passage from A [[ t ]] to A . So the results of Section 2 apply immediately to Iwasawamodules and, in order to keep the paper short, we just consider the case of class groups.We also remark that the analogue of Proposition 1.1 for these Iwasawa algebras has beenproved by T. Ochiai in [18, Section 3] and it suffices for the arithmetical applications wehad in mind. Indeed in Section 3.1 we deal with a global field F of characteristic p > p -adicextensions: this comes from class field theory, since, in the completion F v of F at some place v , the group of 1-units U ( F v ) is isomorphic to Z ∞ p . As hinted above, our strategy to tackle F /F with Gal( F /F ) ≃ Z ∞ p is to work first with Z dp -extensions and then use limits. Theusefulness of Theorem 1.2 in this procedure is illustrated in Section 3.1.1, where we define the HARACTERISTIC IDEALS AND IWASAWA THEORY 3 pro-characteristic ideal f Ch Λ ( A ( F )) dispensing with the crutch of the ad hoc hypothesis [6,Assumption 5.3]. The search for a “good” definition for it was one of the main motivationsfor this work.The arithmetic significance of our pro-characteristic ideal is ensured by a deep result of D.Burns (see [9, Theorem 3.1] and the Appendix [10]), which shows that the characteristic idealof the class group of a Z dp -extension F d /F is generated by a Stickelberger element (by somelanguage abuse we shall call class group of F d the inverse limit of the class groups of thefinite subextensions of F d /F ). Therefore (see Corollary 3.10) our pro-characteristic ideal isgenerated by a Stickelberger element as well and this can be considered as an instance ofIwasawa Main Conjecture for non-noetherian Iwasawa algebras.Next to class groups, [6] and [4] consider the case of Selmer groups of abelian varieties: in[6, Section 3] we employed Fitting ideals of Pontrjagin duals of Selmer groups instead thancharacteristic ideals in order to avoid the difficulties of taking the inverse limit. With someadditional work, Theorem 1.2 permits to define a pro-characteristic ideal for these modules aswell, allowing to formulate a more classical Iwasawa Main Conjecture (details can be foundin [7]). Remark 1.3.
If a pseudo-null A [[ t ]]-module M is finitely generated over A as well, then thestatement of Proposition 1.1 is trivially deduced from the exact sequence0 → M t → M t −→ M → M/tM → Z dp -extension F d /F and any pseudo-null Λ( F d )-module M it is always possible to find (at least one) Z d − p -subextension F d − such that M is finitelygenerated over Λ( F d − ) (where Λ( L ) is the Iwasawa algebras associated with the extension L /F ). Our search for a characteristic ideal via a projective limit does not allow this freedomin the choice of subextensions, hence the need for an “unconditional” result like Theorem 1.2.2. Pseudo-null modules and characteristic ideals
Krull domains.
We begin by reviewing some basic facts and definitions we are goingto need. A comprehensive reference is [8, Chapter VII].An integral domain A is called a Krull domain if A = ∩ A p (where p varies among primeideals of height 1 and A p denotes localization), all A p ’s are discrete valuation rings and any x ∈ A − { } is a unit in A p for almost all p . In particular, one attaches a discrete valuationto any height 1 prime ideal. Furthermore, a ring is a unique factorization domain if and onlyif it is a Krull domain and all height 1 prime ideals are principal ([8, VII, § Torsion modules.
Let A be a noetherian Krull domain. A finitely generated torsion A -module is said to be pseudo-null if its annihilator ideal has height at least 2. A morphism withpseudo-null kernel and cokernel is called a pseudo-isomorphism: being pseudo-isomorphic isan equivalence relation between finitely generated torsion A -modules (torsion is essential here ) and we shall denote it by ∼ A . If M is a finitely generated torsion A -module then there isa pseudo-isomorphism(2.1) M −→ n M i =1 A/ p e i i This is not the definition in [8], but it is equivalent to it: see [8, VII, § For example the map ( p, t ) ֒ → Z p [[ t ]] is a pseudo-isomorphism, but there is no such map from Z p [[ t ]] to( p, t ). HARACTERISTIC IDEALS AND IWASAWA THEORY 4 where the p i ’s are height 1 prime ideals of A (not necessarily distinct) and the p i ’s, n and the e i ’s are uniquely determined by M (see e.g. [8, VII, § elementary A -module and E ( M ) := n M i =1 A/ p e i i ∼ A M is the elementary module attached to M . Definition 2.1.
Let M be a finitely generated A -module: its characteristic ideal is Ch A ( M ) := M is not torsion; n Y i =1 p e i i if M ∼ A n M i =1 A/ p e i i . In particular, M is pseudo-null if and only if Ch A ( M ) = A .We shall denote by Fgt A the category of finitely generated torsion A -modules. Remarks 2.2.1.
