On the non commutative Iwasawa main conjecture for abelian varieties over function fields

TL;DR

This paper proves the Iwasawa main conjecture for semi-stable abelian varieties over certain function fields in characteristic p, using p-adic cohomology and trace formulas under specific assumptions.

Contribution

It establishes the conjecture for a new class of abelian varieties over function fields, extending previous results with novel cohomological methods.

Findings

Proves the Iwasawa main conjecture under specified conditions.

Utilizes p-adic cohomology and trace formulas for the proof.

Assumes the classical μ=0 hypothesis.

Abstract

We establish the Iwasawa main conjecture for semi-stable abelian varieties over a function field of characteristic $p$ under certain restrictive assumptions. Namely we consider $p$-torsion free $p$-adic Lie extensions of the base field which contain the constant $\mathbb Z_p$-extension and are everywhere unramified. Under the classical $\mu=0$ hypothesis we give a proof which mainly relies on the interpretation of the Selmer complex in terms of $p$-adic cohomology [TV] together with the trace formulas of [EL1].

Equations

$$L(A/K,1)=\frac{\#\text{tencyrSh}(A/K)}{\#A(K)\#\hat{A}(K)}.\frac{\#H^0(C,Lie(\mathcal{A}))}{\#H^1(C,Lie(\mathcal{A}))}$$

$$\partial (L(A/K,1))=[R\Gamma (C,T_p(\mathcal{A}))]+[\mathcal{R}\Gamma (\mathcal{C},\mathcal{L}ie(\mathcal{A}))]$$

$$\begin{array}[]{rcl}N_{K_{\infty}}&:=&R\varprojlim_{n}R\Gamma^{Z_{n}}(C_{n},T_{p}(\cal A))\\ L_{K_{\infty}}&:=&R\varprojlim_{n}R\Gamma^{Z_{n}}(C_{n},Lie(\cal A))\end{array}$$

$$\partial {\mathcal{L}}_{A/K_{\infty }}=[N_{K_{\infty }}]+[L_{K_{\infty }}]$$

$$\rho ({\mathcal{L}}_{\mathcal{A}/{\mathcal{K}}_{\infty }})=\mathcal{L}(\mathcal{U},\mathcal{A}/\mathcal{K},{\rho }^{\vee },1)$$

$$\Gamma (\underline{X},-):Mod(E,R)\to {}_{\underline{G}}(\Gamma (E,R)-mod)$$

$$D^{+}(E,{\mathbb{Z}}_p)\to D(\Lambda (G))$$

$$\displaystyle{\small\begin{array}[]{c}R\Gamma(\underline{C}^{\sharp}/{\mathbb{Z}_{p}},Fil^{1}D_{log}(\cal A)(-Z))\mathop{\to}\limits^{\bf 1}R\Gamma(\underline{C}^{\sharp}/{\mathbb{Z}_{p}},D_{log}(\cal A)(-Z))\to R\Gamma(\underline{C},Lie(\cal A)(-Z))\mathop{\to}\limits^{+1}\\ R\Gamma^{\underline{Z}}(\underline{C},T_{p}(\cal A))\to R\Gamma(\underline{C}^{\sharp}/{\mathbb{Z}_{p}},Fil^{1}D_{log}(\cal A)(-Z))\mathop{\to}\limits^{\bf 1-\phi}R\Gamma(\underline{C}^{\sharp}/{\mathbb{Z}_{p}},D_{log}(\cal A)(-Z))\mathop{\to}\limits^{+1}\end{array}}$$

$$P_{K_{\infty }}:=R\underset{⟵}{\lim }R\Gamma (C_n^{♯}/{\mathbb{Z}}_p,D_{log}(\mathcal{A})(-\mathcal{Z}))$$

$$L{\stackrel{L}{\otimes }}_{{\Lambda }_O(G)}(V_{\rho ,{\Lambda }_O(G)}{)}^{\ast }{\stackrel{L}{\otimes }}_{\Lambda (G)}{P}_{K_{\infty }}\simeq R{\Gamma }_{rig,c}(U/L,p{r}^{\ast }{D}^{†}\otimes U({\rho }^{\vee }))$$

$$M_m\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$Ho{m}_{{}_{G}Mod(S)}(M,M^{\prime })\simeq {\underset{⟵}{\lim }}_n{\underset{⟶}{\lim }}_{m}Ho{m}_{{}_{G}Mod(S)}(M^{H_n},M^{\prime H_m}).$$

