On the non commutative Iwasawa Main Conjecture for abelian varieties over function fields

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On the non commutative Iwasawa Main Conjecture for abelian varieties over function fields

Fabien Trihan, David Vauclair

To cite this version:

Fabien Trihan, David Vauclair. On the non commutative Iwasawa Main Conjecture for abelian vari- eties over function fields. Documenta Mathematica, Universität Bielefeld, In press. �hal-02143994�

On the non commutative Iwasawa Main Conjecture for abelian varieties over function fields.

Fabien Trihan and David Vauclair

Abstract. We establish the Iwasawa main conjecture for semistable abelian varieties over a function eld of characteristicpunder certain restrictive assumptions. Namely we considerp-torsion freep-adic Lie extensions of the base eld which contain the constant Zp-extension and are everywhere unramied. Under the usualµ= 0hypothesis, we give a proof which mainly relies on the interpretation of the Selmer complex in terms ofp-adic cohomology [TV] together with the trace formulas of [EL1].

2010 Mathematics Subject Classication: 11S40 (primary), 11R23, 11R34, 11R42, 11R58, 11G05, 11G10 (secondary)

Keywords and Phrases: abelian variety, Iwasawa theory,p-adic coho- mology, syntomic, non-commutative

1 Introduction

1.1 Statement of the main theorem

Consider a function eld K of characteristic p, C the corresponding proper smooth geometrically irreducible curve over a nite eld k and an abelian variety A/K with Néron modelA/C. Assume for simplicity thatA has good reduction everywhere and that the Hasse-Weil L function of A/K does not vanish ats= 1. In this situation the BSD conjecture predicts that each group involved in the right hand term below is nite and that the following formula holds (whereAˆ denotes the dual abelian variety):

L(A/K,1) = #X(A/K)

#A(K)# ˆA(K).#H⁰(C, Lie(A))

#H¹(C, Lie(A)) (1)

Let us restrict our attention to the p-adic valuations on both sides. There are natural perfect complexes of Z^p-modules RΓ(C, Tp(A)), RΓ(C, Lie(A)) whose cohomology groups are thep-part of the groups appearing above. These complexes thus become acyclic after extension of scalars to Qp, hence pro- duce classes in K0(Zp,Qp) ' Z. Looking at L(A/K,1) as an element of K1(Qp)'Qp×, thep-part of (1) simply becomes

∂(L(A/K,1)) = [RΓ(C, Tp(A))] + [RΓ(C, Lie(A))] (2) where ∂ : K1(Q^p) → K0(Z^p,Q^p) is the connecting map of lower K-theory (ie. the p-adic valuationvp :Qp× →Z). As explained in [KT] the complexes appearing on the right hand side can be related top-adic (crystalline and then rigid) cohomology and the trace formulae which are known in this context are then sucient to actually prove (2).

The main purpose of this paper is to establish a similar statement in the setting of non commutative Iwasawa theory. We don't assume that L(A/K,1) 6= 0 anymore and A is now allowed to have semistable reduction at some given set Z of points of C. Consider a Galois extension K∞/K which contains the constant Zp-extension Kk∞/K, is unramied everywhere, and whose Galois groupG= lim

←−nG_n isp-adic Lie withoutp-torsion. In that situation, a general result explained in section 2 will allow us to form perfect complexes of modules over the Iwasawa algebraΛ(G) := lim

←−ⁿZp[G_n]: NK∞ := Rlim

←−ⁿRΓ^Zn(Cn, Tp(A)) L_K∞ := Rlim

←−ⁿRΓ^Zn(C_n, Lie(A))

whereRΓ^Zn(C_n,−)is the functor of cohomology vanishing atZ_n (see. section 4 for a precise denition and the relation to usual Selmer complexes) and is designed to take out the contribution of Z. A signicant dierence with the BSD statement is that here, the cohomology of these complexes are expected to be torsion Λ(G)-modules even if L(A/K,1) = 0. This is well known if K =Kk_∞ (see. Cor. 5.2 or [LLTT] Cor. 2.1.5). To go from this case to the general one, we follow the strategy of [CFKSV], which unfortunately requires an extra assumption, namely the usual µ= 0hypothesis. More precisely, we will prove the following generalization of [LLTT], where only the case K =Kk_∞ was considered.

Theorem 1.1. (Thm. 5.15) Let A/K, K_∞/K and G as above. If the µ- invariant of the Pontrjagin dual of the discrete Selmer group of A overKk_∞ is trivial, then

(i) The cohomology Λ(G)-modules of N_K∞ and L_K∞ are S^∗-torsion, where S^∗ is the canonical Ore set of [CFKSV]. The latter complexes thus produce classes [NK_∞] and [LK_∞] in K0(Λ(G),Λ(G)S^∗) (denoted K0(MH(G))in loc. cit.). .

(ii) There exists a canonical elementL_A/K∞ ∈K1(Λ(G)S^∗)satisfying

∂L_A/K∞ = [N_K∞] + [L_K∞] (3) where∂:K₁(Λ(G)_S^∗)→K₀(Λ(G),Λ(G)_S^∗)denotes the connecting map in lowerK-theory.

(iii) The elementL_A/K∞ veries the interpolation property

ρ(L_A/K∞) =L(U, A/K, ρ^∨,1) (4) for each Artin O-valued representation ρ of G (O/Zp, a totally ram- ied extension), where ρ(−) : K₁(Λ(G)_S^∗) → L^× ∪ {0,∞} denotes the corresponding evaluation map, (−)^∨ denotes the contragredient, and L(U, A/K, ρ, s) is the ρ-twisted Hasse-Weil L-function of A/K without Euler factors atZ =C−U.

We also give a similar statement involving complexes N_K^∗_∞ and L^∗_K∞ which are Λ(G)-dual to NK∞ and LK∞ (see Prop. 2.11 (ii), as well as Prop. 4.16, (ii) for the precise statement).

Here, the construction ofLA/K∞, the proof thatN_K∞ is torsion and the proof of (3) are simultaneous and rely essentially on the main result of [TV] (a sheaed version of [KT] Prop. 5.13) which yields distinguished triangles of perfect complexes of Λ(G)-modules relating at and crystalline cohomology (see Sect. 4.4.4 and Rem. 4.15). The proof of (4) relies on the comparison of crystalline and rigid cohomology [LST] together with the trace formula for the latter and the codescent properties of the Iwasawa complexes along the tower K_∞/K (Prop. 4.16, relying on Thm. 2.11).