An equivalent definition of pseudo-null is that all localizations at primes of height 1are zero. If p and q are two different primes of height 1 (and M is a torsion A -module)we have M ⊗ A A p ⊗ A A q = 0. By the structure theorem recalled in (2.1) it followsimmediately that for a finitely generated torsion A -module M , the canonical map(2.2) M −→ M p (cid:0) M ⊗ A A p (cid:1) (where the sum is taken over all primes of height 1) is a pseudo-isomorphism. Actually,the right-hand side of (2.2) can be used to compute Ch A ( M ): a prime p appears in Ch A ( M ) with exponent the length of the module M ⊗ A A p . The previous remark suggests a generalization of the definition of characteristic idealby means of supernatural divisors . Let M be any torsion A -module (we drop thefinitely generated assumption) and define Ch A ( M ) := Y p p l p ( M ⊗ A A p ) where the product is taken over all primes of height 1 and the exponent of p (i.e., the length of the module M ⊗ A A p ) is a supernatural number (i.e., belongs to N ∪ {∞} ).More precisely, for N a finitely generated torsion A p -module let l p ( N ) denote its length.Then we put l p ( M ⊗ A A p ) := sup { l p ( M α ⊗ A A p ) } , where M α varies among all finitely generated submodules of M . Note that, since A p is flat, M α ⊗ A A p is still a submodule of M ⊗ A A p ; furthermore, the length l p isan increasing function on finitely generated torsion A p -modules (partially ordered byinclusion).2.1.2. Power series.
In the rest of this section, A will denote a Krull domain and B := A [[ t ]]the ring of power series in one variable over A . Proposition 2.3.
Let A be a Krull domain and p ⊂ A a height 1 prime. Then B is also aKrull domain and p B is a height 1 prime of B . We recall that the group of divisors of A is the free abelian group generated by the prime ideals of height1 in A (see [8, VII, § HARACTERISTIC IDEALS AND IWASAWA THEORY 5
This is well-known (actually, one can prove the analogue even with infinitely many variables:see [11]). In order to make the paper as self-contained as possible, and for lack of a suitablereference for the second part of the proposition, we provide a quick proof.
Proof.
Let Q be the fraction field of A . Since A is Krull, we have A = ∩ A q as q varies amongall prime ideals of height 1. Furthermore, each A q is a discrete valuation ring: then [8, VII, § A q [[ t ]] is a unique factorization domain. In particular every A q [[ t ]][ t − ] is a Krull domain and we get(2.3) B = A [[ t ]] = Q [[ t ]] ∩ \ q A q [[ t ]][ t − ] = Q [[ t ]] ∩ \ q \ P ∈ S q (cid:0) A q [[ t ]][ t − ] (cid:1) P (where S q denotes the set of height 1 primes in A q [[ t ]][ t − ]). This shows that B is an intersec-tion of discrete valuation rings. A power series λ = t h P i > c i t i ∈ B (with c = 0) is a unit in A q [[ t ]][ t − ] unless c ∈ q and, in the latter case, λ is still a unit in ( A q [[ t ]][ t − ]) P unless it canbe divided by the generator of P . This proves that B is a Krull domain.Since A p is a discrete valuation ring, its maximal ideal p A p is principal: let π be a uniformizer.Then π is irreducible in A p [[ t ]], hence it generates a height 1 prime ideal P := πA p [[ t ]] = p A p [[ t ]]. By the general theory of Krull domains, P corresponds to a discrete valuation ν P onthe fraction field F rac ( A p [[ t ]]); the restriction of ν P to Q is precisely the discrete valuationassociated with p . Similarly, restricting ν P to F rac ( B ) yields a discrete valuation, with ringof integers D P and maximal ideal m P . The ring D P is the localization of B at m P : hence itis flat over B and, by [8, VII, § m P ∩ B = P ∩ B = p B = 0has height 1. (cid:3) Pseudo-null B -modules. Now assume that A (and hence B ) is Noetherian. In thissection P will be a pseudo-null B -module. We denote by P t the kernel of multiplication by t and remark that in the exact sequence(2.4) P t (cid:31) (cid:127) / / P t / / P / / / / P/tP ,P t and P/tP are finitely generated B -modules, because so is P . The former ones are alsofinitely generated as A -modules, because t acts as 0 on them. Moreover they are torsion A -modules (just take two relatively prime elements f and g in Ann B ( P ): their projections in A via t Ann A ( P t ) and Ann A ( P/tP ) and at least one of them is nonzerosince otherwise t would divide both f and g ). Therefore the characteristic ideals Ch A ( P t ) and Ch A ( P/tP ) are given by Definition 2.1 (there is no need for supernatural divisors here) andboth of them are nonzero.2.2.1.
Preliminaries.
For p a prime of height one in A , define c A p := lim ← A p / p n A p . By a slight abuse of notation, we shall denote by p also the maximal ideals of A p and c A p . Thenatural embedding of A into c A p allows to identify B with a subring of c A p [[ t ]]. Lemma 2.4.
The ring c A p [[ t ]] is a flat B -algebra.Proof. Put S p := A − p . We claim that c A p [[ t ]] is the completion of S − p B with respect to theideal generated by p and t . This is enough, since formation of fractions and completion ofa noetherian ring both generate flat algebras, and the composition of flat morphisms is stillflat. To verify the claim consider the inclusions A p [ t ] / ( p , t ) n ⊂ S − p B/ ( p , t ) n ⊂ c A p [[ t ]] / ( p , t ) n HARACTERISTIC IDEALS AND IWASAWA THEORY 6 and note that they are preserved by taking the inverse limit with respect to n . To concludeobserve that lim ← A p [ t ] / ( p , t ) n = c A p [[ t ]]. (cid:3) The advantage of working over c A p [[ t ]] is that one can apply the Weierstrass PreparationTheorem (for a proof see e.g. [8, VII, § α = P a i t i ∈ c A p [[ t ]] suchthat not all coefficients are in p , there exist u ∈ c A p [[ t ]] ∗ and a monic polynomial β ∈ c A p [ t ]such that α = uβ (the degree of β is equal to the minimum of the indices i such that a i p ).Actually, as it is going to be clear from the proof of Lemma 2.6 below, we shall need just aweaker form of this statement.2.2.2. Characteristic ideals.