$$\begin{array}[]{ccccccc}Cocart(\mathcal{F}/N^{op})&\mathop{\rightarrow}\limits^{F_{1}}&Sect(\mathcal{F}/N^{op})&\mathop{\rightarrow}\limits^{F_{2}}&\prod_{|N|}Mod(\mu_{\lhd}(G_{n},S))&\mathop{\rightarrow}\limits^{F_{3}}&Mod(S)^{|N|}\end{array}$$

$$E\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Gamma (\underline{G},-):Mod(B_G,S)\to {}_{\underline{G}}Mod(S)$$

$$\Gamma (\underline{X},-):Mod(E,R)\to {}_{\underline{G}}Mod(S)$$

$$R\Gamma ({\underline{X}}^{\prime },-)\simeq R\Gamma (\underline{X},R{f}_{\ast }(-))$$

$$R\Gamma (\underline{X},-):D^{+}(Mod(E,R))\to D^{+}({}_{\underline{G}}Mod(S))$$

$$(-{)}_n:D^{+}({}_{\underline{G}}Mod(S))\to D^{+}({}_{G_n}Mod(S))$$

$$\begin{gathered} \underset{⟶}{\lim }:{}_{\underline{G}}Mod({\mathbb{Z}}_p)\to {}_{G}Mod({\mathbb{Z}}_p) \\ (resp.\underset{⟵}{\lim }:{}_{\underline{G}}Mod({\mathbb{Z}}_p)\to Mod(\Lambda )) \end{gathered}$$

$$\begin{array}[]{ccccc}(\displaystyle{\lim_{\longrightarrow}},R\Gamma(\underline{G},-))&:&D^{+}({{}_{G}Mod({\mathbb{Z}_{p}})})&\rightarrow&D^{+}({{}_{\underline{G}}Mod({\mathbb{Z}_{p}})})\\ (\underline{{\operatorname{\Lambda}}}{\mathop{\otimes}\limits^{L}}_{{\operatorname{\Lambda}}}(-),R\displaystyle{\lim_{\longleftarrow}})&:&D^{-}({{}_{\underline{G}}Mod({\mathbb{Z}_{p}})})&\rightarrow&D^{-}({{\operatorname{\Lambda}}})\end{array}$$

$$\begin{array}[]{ccccc}RHom_{\mathbb{Z}_{p}}(-,{\mathbb{Z}_{p}})&:&D^{b}({{}_{G}Mod({\mathbb{Z}_{p}})})&\rightarrow&D^{b}({\operatorname{\Lambda}})\\ RHom_{\mathbb{Z}_{p}}(-,{\mathbb{Z}_{p}})&:&D^{b}({{}_{\underline{G}}Mod({\mathbb{Z}_{p}})})&\rightarrow&D^{b}({{}_{\underline{G}}Mod({\mathbb{Z}_{p}})})\end{array}$$

$$RHo{m}_{{\mathbb{Z}}_p}(M\stackrel{L}{\otimes }{\mathbb{Q}}_p/{\mathbb{Z}}_p,{\mathbb{Q}}_p/{\mathbb{Z}}_p)\simeq RHo{m}_{{\mathbb{Z}}_p}(M,{\mathbb{Z}}_p)$$

$$\begin{array}[]{ccccc}RHom_{{\operatorname{\Lambda}}}(-,{\operatorname{\Lambda}})&:&D^{b}({\operatorname{\Lambda}})&\rightarrow&D^{b}({\operatorname{\Lambda}})\end{array}$$

$$R\Gamma (\underline{X},-)\stackrel{\sim }{\to }R\Gamma (\underline{G},\underset{⟶}{\lim }R\Gamma (\underline{X},-))$$

$$\underline{\Lambda }{\stackrel{L}{\otimes }}_{\Lambda }R\underset{⟵}{\lim }R\Gamma (\underline{X},F)\stackrel{\sim }{\to }R\Gamma (\underline{X},F)$$

$$RHo{m}_{{\mathbb{Z}}_p}(\underset{⟶}{\lim }R\Gamma (\underline{X},F),{\mathbb{Z}}_p)\simeq RHo{m}_{\Lambda }(\underset{⟵}{\lim }R\Gamma (\underline{X},F),\Lambda )$$