1.2 Outline of the paper

Section 2. Dening the complexes NK∞, LK∞ orN_K^∗_∞,L^∗_K∞ occurring in the main conjecture (3) involves forming projective limits along the Galois tower formed by theCn's or alternatively inductive limits and taking duals. In order to perform these operations, a convenient framework is given by the derived cat- egory of normic systemsD^b(GM od(Zp))dened and studied in [Va]. A normic system is a collectionM_n ofG_n-modules together with equivariant morphisms M_n→M_m,M_m→M_n satisfying a natural compatibility (Def. 2.1). The pur- pose of this section is to show that the collection of derived functorsRΓ(C_n,−) comes from a functorRΓ(C,−)with values inD^b(_GM od(Zp)).

We place ourself in a setting which is general enough to handle the various cohomology theories (at, étale, crystalline...) involved here. Namely we show (Lem.-Def. 2.7) that in a ringed topos (E, R)any pro-torsor X = (Xn, Gn)n

gives rise to a functor

Γ(X,−) :M od(E, R)→GM od(Γ(E, R)) (5)

from the category of R-modules of E to that of normic systems of Γ(E, R)- modules. Using [Va], the nal subsection collects the basic properties (descent, codescent, perfectness, duality, under suitable assumptions, see. Thm. 2.11) of the Iwasawa complexes dened using the functor

D⁺(E,Zp)→D(Λ(G)) obtained by composing RΓ(X,−)withRlim

←−n, or alternatively withlim

−→n and thenRHom(−,Z^p).

Section 3. We review the basic facts from K-theory which are needed for our purpose. This involves mainly the determinant functor for perfect complexes from [Kn] and its behaviour with respect to distinguished triangles and local- ization following [FK]. We also discuss the evaluation map which appears in (4).

Section 4. We begin by recalling the denition of the derived functorRΓ^Z(C, .) of global sections vanishing at a closed subschemeZ which naturally appears in the comparison theorem of [TV] and is denotedRΓar,V in [KT] (see Rem. 4.13 for a more precise statement). Next we compare the complex of normic systems RΓ^Z(C, Tp(A))underlyingNK∞ to the usual Selmer complex ofA/K. We take the opportunity to give a tractable denition for normic Selmer complexes and prove the expected duality theorem in this setting.

Finally we recall the comparison result of [TV] (which takes place in the small étale topos ofC) and write down the fundamental distinguished triangles that follow from it, using (5):

RΓ^Z(C, Tp(A))→RΓ(C^]/Z^p, F il¹Dlog(A)(−Z))^1−φ→ RΓ(C^]/Z^p, Dlog(A)(−Z))⁺¹→ RΓ(C^]/Zp, F il¹D_log(A)(−Z))→¹ RΓ(C^]/Zp, D_log(A)(−Z))→RΓ(C, Lie(A)(−Z))⁺¹→

(6) Here φis a semi-linear map such thatφ1=pF rob. By applyingRlim

←−ⁿ, these in turn yield distinguished triangles of perfect complexes of Λ(G)-modules satisfying the derived codescent property. This will be the main ingredient for the proof of 1.1.

Section 5. We put everything together in order to prove the Iwasawa main conjecture. The third term of the rst distinguished triangle above is a k- vector space and the arrow denoted 1thus becomes invertible after inverting S^∗. Whence an endomorphism(1−φ)S^∗1⁻¹_S∗ acting on the localizationP_K∞,S^∗ of

PK∞ :=Rlim

←−RΓ(C_n^]/Zp, Dlog(A)(−Z))

In the case where G= Γ, the base change formula in crystalline cohomology together with a semi-linear argument shows that the rst term in the second distinguished triangle above vanishes as well afterS^∗-localization. In the gen- eral case, an argument using Nakayama's lemma ensures that it is still the case

under the µ= 0 assumption. The endomorphism(1−φ)S^∗1⁻¹_S∗ is thus invert- ible, allowing us to dene L_A/K∞ ∈K1(ΛS^∗)as its determinant, rendering (3) almost tautological. Let us hint the idea of the proof of (4). Using the descent properties of the normic section functor together with the comparison between crystalline and rigid cohomology from [LST], we prove an isomorphism

L⊗^L_ΛO(G)(V_ρ,ΛO(G))^∗⊗^L_Λ(G)P_K∞'RΓ_rig,c(U/L, pr^∗D^†⊗U(ρ^∨)) (7) whereU(ρ^∨)denotes the unipotent convergent isocrystal associated toρ(Prop.

5.16). On the one hand the trace formula in rigid cohomology shows that L(U, A/K, ρ^∨,1) coincides with the alternated product of the determinants of 1−p⁻¹F robacting on theL-valued cohomology of the right hand side. Since we have no morphism fromΛS^∗ toLwe may not use directly the base change property of theDetfunctor together with a localized version of (7) in order to relate the action of1−p⁻¹F robon the right hand side withρ(L). We turn this diculty by investigating carefully the spectral sequence of codescent along Kk_∞/K with the denition of the evaluation map in mind.

Acknowledgements. The rst author is supported by JSPS. Both authors thank the referees for their careful reading and suggestions to simplify the paper.

2 The normic section functor

The purpose of this section is to show that under reasonable conditions the cohomology of a topos along a Galois tower with groupGnaturally gives rise to perfect complexes of Λ(G)-modules satisfying natural properties such as derived descent, codescent and duality along the tower. This goal is achieved in Thm. 2.11.

2.1 Normic systems

We dene and study briey the category of normic systems along a pronite group G. The following denitions slightly generalize those given in [Va].

Definition 2.1. Consider a ring S, a pronite group G, and let (Hn)_n∈N denote the ltered set of the normal open subgroups of G (N an appropriate ltered set of indices). For n≤m in N, we denote Gn =G/Hn andGm,n= Hn/Hm. In this setting we dene the following categories.

(i) GM od(S) the category ofS-modules endowed with a discrete action of G, ie. which are the union of their xed points by the Gn's.