Now we deal with the equality between Ch A ( P t ) and Ch A ( P/tP ). Lemma 2.5.
For any finitely generated torsion A p -module N one has the equality of lengths l A p ( N ) = l c A p ( N ⊗ A p c A p ) . Proof.
Since both A p and c A p are discrete valuation rings and A p / p n ≃ c A p / p n ≃ ( A p / p n ) ⊗ c A p , the statement follows directly from the structure theorem for finitely generated torsion mod-ules over principal ideal domains. (cid:3) Lemma 2.6.
Let P be a pseudo-null B -module. Then P ⊗ B c A p [[ t ]] is a finitely generated c A p -module for any height 1 prime ideal p ⊂ A .Proof. We consider P as an A -module and work separately with primes p belonging or notbelonging to the support of P . If the prime p is not in this support, there is some r ∈ Ann A ( P )which becomes a unit in A p ⊂ c A p [[ t ]], hence P ⊗ B c A p [[ t ]] = 0 and the statement is trivial. Thus,from now on, we assume p ∈ Supp A ( P ) (i.e., Ann A ( P ) ⊂ p ). Since p B is a height 1 primeideal in B , the hypothesis on P yields Ann B ( P ) p B . Hence there exists α ∈ Ann B ( P ) − p B ,i.e., α = X i ≥ a i t i ∈ Ann B ( P ) (with a i ∈ A for any i )with at least one a i p . For such an α , let n be the smallest index such that a n / ∈ p . Then, by[8, VII, § c A p [[ t ]] = α c A p [[ t ]] ⊕ (cid:0) n − M i =0 c A p t i (cid:1) . Now one just uses P ⊗ B α c A p [[ t ]] = α · ( P ⊗ B c A p [[ t ]]) = 0. (cid:3) Proposition 2.7.
Let P be a pseudo-null B -module. Then Ch A ( P t ) = Ch A ( P/tP ) .Proof. By Remark 2.2 and Lemma 2.5, we need to show that l c A p ( P t ⊗ A c A p ) = l c A p (( P/tP ) ⊗ A c A p )for any height 1 prime ideal p of A . By Lemma 2.4, the functor ⊗ B c A p [[ t ]] is exact. Applyingit to (2.4), we get an exact sequence(2.5) P t ⊗ B c A p [[ t ]] ֒ → P ⊗ B c A p [[ t ]] t −→ P ⊗ B c A p [[ t ]] ։ ( P/tP ) ⊗ B c A p [[ t ]] . HARACTERISTIC IDEALS AND IWASAWA THEORY 7
Lemma 2.6 shows that all terms of (2.5) are finitely generated c A p -modules. Hence, the firstand last term of the sequence have the same length. Finally, just observe that if N is a B -module annihilated by t then N ⊗ B c A p [[ t ]] = N ⊗ A c A p . (cid:3) Example 2.8. If P happens to be finitely generated over A then the statement of the propo-sition is obvious. We give a few examples of pseudo-null B := Z p [[ s, t ]]-modules which are notfinitely generated as A := Z p [[ s ]]-modules, providing non-trivial examples in which the abovetheorem applies. However we remark that the main consequence of Lemma 2.6 is exactlythe fact that we can ignore the issue of checking whether a pseudo-null B -module is finitelygenerated over A or not. P = B/ ( p, s ) . Then P ≃ F p [[ t ]] is not finitely generated over A . In this case P t = 0and P/tP ≃ F p (both A -pseudo-null), so that Ch A ( P t ) = Ch A ( P/tP ) =
A . P = B/ ( s, pt ) . Then P ≃ Z p [[ t ]] / ( pt ) is not finitely generated over A and elements in P can be written as m = X i ≥ a i t i a ∈ Z p and a i ∈ { , ..., p − } ∀ i ≥ . Moreover P t = p Z p [[ t ]] / ( pt ) ≃ p Z p ≃ Z p ≃ A/ ( s )and P/tP = Z p [[ s, t ]] / ( t, s, pt ) ≃ Z p ≃ A/ ( s ) , so both have characteristic ideal ( s ) (as A -modules). With P = B/ ( p, st ), a similar reasoning shows that Ch A ( P/tP ) = Ch A ( P t ) = ( p ). Remark 2.9.
The hypothesis that P is pseudo-null is necessary: if M is a torsion B -modulethen it is not true, in general, that Ch A ( M t ) = Ch A ( M/tM ). We give an easy example: letagain B = Z p [[ s, t ]] with A = Z p [[ s ]], and consider M = Z p [[ s, t ]] / ( p + s + t ), which is atorsion B -module. Observe that M t is trivial (so Ch A ( M t ) = A ) and M/tM = Z p [[ s, t ]] / ( t, p + s ) ≃ A/ ( p + s )has characteristic ideal over A equal to ( p + s ). Moreover Ch A ( M/tM ) = ( p + s ) is theimage of Ch B ( M ) = ( p + s + t ) under the projection π : B → A , t
0. Hence, in this case, Ch A ( M t ) π ( Ch B ( M )) = Ch A ( M/tM )which anticipates the general formula of Proposition 2.10.As mentioned in the Introduction, the following proposition will be crucial in the study ofcharacteristic ideals for Iwasawa modules under descent.