$$D^{+}(C_{FL},R^{\prime })\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$De{t}_R:D^p(R{)}^{is}\simeq K^p(R{)}^{is}\to {\mathcal{C}}_{\mathcal{R}}$$

$$P\oplus Q^{\prime }\oplus M\simeq P^{\prime }\oplus Q\oplus M.$$

Full text

1. 1 Introduction

2. 1.1 Statement of the main theorem

3. 1.2 Outline of the paper

4. 2 The normic section functor

5. 2.1 Normic systems

6. 2.2 From sheaves to normic systems

7. 2.3 Descent, codescent and duality

8. 2.4 Examples

9. 3 K-theory for $\Lambda(G)$

10. 3.1 Review of $K_{0}$ and $K_{1}$

11. 3.2 The determinant functor

12. 3.3 Localization

13. 3.4 The evaluation map at Artin representations

14. 4 Selmer complexes and crystalline cohomology

15. 4.1 Preliminaries

1. Introduction

1.1. Statement of the main theorem

Consider a function field $K$ of characteristic $p$, $C$ the corresponding proper smooth geometrically irreducible curve over a finite field $k$ and an abelian variety $A/K$ with Néron model $\cal A/C$. Assume for simplicity that $A$ has good reduction everywhere and that the Hasse-Weil $L$ function of $A/K$ does not vanish at $s=1$. In this situation the BSD conjecture predicts that each group involved in the right hand term below is finite and that the following formula holds (where $\hat{A}$ denotes the dual abelian variety):

[TABLE]

Let us restrict our attention to the $p$-adic valuations on both sides. There are natural perfect complexes of ${\mathbb{Z}_{p}}$-modules $R\Gamma(C,T_{p}(\cal A))$, $R\Gamma(C,Lie(\cal A))$ whose cohomology groups are the $p$-part of the groups appearing above. These complexes thus become acyclic after extension of scalars to ${\mathbb{Q}_{p}}$ hence produce classes in $K_{0}({\mathbb{Z}_{p}},{\mathbb{Q}_{p}})\simeq{\mathbb{Z}}$. Looking at $L(A/K,1)$ as an element of $K_{1}({\mathbb{Q}_{p}})\simeq{\mathbb{Q}_{p}}^{\times}$ the $p$-part of (1) simply becomes

[TABLE]

where $\partial:K_{1}({\mathbb{Q}_{p}})\to K_{0}({\mathbb{Z}_{p}},{\mathbb{Q}_{p}})$ is the connecting map of lower $K$-theory (ie. the $p$-adic valuation $v_{p}:{\mathbb{Q}_{p}}^{\times}\to{\mathbb{Z}}$). As explained in [KT] the complexes appearing on the right hand side can be related to $p$-adic (crystalline and then rigid) cohomology and the trace formulae which are known in this context are then sufficient 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 $L(A/K,1)\neq 0$ anymore and $A$ is now allowed to have semistable reduction at some given set $Z$ of points of $C$. Consider a Galois extension $K_{\infty}/K$ which contains the constant ${\mathbb{Z}_{p}}$-extension $Kk_{\infty}/K$, is unramified everywhere, and whose Galois group $G=\varprojlim_{n}G_{n}$ is $p$-adic Lie without $p$-torsion. In that situation, a general result explained in section 2 will allow us to form perfect complexes of modules over the Iwasawa algebra $\Lambda(G):=\varprojlim_{n}{\mathbb{Z}_{p}}[G_{n}]$:

[TABLE]

where $R\Gamma^{Z_{n}}(C_{n},-)$ is the functor of cohomology vanishing at $Z_{n}$ (see. section 4 for a precise definition and the relation to usual Selmer complexes) and is designed to take out the contribution of $Z$. A significant difference with the BSD statement is that here, the cohomology of these complexes are expected to be torsion $\Lambda(G)$-modules even if $L(A/K,1)=0$. This is well known if $K=Kk_{\infty}$ (see. Cor. LABEL:torsionar 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 $\mu=0$ hypothesis. More precisely, we will prove the following generalization of [LLTT], where only the case $K=Kk_{\infty}$ was considered.