(ii) _GnM od(S) the full subcategory of GM od(S) whose objects are those on which H_n acts trivially. For m ≥ n, the inclusion functor GnM od(S) →

G_mM od(S) has a left and a right adjoint, described respectively as the coin- variants functor(−)G_m,n and the xed points functor(−)^Gm,n.

(iv)GM od(S)the category of M od(S)-valued normic systems is dened a fol- lows:

- an object is a triple M = ((Mn)_n∈N,(jn,m:Mn →Mm)_n≤m,(km,n:Mm→ Mn)_n≤m)satisfying the following properties:

(N orm1) Mn is an object of GnM od(S), jn,m and km,n are morphisms of

GM od(S).

(N orm2) If n1 ≤ n2 ≤n3 then jn₂,n₃ ◦jn₁,n₂ =jn₁,n₃ and kn₂,n₁◦kn₃,n₂ = kn₃,n₁.

(N orm3)If n≤m,jn,m◦km,n:Mm→Mmcoincides with the endomorphism P

g∈Gm,ncg ofMm. Note that this is not only a morphism inM od(S), but also in GM od(S)sinceHnCG.

- a morphism φ: M → M⁰ is a collection of morphisms of GM od(S), (φn : Mn→M_n⁰)n∈N such that the following squares commute for each couplen≤m:

Mm

φ_m //M_m⁰ Mm

k_m,n

φ_m //M_m⁰

k_m,n

M_n

jn,m

OO

φ_n //M_n⁰

jn,m

OO

M_n ^φn //M_n⁰

Let us make some remarks about the categoryGM od(S)of discreteG-objects.

- ForM, M⁰∈G M od(S):

Hom_{GM od(S)}(M, M⁰)'lim

←−n

lim−→

m

Hom_{GM od(S)}(M^Hn, M^0Hm).

- The category GM od(S) is a full subcategory of M od(S[G]). The inclusion functor has a right adjointM 7→ ∪nM^Hn. In particular,GM od(S)has enough injectives sinceM od(S[G])has.

We now turn to some properties of the categoryGM od(S)of normic systems.

Proposition 2.2. Small inductive and projective limits exist inGM od(S)and commute to the component functors GM od(S) → G_nM od(S). In particular,

GM od(S)is an abelian category.

Proof. In order to form the inductive (resp. projective) limit indexed by a set I in GC, it suces to form the limit of the components and endow them with the jn,m and km,n provided by functoriality. This prove the rst statement.

The second one follows since being abelian is a property of limits ([KS], Def.

8.2.8 and Def. 8.3.5).

The following lemma, inspired by [TW] has been pointed out by B. Kahn.

Lemma 2.3. (i) If G is in fact a nite group, then the category GM od(S) is equivalent to M od(µ_C(G, S)) where µ_C(G, S) is the normal Mackey al- gebra, dened as the quotient of the free associative algebra S{{cg, g ∈ G},{jn,m, km,n, n≤m}} by the following relations:

-cg₁cg₂ =cg₁g₂,cgjn,m=jn,mcg,cgkm,n=km,ncg, forg, g1, g2∈Gandn≤m in N.

- jn₂,n₃jn₁,n₂ =jn₁,n₃,kn₂,n₁kn₃,n₂ =kn₃,n₁, for n1≤n2≤n3 inN. - cgjn,n =jn,n,jn,n=kn,n, forn∈N andg∈Hn.

- jn,mkm,n=P

g∈Gm,ncgjm,m, for n≤minN. - P

n∈Njn,n= 1.

(ii) In general, GM od(S) is equivalent to the category of cocartesian sec- tions of the cobered category above N^op associated to the covariant pseudo- functor on N^op mapping n to M od(µ_C(Gn, S)), and m ≥ n to the functor M od(µ_C(G_m, S)) → M od(µ_C(G_n, S)) given by extension of scalars through the morphism µ_C(G_m, S)→µ_C(G_n, S) dened by killing the j_a,b's and k_b,a's with bn.

Proof. (i)Let us describe the functorGM od(S)→M od(µ_C(G, S)). A normic system M is sent to ⊕nM_n, together with its obvious structure of µ_C(G, S)- module: c_g acts on every components whereas j_n,m (resp. k_m,n) sends the n-th (resp. m-th) component into them-th (resp. n-th) one. By denition, a morphism of normic systemsM →M⁰consists in a collection of morphismsφ_n: Mn →M_n⁰, compatible with the cg's, thejn,m's and the km,n's. It thus gives rise to a morphism ⊕nMn → ⊕nM_n⁰, compatible with the action ofµ_C(G, S) and this is clearly compatible to composition. We thus have dened the desired functor GM od(S)→M od(µ_C(G, S)). It now remains to notice that both the full faithfulness and the essential surjectivity of this functor immediately follow from the isomorphism of algebrasµ_C(G, S)'Q

n∈Njn,nµ_C(G, S).

(ii)We apply(i)to the caseG=Gn. Lettingnvary, this gives an equivalence of cobered categories. Whence the result, sinceGM od(S)is clearly equivalent to the category of cocartesian sections of the cobered category associated to n7→GnM od(S).

The following corollary answers a question of [Va].

Corollary 2.4. The categoryGM od(S) has enough injectives.

Proof. Let F/N^op denote the bered categoryn 7→M od(µ_C(Gn, S)) consid- ered previously and letSect(F/N^op) (resp. Cocart(F/N^op)) denote its cate- gory of sections (resp. cocartesian sections). Denoting|N|the discrete category underlying N, there are three obvious forgetful functors

Cocart(F/N^op) →^F1 Sect(F/N^op)

F₂

→ Q

|N|M od(µ_C(Gn, S))

F3

→ M od(S)^|N|

each of which is exact and whose composition is faithful. The result will follow formally if one proves that each of them has a right adjoint. ForF3, this is easy and left to the reader. ForF2, this results from [SGA4], Vbis, 1.2.10 (note that hereF/N^opis indeed bibered). ForF1, we use the following general easy fact concerning an arbitrary cobered categoryF/S with cocleavagef 7→f!: Fact: Assume that for anyf :X→Y the following properties are veried:

- The functor "composition withf" :S/X → S/Y is conal.