Proposition 2.10.
Let π : B → A be the projection given by t and let M be a finitelygenerated torsion B -module. Then (2.6) Ch A ( M t ) π ( Ch B ( M )) = Ch A ( M/tM ) . Moreover, Ch A ( M t ) = 0 ⇐⇒ π ( Ch B ( M )) = 0 ⇐⇒ Ch A ( M/tM ) = 0 and in this case M t and M/tM are A -modules of the same rank. HARACTERISTIC IDEALS AND IWASAWA THEORY 8
Proof.
Recall that the structure theorem (2.1) provides a pseudo-isomorphism between M and its associated elementary module E ( M ). As noted above, being pseudo-isomorphic is anequivalence relation for torsion modules: therefore one has a (non-canonical) sequence E ( M ) (cid:31) (cid:127) / / M / / / / P where P is pseudo-null over B and the injectivity on the left comes from the fact that ele-mentary modules have no nontrivial pseudo-null submodules (just use the valuation on B p tocheck that the annihilator of any x ∈ B/ p e − { } must be contained in p ). The snake lemmasequence coming from the diagram E ( M ) (cid:31) (cid:127) / / t (cid:15) (cid:15) M / / / / t (cid:15) (cid:15) P t (cid:15) (cid:15) E ( M ) (cid:31) (cid:127) / / M / / / / P reads as(2.7) E ( M ) t ֒ → M t −→ P t −→ E ( M ) /tE ( M ) −→ M/tM ։ P/tP .
As we remarked at the beginning of Section 2.2, both P t and P/tP are finitely generatedtorsion A -modules. It is also easy to see that all modules in the sequence (2.7) are finitelygenerated over A . Now observe that ( B/ p e ) t is zero if p = ( t ) and isomorphic to A if p = ( t );similarly, ( B/ p e ) /t ( B/ p e ) is either pseudo-null or isomorphic to A . Thus, putting E ( M ) = ⊕ B/ p e i i , we find E ( M ) t ≃ A r and E ( M ) /tE ( M ) = ⊕ B/ ( p e i i , t ) ≃ ⊕ A/ ( π ( p i ) e i ) ≃ A r ⊕ • , where r := { i | p i = tB } and • is a pseudo-null B -module. Moreover (2.7) shows that E ( M ) /tE ( M ) is A -torsion if and only if M/tM is A -torsion and E ( M ) t is A -torsion if andonly if M t is A -torsion. Therefore we have two cases: if r >
0, then ( t ) divides Ch B ( M ), so π ( Ch B ( M )) = 0 and, since M t and M/tM are not A -torsion, Ch A ( M t ) = Ch A ( M/tM ) = 0 as well (the statement on A -ranks isimmediate from (2.7): e.g., apply the exact functor ⊗ A F rac ( A )); if r = 0, then, because of the equivalent conditions above, all the characteristic idealsinvolved in (2.6) are nonzero; moreover we have Ch A ( E ( M ) /tE ( M )) = π ( Ch B ( E ( M ))) = π ( Ch B ( M ))and (2.6) follows from the sequence (2.7), Proposition 2.7 and the multiplicativity ofcharacteristic ideals. (cid:3) Corollary 2.11.
In the above setting assume that
M/tM is a finitely generated torsion A -module. Then M is a pseudo-null B -module if and only if Ch A ( M t ) = Ch A ( M/tM ) . More-over if M/tM ∼ A , then M ∼ B .Proof. The “only if” part is provided by Proposition 2.7. For the “if” part we assumethe equality of characteristic ideals (which are nonzero by hypothesis). By (2.6) we have π ( Ch B ( M )) = A , hence there is some f ∈ Ch B ( M ) such that π ( f ) = 1. But then f = P i ≥ c i t i with c = 1, which is an obvious unit in B = A [[ t ]]. Therefore Ch B ( M ) = B ,i.e., M is pseudo-null over B . For the last statement just note that Ch A ( M/tM ) = A yields Ch A ( M t ) π ( Ch B ( M )) = A , so Ch A ( M t ) = A as well. (cid:3) Remarks 2.12.
HARACTERISTIC IDEALS AND IWASAWA THEORY 9 When R ≃ Z p [[ t , . . . , t d ]] (i.e., the Iwasawa algebra for a Z dp -extension of global fields),the statement of the previous corollary appears in [20, Lemme 4]. Note anyway thatthe proof given there relies on the choice of a Z d − p -subextension (i.e., on the strategymentioned in Remark 1.3). The possibility of lifting pseudo-nullity from
M/tM to M has been used to provesome instances of Greenberg’s Generalized Conjecture (for statement and examplessee, e.g., [2], [3] and [19]).2.3. Pro-characteristic ideals.
We can now define an analogue of characteristic ideals forfinitely generated modules over certain non-noetherian Krull domains Λ. We need Λ to be theinverse limit of noetherian Krull domains and we limit ourselves to finitely generated modulesbecause characteristic ideals are usually defined only for them.Let { Λ d } d > be an inverse system of noetherian Krull domains such thatΛ d ≃ Λ d +1 / p d +1 and Λ d +1 ≃ lim ←− n Λ d +1 / p nd +1 for any d > p d +1 a principal prime ideal of Λ d +1 of height 1). Let Λ := lim ←− d Λ d and note that, by hy-pothesis, Λ d +1 ≃ Λ d [[ t d +1 ]], where the variable t d +1 corresponds to a generator of the ideal p d +1 . Take a finitely generated Λ-module M which can be written as the inverse limit ofΛ d -modules M = lim ←− d M d (all the relevant arithmetic applications to Iwasawa modules satisfythis requirement). Theorem 2.13.