-f! commutes to projective limits indexed byS_/X.

Then the inclusion functor Cocart(F/S) → Sect(F/S) has a right ad- joint, which takes a section X 7→ s(X) to the cocartesian section X 7→

lim←−f:Z→Xf!s(Z).

Remark 2.5. One may show that GM od(S)has enough projectives as well.

2.2 From sheaves to normic systems As before letG= lim

←−nG_ndenote a pronite group and letG_n=G/H_n. Recall that the classifying topos B_G is the category of discrete leftG-sets. Consider another arbitrary topos E together with its structural morphismπ:E →Set and assume given a projective system X = (X_n) ofG-objects ofE such that X_n is a torsor of E under π⁻¹G_n. In this situation we have for m ≥ n a commutative diagram of topoi as follows:

E

p∞

ss ^pmuu ^pn|| ^π

BG q_m //BG_m q_m,n //BG_n //Set

Here the horizontal arrows are by functoriality of the classifying topos with respect to the group (their inverse and direct images functors are ination and xed points by the adequate subgroup) and the oblique morphisms, dened by X, are as follows:

- If Y is in BG_n and Y⁰ denotes the underlying set, the formula g.(x, y) = (gx, gy)denes an action ofπ⁻¹Gn onXn×π⁻¹Y⁰andp⁻¹_n Y =π⁻¹Gn\(Xn× π⁻¹Y⁰) is the coinvariant object of this action. If Z is an object of E, then p_n,∗Z is the setHom(Xn, Z) endowed with the left action of Gn induced by the inverse action onXn: (g.f)(x) =f(g⁻¹x).

- If Y is inBG then the formula g.(x, y) = (gx, gy)denes an action ofp⁻¹_∞G onXm×π⁻¹(Y^Hn)⁰ for eachm, nin N andp⁻¹_∞Y = lim

−→ⁿlim

←−^mπ⁻¹G\(Xm× π⁻¹(Y^Hn)⁰) (Note that p⁻¹_∞ is exact, since lim

←−^m is essentially constant while

lim−→ⁿ is ltered). IfZ is an object ofEthenp_∞,∗Zis the setlim

−→ⁿHom(Xn, Z) endowed with the left action of Ginduced by the inverse action of Gonf. Endowing the setGn with its left action by translations turns it into an object of BG (resp. BG_m, if m ≥n) whose image by p_∞ (resp. pm) is nothing but Xn.

For the purpose of what follows we will consider a ringS. Endowing it with a trivial action ofG, we may as well view it as a ring ofBG_n orBG.

Definition 2.6. We dene a functor

Γ(G,−) :M od(BG, S)→GM od(S)

by sending anS-moduleM ofBG to(Mn, jn,m, km,n), whereMn=M^Hn,jn,m

is the inclusion and k_m,nthe trace map.

Lemma + definition2.7. LetG,S,E andX be as above. Consider further- more a ring R of E and a ring homomorphismS →π_∗R. Letting M od(E, R) denote the category ofR-modules ofE, we dene the functor of Normic sections alongX

Γ(X,−) :M od(E, R)→GM od(S)

by composing p∞,∗ : M od(E, R) → M od(BG, S) with the functor Γ(G,−) : M od(BG, S)→GM od(S)of Def. 2.6.

For anyF inM od(E, R), the object((M_n),(j_n,m),(k_m,n)) := Γ(X, F)satises the following properties:

- M_n is the restriction of scalars to S of the R(X_n)-module F(X_n) endowed with the action ofG_ncoming from the right action ofG_n onX_n: (g, x)7→g⁻¹x. -j_n,m:M_n →M_m is the restriction alongX_m→X_n. and induces an isomor- phism M_n 'Mm^Gm,n (ie. F(X_n)→F(X_m)^Gm,n).

Proof. By denition we haveM_n=M^Hn=q_n,∗M, whereM =p_∞,∗F. Now, q_np_∞'p_n, thereforeM_n'p_n,∗F=F(X_n). The second property is clear.

Remark 2.8. If the topos E is locally connected, it is possible to build trace maps along nite locally free morphisms, such as Xm → Xn. One may then show thatkm,n coincides with the trace map.

Lemma2.9. Consider another toposE⁰with structural morphismπ⁰:E⁰→Set and a morphism of ringed topoif : (E⁰, R⁰)→(E, R). Let us moreover denote X⁰ = (f⁻¹Xn) the projective system of torsors of E⁰ deduced from X and consider the morphism S→π_∗⁰R⁰ induced byh:S→π_∗R andR→f_∗R⁰. The functorΓ(X⁰,−)associated to these data is subject to a canonical isomorphism

RΓ(X⁰,−)'RΓ(X, Rf_∗(−)) of functors D⁺(E⁰, R⁰)→D⁺(GM od(S)).

Proof. This simply follows from the fact that the morphism of ringed topoi (E⁰, R⁰)→(BG, S)dened byX coincides with the one obtained by composing f with the morphism(E, R)→(BG, S)dened byX.

The derived functor

RΓ(X,−) :D⁺(M od(E, R))→D⁺(GM od(S))

will be our tool to deduce the fundamental distinguished triangles (which are the main ingredient in our proof of the main conjecture) from the comparison isomorphism of [TV]. It might be worth to emphasize that thenth component functor

(−)n:D⁺(_GM od(S))→D⁺(_GnM od(S)) sendsRΓ(X, F)toRΓ(Xn, F_|Xn).

2.3 Descent, codescent and duality

We now assume that the ring S of Def. 2.6 isZp, and we discuss how to pass from normic systems to Iwasawa modules.

In order to be able to invoke directly the results of [Va], we now assume that Λ :=Z^p[[G]] := lim

←−Z^p[Gn]is Noetherian of nite global dimension d+ 1and has an open pro-p-subgroup. As is well known, these condition is in particular veried ifGis a compactp-adic Lie group of dimensiondwithoutp-torsion.