With the above notations if, for any d ≫ , ( M d ) t d (the p d -torsion submodule of M d ) is a pseudo-null Λ d − -module; Ch Λ d − ( M d /t d M d ) ⊆ Ch Λ d − ( M d − ) ,then the pro-characteristic ideal of M over Λ is well defined as f Ch Λ ( M ) := lim ←− d ( π ΛΛ d ) − ( Ch Λ d ( M d )) (where π ΛΛ d : Λ → Λ d is the natural map provided by the inverse limit defining Λ ).Proof. We can assume that the M d are torsion Λ d -modules (at least for d ≫ Ch Λ d ( M d ) are zero and there is nothing to prove. By Proposition 2.10, applied to A = Λ d − , B = Λ d and M = M d , we get Ch Λ d − (( M d ) t d ) π Λ d Λ d − ( Ch Λ d ( M d )) = Ch Λ d − ( M d /t d M d ) . For d ≫ π Λ d Λ d − ( Ch Λ d ( M d )) ⊆ Ch Λ d − ( M d − ) , which shows that the generators of the ideals Ch Λ d ( M d ) form a coherent sequence with respectto the maps defining Λ. Hence this sequence defines an element in Λ which can be consideredas a generator for the pro-characteristic ideal f Ch Λ ( M ) := lim ←− d ( π ΛΛ d ) − ( Ch Λ d ( M d )) . (cid:3) Our pro-characteristic ideal maintains two classical properties of characteristic ideals.
Corollary 2.14.
Let M , M ′ and M ′′ be finitely generated Λ -modules which verify the hy-potheses of Theorem 2.13. HARACTERISTIC IDEALS AND IWASAWA THEORY 10 The pro-characteristic ideals are multiplicative, i.e., if there is an exact sequence (2.8) M ′ (cid:31) (cid:127) / / M / / / / M ′′ , then f Ch Λ ( M ) = f Ch Λ ( M ′ ) f Ch Λ ( M ′′ ) . f Ch Λ ( M ) = 0 if and only if M d is a finitely generated torsion Λ d -module for d ≫ .Proof. For any d > M ′ d (cid:31) (cid:127) / / M d / / / / M ′′ d , for which the equality Ch Λ d ( M d ) = Ch Λ d ( M ′ d ) Ch Λ d ( M ′′ d ) holds. The previous theorem allowsto take limits on both sides maintaining the equality. Obvious. (cid:3)
Remark 2.15.
In the previous corollary it is enough to assume that M ′ and M ′′ verify thehypotheses of Theorem 2.13. Indeed, using the snake lemma exact sequence( M ′ d ) t d ֒ → ( M d ) t d → ( M ′′ d ) t d → M ′ d /t d M ′ d → M d /t d M d ։ M ′′ d /t d M ′′ d , one immediately has that( M ′ d ) t d and ( M ′′ d ) t d ∼ Λ d − ⇒ ( M d ) t d ∼ Λ d − Ch Λ d − ( M d /t d M d ) = Ch Λ d − ( M ′ d /t d M ′ d ) Ch Λ d − ( M ′′ d /t d M ′′ d ) ⊆ Ch Λ d − ( M ′ d − ) Ch Λ d − ( M ′′ d − ) = Ch Λ d − ( M d − ) . Class groups in global fields
For the rest of the paper we adjust our notations a bit to be more consistent with the usualones in Iwasawa theory. We fix a prime number p and a global field F (note that for now weare not making any assumption on the characteristic of F ). For any finite extension E/F let M ( E ) be the p -adic completion of the group of divisor classes of E , i.e., M ( E ) := ( E ∗ \ I E / Π v O ∗ E v ) ⊗ Z p where I E is the group of finite ideles of E , v varies over all non-archimedean places of E and O E v is the ring of integers of the completion of E at v . When L /F is an infinite extension, weput M ( L ) := lim ← M ( E ) as E runs among finite subextensions of L /F (the limit being takenwith respect to norm maps). Class field theory yields a canonical isomorphism(3.1) M ( E ) ∼ −→ X ( E ) := Gal( L ( E ) /E ) , where L ( E ) is the maximal unramified abelian pro- p -extension of E . Passing to the limitshows that (3.1) is still true for infinite extensions.Finally, for any infinite Galois extension L /F , let Λ( L ) := Z p [[Gal( L /F )]] be the associatedIwasawa algebra. We shall be interested in the situation where Gal( L /F ) is an abelian p -adicLie group: in this case, both M ( L ) and X ( L ) are Λ( L )-modules (the action of Gal( L /F )on X ( L ) is the natural one via inner automorphisms of Gal( L ( L ) /F ) ) and these structuresare compatible with the isomorphism (3.1). Furthermore, if Gal( L /F ) ≃ Z dp then Λ( L ) ≃ Z p [[ t , .., t d ]] is a Krull domain. Lemma 3.1.
Let F /F be a Z dp -extension, ramified only at finitely many places. If d > , onecan always find a Z p -subextension F /F such that none of the ramified places splits completelyin F . HARACTERISTIC IDEALS AND IWASAWA THEORY 11
Proof.
Let S denote the set of primes of F which ramify in F and, for any place v in S let D v ⊂ Gal( F /F ) =: Γ be the corresponding decomposition group. Getting F amountsto finding α ∈ Hom(Γ , Z p ) such that α ( D v ) = 0 for all v ∈ S . By hypothesis, for such v ’s the vector spaces D v ⊗ Q p are non-zero, hence their annihilators are proper subspaces ofHom(Γ ⊗ Q p , Q p ) and since a Q p -vector space cannot be union of a finite number of propersubspaces, we deduce that the required α exists. (cid:3) The following lemma is mostly a restatement of [13, Theorem 1].
Lemma 3.2.
Let F /F be a Z dp -extension, ramified only at finitely many places, and F ′ ⊂ F a Z d − p -subextension, with d > . Let I be the kernel of the natural projection Λ( F ) → Λ( F ′ ) . Then X ( F ) /IX ( F ) is a finitely generated torsion Λ( F ′ ) -module and X ( F ) is a finitelygenerated torsion Λ( F ) -module. This holds also for d = 2 , provided that no ramified place in F /F is totally split in F ′ .Proof. The idea is to proceed by induction on d . Choose a filtration F =: F ⊂ F ⊂ · · · ⊂ F d − := F ′ ⊂ F d := F so that Gal( F i / F i − ) ≃ Z p for all i and no ramified place in F /F is totally split in F (byLemma 3.1, this can always be achieved when d > F i )-module M is in Fgt Λ( F i ) if M/I ii − M is in Fgt Λ( F i − ) (where I ii − is the kernel of the pro-jection Λ( F i ) → Λ( F i − ) ) and Greenberg’s proof shows that X ( F i − ) ∈ Fgt Λ( F i − ) implies X ( F i ) /I ii − X ( F i ) ∈ Fgt Λ( F i − ) . So it is enough to prove that X ( F ) is a finitely generatedtorsion Λ( F )-module. This follows from Iwasawa’s classical proof ([15], exposed e.g. in [22];the function field version can be found in [17]). (cid:3) Remarks 3.3.1.
In a Z p -extension of a global field, only places with residual characteristic p can ramify:thus the finiteness hypothesis on the ramification locus is automatically satisfied unless char ( F ) = p . Note, however, that in the latter case this hypothesis is needed (see,e.g. [12, Remark 4]). Among all Z p -extensions of F there is a distinguished one, namely, the cyclotomic Z p -extension F cyc if F is a number field and the arithmetic Z p -extension F arit (arising fromthe unique Z p -extension of the constant field) if F is a function field. The conditionon F ′ (when d = 2) is satisfied if it contains either F cyc or F arit . For d = 1, we have F ′ = F and Λ( F ′ ) = Z p . Thus the analogue of Lemma 3.2 wouldstate that X ( F ) /IX ( F ) is finite. This holds quite trivially if F is a global function fieldand F = F arit (note also that if char ( F ) = ℓ = p then F arit is the only Z p -extensionof F , see e.g. [5, Proposition 4.3]). In this case the maximal abelian extension of F contained in L ( F ) is exactly L ( F ), hence X ( F ) /IX ( F ) ≃ Gal( L ( F ) /F arit ) which isknown (e.g. by class field theory) to be finite.3.1. Iwasawa theory for class groups in function fields.
In this section F will be aglobal function field of characteristic p and F arit its arithmetic Z p -extension as defined above.Let F /F be a Z ∞ p -extension unramified outside a finite set of places S , with Γ := Gal( F /F )and associated Iwasawa algebra Λ := Λ( F ). We fix a Z p -basis { γ i } i ∈ N for Γ and for any d > F d ⊂ F be the fixed field of { γ i } i>d . Also, we assume that our basis is such that noplace in S splits completely in F (Lemma 3.1 shows that there is no loss of generality in thisassumption). Remark 3.4. If F contains F arit we can take the latter as F . The additional hypothesison F appears also in [16, Theorem 1.1]: the authors enlarge the set S and the extension F d in order to get a Z p -extension verifying that hypothesis and use this to get a monomial HARACTERISTIC IDEALS AND IWASAWA THEORY 12
Stickelberger element. This is a crucial step in the proof of the Main Conjecture provided in[10].For notational convenience, let t i := γ i −
1. Then the subring Z p [[ t , . . . , t d ]] of Λ is canonicallyisomorphic to Λ( F d ) and, by a slight abuse of notation, the two shall be identified in thefollowing. In particular, for any d > F d ) = Λ( F d − )[[ t d ]] and we can apply theresults of Section 2. We shall denote by π dd − the canonical projection Λ( F d ) → Λ( F d − ) withkernel I dd − = ( t d ) (the augmentation ideal of F d / F d − ) and by Γ dd − the group Gal( F d / F d − ).For two finite extensions L ⊃ L ′ ⊃ F , the degree maps deg L and deg L ′ fit into the commu-tative diagram (with exact rows)(3.2) A ( L ) (cid:31) (cid:127) / / N LL ′ (cid:15) (cid:15) M ( L ) deg L / / / / N LL ′ (cid:15) (cid:15) Z p (cid:15) (cid:15) A ( L ′ ) (cid:31) (cid:127) / / M ( L ′ ) deg L ′ / / / / Z p , where N LL ′ denotes the norm and the vertical map on the right is multiplication by [ F L : F L ′ ](the degree of the extension between the fields of constants). For an infinite extension L /F contained in F , taking projective limits one gets an exact sequence(3.3) A ( L ) (cid:31) (cid:127) / / M ( L ) deg L / / Z p . Remark 3.5.