- Limits. Forgetting the km,n's (resp. thejn,m's), inating fromGn-modules to discreteG-modules (resp. abstract Λ-modules) and then forming the limit of the resulting inductive (resp. projective) system gives rise to a functor which will abusively be denoted

lim−→:_GM od(Zp)→GM od(Zp)(resp. lim

←−:_GM od(Zp)→M od(Λ)) The rst is exact and thus passes to derived categories while the second is only left exact, but right derivable. From now on, we always assume thatRlim

←−has nite cohomological dimension. This is e.g. the case if N has a numerable conal subset (as is always the case in practice).

Proposition 2.10. There are canonical adjunctions

(lim−→, RΓ(G,−)) : D⁺(GM od(Zp)) → D⁺(GM od(Zp)) (Λ⊗^L_Λ(−), Rlim

←−) : D⁻(_GM od(Zp)) → D⁻(Λ)

Here, Λ denotes the natural normic system of right Λ-modules (Zp[G_n]) and Λ⊗Λ(−) :M od(Λ)→GM od(Z^p)denotes the induced right exact, left derivable functor.

Proof. The derived version are easily deduced from the obvious ordinary ad- junctions using that GM od(Z^p) and GM od(Z^p) have enough injectives (cf.

Prop. 2.4). In [Va], this was not known and a niteness assumption was thus needed to avoid derivinglim

←−(cf. loc. cit. Prop. 4.2. and Rem. 4.3).

In the above proposition, it is possible to make the adjunction morphisms involved functorial at the level of the complexes. Also, they can still be provided a functorial cone, as in loc. cit.

- Duality. Consider a Zp-module I. If M (resp. M_n) is a Zp-module endowed with a left action of G (resp. G_n) then Hom_Zp(M, I) (resp.

Hom_Zp(M_n, I)) is endowed a left action of G (resp. G_n) as well by re- versing the action on M (ie. (g.f)(m) = f(g⁻¹m)). If the action of G is discrete the action of G on Hom_Zp(M, I) extends to an action of Λ. If now M = (Mn, jn,m, km,n) is a normic system then Hom_Zp(M , I) :=

(Hom_Zp(Mn, I), Hom_Zp(km,n, I), Hom_Zp(jn,m, I))is a normic system as well.

Using an injective resolution ofZp we get duality functors RHom_Zp(−,Zp) : D^b(_GM od(Zp)) → D^b(Λ) RHom_Zp(−,Zp) : D^b(_GM od(Zp)) → D^b(_GM od(Zp)) It might be useful to point out the following relation to Pontryagin duality:

RHom_Zp(M⊗^LQp/Zp,Qp/Zp)'RHom_Zp(M,Zp)

If nowM is aΛ-module we viewHom_Λ(M,Λ)as a leftΛ-module via the right action of Λon itself and the involutiong 7→g⁻¹. Using projective resolutions ofM we get a functor

RHomΛ(−,Λ) : D^b(Λ) → D^b(Λ) Theorem 2.11. Let E,X_n be as in the previous paragraph.

(i) (Descent)There is a canonical isomorphism RΓ(X,−)→^∼RΓ(G,lim

−→RΓ(X,−)) of functors D⁺(E,Zp)→D⁺(GM od(Zp)).

(ii) Consider a bounded complex F of Zp-modules of E (resp. and assume that there existsq0) such that H^q(Xn, F_|Xn)is nitely generated overZp (resp.

trivial) for any n∈N,q≥0 (resp. q≥q0).

(Codescent) Functorially in F, There is a canonical isomorphism in D⁺(GM od(Zp)):

Λ⊗^LΛRlim

←−RΓ(X, F)→^∼RΓ(X, F) (P erf ectness) Rlim

←−RΓ(X, F)andRHom_Zp(lim

−→RΓ(X, F),Z^p) are inD^p(Λ) (the derived category of perfect complexes).

(Duality) Functorially inF, There is a canonical isomorphism inD^p(Λ): RHom_Zp(lim

−→RΓ(X, F),Zp)'RHom_Λ(lim

←−RΓ(X, F),Λ)

Proof. (i) We are going to check that the adjunction morphism of (lim−→, RΓ(G,−)) is an isomorphism. Replacing F by an injective resolution and truncating, one reduces to the case where F is an injective Zp-module of E placed in degree 0. Let M := Γ(X, F). Then ∀n, Mn is injective in

GnM od(Zp)(indeedMncorresponds topn∗F) and the descent map induces an isomorphismMn→Mm^Gm,n. Similarly lim−→nMn is injective and the adjunction morphism of descent occurring inD⁺(GM od(Zp))is represented at level nby the isomorphismΓ(Xn, F_|Xn)→Γ(Hn,lim

−→^mΓ(Xm, F_|Xm)). (ii)Under the stated assumptions,

-(Codescent)and(P erf ectness)follow from (i), by [Va] 4.4(ii)et 4.6(i). -(Duality)follows from (Codescent), by loc. cit. 4.6(iii).

Remark 2.12. Properties (Codescent) and (P erf ectness) are a crucial tool in this paper. It seems plausible that they hold without assuming that G has nite cohomological dimension but we have not tried to check this.

2.4 Examples

Start with a projective system ofC-schemesCn which are étale surjective over C and assume that theCn's are endowed with compatible actions of the Gn's such that each Gn×Cn → Cn ×CCn, (g, x) 7→ (gx, x) is an isomorphism.

(In view of our applications, let us notice that ifC andCn are proper smooth curves over a eld with respective function elds K and Kn then the latter condition holds as soon as the unramied extension Kn/K is Galois). These data represent a projective system of torsors, say X of the small étale topos E=C_et and we are thus in the situation of the previous paragraphs.

Of course these data also produce a projective system of torsors, sayX⁰, of the big, say at, topos E⁰ =C_{F L}. Note that if:C_{F L}→C_et denotes the natural morphism, thenX⁰=⁻¹X.

Consider now the small étale crystalline toposE⁰⁰ ofC/Z^pand the projection morphism u : E⁰⁰ = (C/Z^p)crys,et → Cet = E. Then we dene a projective system of torsors ofE⁰⁰ by pulling back viau: X⁰⁰:=u⁻¹X.