The map deg L above becomes zero exactly when L contains the unramified Z p -subextension F arit .By (3.1), Lemma 3.2 shows that M ( F d ) is a finitely generated torsion Λ( F d )-module, so thesame holds for A ( F d ). Hence, by Proposition 2.10, one has, for all d > Ch Λ( F d − ) ( A ( F d ) t d ) π dd − ( Ch Λ( F d ) ( A ( F d ))) = Ch Λ( F d − ) ( A ( F d ) /t d A ( F d ))and note that A ( F d ) t d = A ( F d ) Γ dd − , A ( F d ) /t d A ( F d ) = A ( F d ) /I dd − A ( F d ) . Consider the following diagram(3.5) A ( F d ) (cid:31) (cid:127) / / t d (cid:15) (cid:15) M ( F d ) deg / / / / t d (cid:15) (cid:15) Z pt d (cid:15) (cid:15) A ( F d ) (cid:31) (cid:127) / / M ( F d ) deg / / / / Z p (note that the vertical map on the right is 0) and its snake lemma sequence(3.6) A ( F d ) Γ dd − (cid:31) (cid:127) / / M ( F d ) Γ dd − deg / / Z p (cid:15) (cid:15) Z p M ( F d ) /I dd − M ( F d ) deg o o o o A ( F d ) /I dd − A ( F d ) . o o For d ≥ Z p is a torsion Λ( F d − )-module), (3.6) and Lemma 3.2 showthat A ( F d ) /I dd − A ( F d ) is in Fgt Λ( F d − ) as well. By Proposition 2.10 it follows that no termin (3.4) is trivial. HARACTERISTIC IDEALS AND IWASAWA THEORY 13
Totally ramified extensions and the Main Conjecture.
The main examples we have inmind are extensions satisfying the following
Assumption 3.6.
The (finitely many) ramified places of F /F are totally ramified.In what follows an extension satisfying this assumption will be called a totally ramified exten-sion . A prototypical example is the a -cyclotomic extension of F q ( T ) generated by the a -torsionof the Carlitz module ( a an ideal of F q [ T ], see e.g. [21, Chapter 12]). As usual in Iwasawatheory over number fields, most of the proofs will work (or can be adapted) simply assumingthat ramified primes are totally ramified in F / F e for some e >
0, but, in the function fieldsetting, one would need some extra hypothesis on the behaviour of these places in F e /F .Under this assumption any Z p -subextension can play the role of F . Moreover M ( F )is defined using norm maps and norms are surjective on class groups in totally ramifiedextensions, so M ( F d ) = N FF d ( M ( F )) := M ( F ) d and M ( F ) = lim ←− d M ( F ) d = lim ←− d M ( F d )(in the notations of Theorem 2.13). The same holds for the modules A ( F ) and A ( F d ).Let L ( F d − ) be the maximal abelian extension of F d − contained in L ( F d ), we have F d L ( F d − ) ⊆ L ( F d − ) and Gal( L ( F d ) /L ( F d − )) = I dd − M ( F d )(see [24, Lemma 13.14]). Galois theory provides a surjectionGal( L ( F d − ) / F d ) ։ Gal( F d L ( F d − ) / F d ) , i.e., M ( F d ) /I dd − M ( F d ) ։ M ( F d − ) , which yields(3.7) Ch Λ d − ( M ( F d ) /I dd − M ( F d )) ⊆ Ch Λ d − ( M ( F d − )) . The same relation holds for the characteristic ideals of the A ( F d ) for d >
3, because of (3.6).In particular if we have only one ramified prime, the surjection above is an isomorphism (justadapt the proof of [24, Lemma 13.15]) and (3.7) is an equality. This takes care of hypothesis in Theorem 2.13.A little modification of the proof of [6, Lemma 5.7] (note that [6, Lemmas 5.4 and 5.6] stillhold in the present setting), shows that elements of M ( F d ) Γ dd − are represented by divisorssupported on ramified primes. Hence M ( F d ) Γ dd − (and A ( F d ) Γ dd − ) are finitely generated Z p -modules, i.e., pseudo-null Λ( F d − )-modules for d >
3. From (3.4) we obtain
Corollary 3.7.
Let F d be a Z dp -extension of F contained in a totally ramified extension.Then, for any Z d − p -extension F d − contained in F d , one has (3.8) π dd − ( Ch Λ( F d ) ( A ( F d ))) = Ch Λ( F d − ) ( A ( F d ) /I dd − A ( F d )) ⊆ Ch Λ d − ( A ( F d − )) . Hence the modules A ( F d ) verify the hypotheses of Theorem 2.13 and we can define Definition 3.8.
Let F /F be a totally ramified Z ∞ p -extension. The pro-characteristic ideal of A ( F ) is f Ch Λ ( A ( F )) := lim ←− F d ( π F d ) − ( Ch Λ( F d ) ( A ( F d ))) . HARACTERISTIC IDEALS AND IWASAWA THEORY 14
Remark 3.9.
Definition 3.8 only depends on the extension F /F and not on the filtrationof Z dp -extension we choose inside it. Indeed take two different filtrations {F d } and {F ′ d } anddefine a new filtration containing both by putting F ′′ := F and F ′′ n = F n F ′ n ∀ n ≥ F ′′ n is not, in general, a Z np -extension and F ′′ n / F ′′ n − is a Z ip -extension with 0 ≤ i ≤ {F ′′ n } . This answers questions a and b of [6, Remark 5.11]: we had a similar definitionthere but it was based on the particular choice of the filtration.We recall that, in [9, Theorem 3.1] (and [10]), the authors prove an Iwasawa Main Conjec-ture (IMC) at “finite level”, which (in our simplified setting and notations) reads as(3.9) Ch Λ( F d ) ( A ( F d )) = ( θ F d /F,S ) , where θ F d /F,S is the classical Stickelberger element (defined e.g. in [6, Section 5.3]). By [23,Proposition IV.1.8], the elements θ F d /F,S form a coherent sequence with respect to the maps π de , so, taking inverse limits in (3.9), one obtains Corollary 3.10 (IMC in non-noetherian algebras) . In the previous setting we have f Ch Λ ( A ( F )) = lim ←− F d ( θ F d /F,S ) := ( θ F /F,S ) , as ideals in Λ . More details on the statement and its proof (now independent from the filtration {F d } d > )can be found in [6, Section 5]. Remark 3.11.
A different approach, using a more natural filtration of global function fieldsfor the Carlitz p -cyclotomic extension of F q ( T ) and Fitting ideals of class groups, will becarried out in [1]. It leads to a similar version of the Iwasawa Main Conjecture in the algebraΛ, but it has the advantage of having more direct and relevant arithmetic applications (see[1, Section 6]). References [1] B. Angl´es - A. Bandini - F. Bars - I. Longhi,
Iwasawa Main Conjecture for the Carlitz cyclotomic extensionand applications , in progress.[2] A. Bandini,
Greenberg’s conjecture for Z dp -extensions , Acta Arith. (2003), no. 4, 357–368.[3] A. Bandini, Greenberg’s conjecture and capitulation in Z dp -extensions , J. Number Theory (2007),121–134.[4] A. Bandini - I. Longhi, Control theorems for elliptic curves over function fields , Int. J. Number Theory (2009), no. 2, 229–256.[5] A. Bandini - I. Longhi, Selmer groups for elliptic curves in Z dl -extensions of function fields of characteristic p , Ann. Inst. Fourier (2009), no. 6, 2301–2327.[6] A. Bandini - F. Bars - I. Longhi, Aspects of Iwasawa theory over function fields , arXiv:1005.2289 [math.NT](2011), to appear in the EMS Congress Reports.[7] A. Bandini - F. Bars - I. Longhi,
Characteristic ideals and Selmer groups , arXiv:1404.2788 [math.NT](2014).[8] N. Bourbaki,
Commutative algebra. Chapters 1-7 , Elements of Mathematics, Springer-Verlag, Berlin, 1998.[9] D. Burns,
Congruences between derivatives of geometric L -functions , Invent. Math. (2011), no. 2,221–256.[10] D. Burns - K. F. Lai - K.-S. Tan, On geometric main conjectures , Appendix to [9].[11] R. Gilmer,
Power series rings over a Krull domain , Pacific J. Math. (1969) 543–549.[12] R. Gold and H. Kisilevsky, On geometric Z p -extensions of function fields , Manuscr. Math. (1988),145–161. HARACTERISTIC IDEALS AND IWASAWA THEORY 15 [13] R. Greenberg,
The Iwasawa invariants of Γ -extensions of a fixed number field , Amer. J. Math. (1973),204–214.[14] R. Greenberg, On the structure of certain Galois groups , Invent. Math. (1978), 85–99.[15] K. Iwasawa, On Γ -extensions of Algebraic Number Fields , Bull. Amer. Math. Soc. (1959), 183–226.[16] K-L. Kueh - K.F. Lai - K.-S. Tan, Stickelberger elements for Z dp -extensions of function fields , J. NumberTheory (2008), 2776-2783.[17] C. Li - J. Zhao, Iwasawa theory of Z dp -extensions over global function fields , Exposition. Math. (1997),no. 4, 315-337.[18] T. Ochiai, Euler system for Galois deformations , Ann. Inst. Fourier (2005), no. 1, 113–146.[19] M. Ozaki, Iwasawa invariants of Z p -extensions over an imaginary quadratic field , Adv. Studies in PureMath. (2001), 387–399.[20] B. Perrin-Riou, Arithm´etique des courbes elliptiques et th´eorie d’Iwasawa , M´em. Soc. Math. Fr. (Nouv.S´er.) , (1984).[21] M. Rosen, Number theory in function fields , GTM 210 , Springer-Verlag, New york, 2002.[22] J.-P. Serre,
Classes des corps cyclotomiques (d’apr`es K. Iwasawa) , S´eminaire Bourbaki, Exp. No. ,(1958-59).[23] J. Tate,
Les conjectures de Stark sur les Fonctions L d’Artin en s = 0, Progress in Mathematics ,Birkh¨auser, 1984.[24] L. Washington, Introduction to cyclotomic fields. Second edition , GTM 83 , Springer-Verlag, New York,1997.
Dipartimento di Matematica e Informatica, Universit`a degli Studi di Parma, Parco Area delleScienze, 53/A - 43124 Parma (PR), Italy
E-mail address : [email protected] Departament Matem`atiques, Edif. C, Universitat Aut`onoma de Barcelona, 08193 Bellaterra,Catalonia
E-mail address : [email protected] Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren Ai Road,Dushu Lake Higher Education Town, Suzhou Industrial Park, Suzhou, Jiangsu, 215123, China
E-mail address ::