As explained in the previous paragraph, these data give rise to compatible normic section functors, ie. to an essentially commutative diagram

D⁺(CF L, R⁰) ^R∗ //

RΓ(X⁰,−) ((

D⁺(Cet, R)

RΓ(X,−)

D⁺((C/Zp)crys,et, R⁰⁰)

Ru_∗

oo

RΓ(X⁰⁰,−)

uu

D⁺(_GM od(Zp))

whenever one is givenZ^p-algebrasR,R⁰,R⁰⁰inE,E⁰,E⁰⁰and homomorphisms R→_∗R⁰,R→_∗R⁰⁰.

Then-component of the left (resp. middle, resp. right) normic section functor computes the cohomology of the localized toposCF L/X_n⁰, (resp. Cet/Xn, resp.

(C/Zp)crys,et/X_n⁰⁰), ie. of Cn,F L (resp. Cn,et, resp. (Cn/Zp)crys,et). We will thus use the more suggestive notations RΓ(C_{F L},−) (resp. RΓ(C_et,−), resp.

RΓcrys,et(C/Zp,−)).

3 K-theory for Λ(G)

For the convenience of the reader, we review the K-groups and determinant functors which are needed for our purpose. Unless specied otherwise, R de- notes a (unitary) ring.

3.1 Review of K₀ andK₁

The following denitions can be found in [Ba], ch. VII, IX.

- K0(R)is the abelian group dened by generators [P], where P is a nitely generated projective R-modules, and relations

i. [P] = [Q] ifP is isomorphic toQas aR-module.

ii. [P⊕Q] = [P] + [Q]

-K1(R)is the abelian group dened by generators[P, f], whereP is a nitely generated projectiveR-module andf is an automorphism of P with relations (the group law is denoted multiplicatively):

i. [P, f] = [Q, g]ifP is isomorphic toQas aR-module via an isomorphism which is compatible withf andg.

ii. [P⊕Q, f⊕g] = [P, f][Q, g]

iii. [P, f g] = [P, f][P, g]

Note that if the ring R is Noetherian and regular, then forgetting the word projective does not change the denitions (cf [Ba], IX, Proposition 2.1).

- The functor HomR(−, R) realizes an anti-equivalence between nitely gen- erated projective left modules and nitely generated projective right modules.

As a result, one gets an isomorphism Ki(R) ' Ki(R^op), if R^op denotes the opposed ring. Both this isomorphism and its inverse will be denoted(−)^∗(eg.

[P, f]^∗= [P^∗, f^∗]iff^∗ denotes the transpose off).

- Morita equivalence. Let us x a nitely generated projective rightR-module V and letV^∗=Hom_R(V, R). ThenV (resp. V^∗) is naturally endowed with a natural structure of((End_R(V), R)-bimodule (resp. (R, End_R(V))-bimodule).

Since V is projective and nitely generated, one has a canonical isomorphism of (R, R)-bimodules: V^∗ ⊗_EndR(V)V ' R (resp. of (EndR(V), EndR(V))- bimodules: V ⊗RV^∗'EndR(V).

The functor V^∗⊗_EndR(V)(−) : M od(EndR(V))→M od(R)is thus an equiv- alence of categories, with V ⊗R(−)as quasi-inverse. In particular, there is a canonical isomorphismKi(EndR(V))'Ki(R),i= 0,1.

3.2 The determinant functor

LetP(R)denote the category of strictly perfect (ie. bounded with projective nitely generated objects) complexes,K^p(R)its homotopy category andD^p(R) its essential image in the derived category (which is naturally equivalent, and identied, to K^p(R)). If D^p(R)^is denotes the subcategory of D^p(R) where morphisms are the isomorphisms of the latter we have by [Kn], th. 2.3, 2.12 a canonical functor

Det_R:D^p(R)^is'K^p(R)^is→ CR (8) where C_R is the Picard category (ie. a category together with an endobifunc- tor, referred to as the product, endowed with associativity and commutativity isomorphisms satisfying natural compatibilities, unit objects and where every object and every morphism is invertible; cf [Kn] appendix A) considered in [FK] 1.2:

• An object of C_R is a couple (P, Q) of nitely generated projective R- modules.

• M or((P, Q),(P⁰, Q⁰))is empty if[P]−[Q]6= [P⁰]−[Q⁰]in K0(R). Else, there exists anR-moduleM such that

P⊕Q⁰⊕M 'P⁰⊕Q⊕M.

We setIM :=Isom(P⊕Q⁰⊕M, P⁰⊕Q⊕M)andGM :=Aut(P⁰⊕Q⊕M) and we dene the set of morphisms from(P, Q)to(P⁰, Q⁰)asK1(R)×^GM I_M where the right hand side denotes the quotient ofK₁(R)×I_M by the action ofG_M given by

(x, y)7→(x¯g, g⁻¹y))

wherex∈K1(R),y∈IM,g∈GM and¯g is its image inK1(R). As seen easily, this set does not depend onM, up to a canonical isomorphism, and this fact can be used to dene composition in a natural way. Note that by denition, one has a canonical identicationAutC_R((P, Q)) =K1(R) for any object(P, Q).

• The product is dened as

(P, Q).(P⁰, Q⁰) := (P⊕P⁰, Q⊕Q⁰).

and admits naturally(0,0) as a unit. Every object (P, Q)in CR admits (P, Q)⁻¹:= (Q, P)as a natural inverse.

It follows immediately from its construction that the functor(8)is compatible with base change, ie. the diagram

D^p(R⁰) −−−−→^DetR0 C_R⁰

R⁰⊗^L_R(−)

x









x









R⁰⊗_R(−)

D^p(R) −−−−→^DetR CR

is naturally pseudo-commutative for anyR⁰/R. Let us quickly review the con- struction ofDet_Rfollowing [FK] 1.2.

• A strictly perfect complex P is sent to DetR(P) := (P⁺, P⁻) where P⁺:=⊕_i∈2ZPⁱ andP⁻:=⊕_i∈1+2ZPⁱ.

• Any exact sequence0→P⁰→P→P⁰⁰→0ofP(R)induces a canonical isomorphismDet_R(P)'Det_R(P⁰).Det_R(P⁰⁰). Using this, one constructs a canonical trivialization can_P : Det_R(0) ' Det_R(P), for any strictly perfect complexP which is acyclic.

• If Cone(f) denotes the mapping cone of a morphism f : P → P⁰ of strictly perfect complexes then HomC_R(DetR(P), DetR(P⁰)) identies withHomCR(DetR(0), DetR(Cone(f))) (indeed Cone(f)⁺ 'P⁻⊕P⁰⁺

and Cone(f)⁻ ' P⁺ ⊕ P⁰⁻). In particular, when f is a quasi- isomorphism, then can_Cone(f) : Det_R(0) → Det_R(Cone(f)) induces a morphismDet_R(P)→ Det_R(P⁰), which we denote Det_R(f). One may check that it is compatible with composition and only depends on the homotopy class off so that the functor Det_R is nally dened.

We will make essential use of the homorphism

DetR:Aut_Dp(R)(P)→AutC_R(DetR(P)) =K1(R) (9) which is induced by(8)for any perfect complexP. Of course ifPis reduced to a (nitely generated projective) moduleP⁰placed in degree zero thenDetR(P) = (P⁰,0) andDetR(f) : (P⁰,0)→(P⁰,0) is the class off⁰ :P⁰ →P⁰. In that case(9)is thus nothing but the tautological mapAutR(P⁰)→K1(R). Let us now state some multiplicative properties.

• Consider a morphism between exact sequences of strictly perfect com- plexes

0 −−−−→ P₁⁰ −−−−→ P₁ −−−−→ P₁⁰⁰ −−−−→ 0

f⁰







 y

f







 y

f⁰⁰







 y

0 −−−−→ P₂⁰ −−−−→ P₂ −−−−→ P₂⁰⁰ −−−−→ 0

where vertical arrows are quasi-isomorphisms. The following square com- mutes:

DetR(P1) ' DetR(P₁⁰).DetR(P₁⁰⁰)

DetR(f)







 y







 y

DetR(f⁰).DetR(f⁰⁰)

DetR(P2) ' DetR(P₂⁰).DetR(P₂⁰⁰)

• Consider an automorphismf of someP inK^p(R). Then:

-Det_R(f[1]) =Det_R(f)⁻¹in Aut_CR(Det_R(P)) =K₁(R).

-f is the homotopy class of some automorphism of the complexP. Let us x such one and denotef^q :P^q →P^q its component of degreeq. Then in K1(R) =Aut_CR(DetR(P)) = Aut_CR(DetR(P^q)), we have DetR(f) = QDetR(f^q)^(−1)q.

- If P is cohomologically perfect (ie. each H^q is an object of D^p(R)) then Det_R(f) = QDet_R(H^q(f))^(−1)q in K₁(R) = Aut_CR(Det_R(P)) = Aut_CR(Det_R(H^q(P))).

3.3 Localization

The mainK-group of interest for Iwasawa theory is a relative one. We recall its denition ([FK] 1.3). Consider a strictly full triangulated subcategory Σ of D^p(R). The group K₁(R,Σ)is then dened by generators and relations as follows:

• Generators: [C, a]whereCis an object ofΣanda:DetR(0)→DetR(C) is a trivialization ofC.

• Relations: LetC, C⁰, C”be objects ofΣ.

˘ IfC'0 then[C, canC] = 1.

˘ If f :C 'C⁰, then compatible trivializations ofC and C⁰ give rise to the same element in K1(R,Σ) (ie. [C⁰, DetR(f)◦a] = [C, a] if a:DetR(0)'DetR(C)).

˘ If 0 → C⁰ → C → C⁰⁰ → 0 is an exact sequence of P(R), and a: 1'DetR(C),a⁰: 1'DetR(C⁰), then

[C, a] = [C⁰, a⁰].[C⁰⁰, a⁰⁰] where a: 1^a

0.a⁰⁰

→ DetR(C⁰).DetR(C⁰⁰)'DetR(C). There is a localization exact sequence (cf [FK] 1.3.15)

K1(R) −−−−→ K1(R,Σ) −−−−→^∂ K0(Σ)−−−−→ K0(R)

(here K0(Σ) denotes the Grothendieck group of the triangulated categoryΣ) where:

- the rst map sends[P, f]∈K1(R)to the complex[P→^f P]placed in degrees [−1,0], together with the trivialization which is represented by the identity of P.

-∂ sends[C, f]∈K1(R,Σ)to the element−[C]∈K0(Σ).

- the last map sends the class of a strictly perfect complex[C]to the alternated sumP(−1)ⁱ[Cⁱ].

As checked easily, this localization sequence is functorial with respect to(R,Σ): if R⁰ is an R-algebra, and if Σ⁰ is a strictly full triangulated subcategory of D^p(R⁰) containing the essential image of Σ under the functor R⁰ ⊗^L_R(−) : D^p(R) →D^p(R⁰) then one has a morphism K1(R,Σ)→ K1(R⁰,Σ⁰) which is compatible with the obvious functoriality maps.

If now T ⊂Ris a left denominator set (cf [FK], 1.3.6) then we can apply the above constructions to the full triangulated subcategoryΣ_T ofD^p(R)consisting of complexes whose image under the functor R_T ⊗^L_R(−) : D^p(R)→D^p(R_T) become acyclic. We note that if R is Noetherian and regular then K₀(Σ_T)is isomorphic to the Grothendieck group of nitely generatedT-torsion modules, by sending[C]toP(−1)ⁱHⁱ(C). For a generalRwe have the following result.

Proposition 3.1. ([FK], 1.3.7) There is a canonical isomorphism of groups K₁(R,Σ_T)'K₁(R_T)

sending [C, a] to the isomorphism Det_RT(0)^a→^TDet_RTC_T

can⁻¹

→CTDet_RT(0) viewed as an element of K₁(R_T) = Aut_CRT(Det_RT(0)) (here a_T and C_T are deduced from a and C by localization). This isomorphism is functorial with respect to (R, T).

Let us mention an alternative characterization of this isomorphism. Con- sider an endomorphism f of a strictly perfect complex P such that fT is a quasi-isomorphism. Identifying Hom_CR(DetR(P), DetR(P)) with Hom(DetR(0), DetR(Cone(f))) sends the identity ofDetR(P)to a morphism triv:DetR(0)→DetR(Cone(f)). Then the class[Cone(f), triv]∈K1(R,ΣT) corresponds toDetR_T(fT)⁻¹∈K1(RT)(both are indeed described by the same endomorphism ofP_T⁺⊕P_T⁻ coming fromcan_Cone(f)T).

Crucial to us will be the following:

Lemma 3.2. Consider morphisms a, b:C →C⁰ in D^p(R)whose localizations aT, bT are isomorphisms in D^p(RT). InK0(ΣT), one has the equality

∂DetR_T(aTb⁻¹_T ) = [Cone(a)]−[Cone(b)]

as long as the following condition holds:

(f rac) InD^p(R), there exists a commutative square of the form C −−−−→^a C⁰

b







 y

d







 y C⁰ −−−−→^c C⁰

in which canddalso become isomorphisms after localization by T.

Proof. An easy diagram chasing shows that[Cone(c)]−[Cone(a)] = [Cone(d)]−

[Cone(b)]. Since DetRT(aTb⁻¹_T ) = DetRT(dT)⁻¹DetRT(cT) and ∂ is a ho- momorphism, it thus suces to prove that ∂Det_RT(c_T) = [Cone(c)] and

∂Det_RT(d_T) = [Cone(d)]. But this is clear from the alternative characteri- zation of the isomorphismK₁(R,Σ_T)'K₁(R_T)mentioned above.

We do not know whether or not the condition(f rac)always holds, neither if this computation of ∂DetR_T(aTb⁻¹_T )is always correct. We thus content ourselves with the following sucient condition.

Lemma 3.3. AssumeR is Noetherian and considera, b:C→C⁰ inD^p(R). If there existsf in the center of Rsuch that R[_f¹]⊗^L_RCone(b) = 0, then one can nd c, d:C⁰→C⁰ such that da=cb. In fact one can chosed=fⁿ fornlarge enough.

Proof. Since cohomology modules ofCone(b)are nitely generated and almost all of them are zero, it is possible to ndnsuch that fⁿ :Cone(b)→Cone(b) is zero. But then, the long exact sequence ofExt's

· · · →Hom(C⁰, C⁰)^Hom(b,C

0)

→ Hom(C, C⁰)→Ext¹(Cone(b), C⁰)→ · · · shows thatfⁿahas a trivial image inExt¹(Cone(b), C⁰)and thus comes from some c∈Hom(C⁰, C⁰)as claimed.

3.4 The evaluation map at Artin representations

Consider a pronite groupG= lim

←−ⁿG_n and a closed normal subgroupH such that Γ :=G/H is isomorphic toZp. We use the following notations.

- We let Λ = Λ(G) := lim

←−ⁿZp[Gn] denotes the Iwasawa algebra of G. We assume that this ring is left Noetherian and regular (which will be the case when considering the Galois group of ap-adic Lie extension as in the introduction).

- If O is the ring of integers of a nite extension L of Qp, then ΛO(G) :=

lim←−nO[G_n]'O⊗_ZpΛ(G)andΛ_L(G) :=L⊗OΛ_O(G)have similar properties.

- S and S^∗ denote the canonical Ore sets dened in [CFKSV]. Recall that an element f of Λ(G) is in S if and only if Λ(G)/f is a nitely generated

Λ(H)-module whereasS^∗=∪kp^kS. As usual,ΛS = Λ(G)S andΛS^∗= Λ(G)S^∗

denote the corresponding localizations of Λ. If G= Γthen QO(Γ) := Λ(Γ)S^∗

is the fraction eld of ΛO(Γ).

-MH(G)denotes the category of S^∗-torsion nitely generated Λ(G)-modules.

We recall that a nitely generated Λ(G) module M is S-torsion (resp. S^∗- torsion) if and only if it is a nitely generatedΛ(H)-module (resp. modulo its p-torsion). In this context, the localization exact sequence reads:

K₁(Λ) −−−−→ K₁(Λ_S^∗) −−−−→^∂ K₀(M_H(G))−−−−→ K₀(Λ) - If R is one of the previous (localized) Iwasawa algebras, one often prefers to endow the dual of a left module with a left action, deduced from the right one via the involution g7→ g⁻¹. This is our convention for duality of normic systems and their limit modules. In this paragraph though, we leave right modules on the right, for the sake of clarity.

Consider a free O-module V of nite rank. For any O-algebra R (eg. R = ΛO(G)), we denote VR (resp. RV) the right (resp. left) R-module V ⊗OR (resp. R⊗O V) deduced from V by extending scalars from O to R. Also, (RV)^∗ := HomR(RV , R) (resp. (VR)^∗ := HomR(VR, R)) is systematically given its right (resp. left)R-module structure coming from the(R, R)-bimodule structure ofR.

It might be useful to recall that one has a canonical isomorphism of left (resp.

right) R-modules (V_R)^∗ ' R(V^∗) (resp. (_RV)^∗ ' (V^∗)_R) and canonical iso- morphisms of O-algebras End_R((V_R)^∗) ' End_R(V_R)^op and End((_RV)^∗) ' End(_RV)^op (R-algebras). If nowRis a central O-algebra (eg. ifR= Λ_O(G)) we have moreover canonical isomorphisms ofO-algebrasEnd_R(_RV)'R^op⊗_O End_O(V), End_R(V_R)'End_O(V)⊗_OR. These isomorphisms are subject to natural compatibilities, such as the commutativity of the following diagram:

EndR((RV)^∗) ' EndR(RV)^op ' (R^op⊗OEndO(V))^op

|o |o

EndR((V^∗)R) ' EndO(V^∗)⊗OR ' EndO(V)^op⊗OR

Consider now anO-Artin representationρ:G→AutO(V). ByO-Artin repre- sentation we mean that V is as above and ρhas a nite image. Consider the unique homomorphism ofZp-algebras

Φ : Λ(G)→EndΛ_O(G)(VΛ_O(G)) sendingg∈G⊂ΛO(G)tov⊗λ7→ρ(g)(v)⊗gλ. By functoriality ofK₁, one has a commutative diagram:

K1(Λ(G))

twρ

OO

K1(ρ) ++

K₁(Φ) //K1(End_ΛO(G)(V_ΛO(G)))

K₁(_V)

∼

M orita//K1(ΛO(G))

K₁()

K₁(End_O(V)) ^{M orita}_∼ //K₁(O)