Theorem 1.1**.**

(Thm. LABEL:main-theorem) Let $A/K$, $K_{\infty}/K$ and $G$ as above. If the $\mu$-invariant of the Pontrjagin dual of the discrete Selmer group of $A$ over $Kk_{\infty}$ is trivial, then

1. (i)

The cohomology $\Lambda(G)$-modules of $N_{K_{\infty}}$ and $L_{K_{\infty}}$ are $S^{*}$-torsion, where $S^{*}$ is the canonical Ore set of [CFKSV]. The latter complexes thus produce classes $[N_{K_{\infty}}]$ and $[L_{K_{\infty}}]$ in $K_{0}(\Lambda(G),\Lambda(G)_{S^{*}})$ (denoted $K_{0}(\mathfrak{M}_{H}(G))$ in loc. cit.). . 2. (ii)

There exists a canonical element ${\cal L}_{A/K_{\infty}}\in K_{1}(\Lambda(G)_{S^{*}})$ satisfying

[TABLE]

where $\partial:K_{1}(\Lambda(G)_{S^{*}})\to K_{0}(\Lambda(G),\Lambda(G)_{S^{*}})$ denotes the connecting map in lower $K$-theory. 3. (iii)

The element ${\cal L}_{A/K_{\infty}}$ verifies the interpolation property

[TABLE]

for each Artin $\cal O$-valued representations $\rho$ of $G$ ($\cal O/{\mathbb{Z}}_{p}$, a totally ramified extension), where $\rho(-):K_{1}(\Lambda(G)_{S^{*}})\to L^{\times}\cup\{0,\infty\}$ denotes the corresponding evaluation map, $(-)^{\vee}$ denotes the contragredient, and $L(U,A/K,\rho,s)$ is the $\rho$-twisted Hasse-Weil $L$-function of $A/K$ without Euler factors at $Z=C-U$.

We also give a similar statement involving complexes $N_{K_{\infty}}^{*}$ and $L_{K_{\infty}}^{*}$ which are ${\operatorname{\Lambda}}(G)$-dual to $N_{K_{\infty}}$ and $L_{K_{\infty}}$ (see Prop. 2.11 $(ii)$, as well as Prop. LABEL:La-duality, (ii) for the precise statement).

Here, the construction of $\cal L_{A/K_{\infty}}$, the proof that $N_{K_{\infty}}$ is torsion and the proof of (3) are simultaneous and rely essentially on the main result of [TV] (a sheafified version of [KT] Prop. 5.13) which yields distinguished triangles of perfect complexes of $\Lambda(G)$-modules relating flat and crystalline cohomology (see Sect. LABEL:paradtinfty and Rem. LABEL:remTV). 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 $K_{\infty}/K$ (Prop. LABEL:La-duality, relying on Thm. 2.11).

1.2. Outline of the paper

Section 2. Defining the complexes $N_{K_{\infty}}$, $L_{K_{\infty}}$ or $N_{K_{\infty}}^{*}$, $L_{K_{\infty}}^{*}$ occurring in the main conjecture (3) involves forming projective limits along the Galois tower formed by the $C_{n}$’s or alternatively inductive limits and taking duals. In order to perform these operations, a convenient framework is given by the derived category of normic systems $D^{b}({{}_{\underline{G}}}({\mathbb{Z}_{p}}\hbox{-}mod))$ defined and studied in [Va]. A normic system is a collection $M_{n}$ of $G_{n}$-modules together with equivariant morphisms $M_{n}\to M_{m}$, $M_{m}\to M_{n}$ satisfying a natural compatibility (Def. 2.1). The purpose of this section is to show that the collection of derived functors $R\Gamma(C_{n},-)$ comes from a functor $R\Gamma(\underline{C},-)$ with values in $D^{b}({{}_{\underline{G}}}({\mathbb{Z}_{p}}\hbox{-}mod))$.

We place ourself in a setting which is general enough to handle the various cohomology theories (flat, étale, crystalline…) involved here. Namely we show (Lem.-Def. 2.7) that in a ringed topos $(E,R)$ any pro-torsor $\underline{X}=(X_{n},G_{n})_{n}$ gives rise to a functor

[TABLE]

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

[TABLE]

obtained by composing $R\Gamma(\underline{X},-)$ with $R\varprojlim_{n}$, or alternatively with $\varinjlim_{n}$ and then $RHom(-,{\mathbb{Z}_{p}})$.

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 localization following [FK]. We also discuss the evaluation map which appears in (4).

Section 4. We begin by recalling the definition of the derived functor $R\Gamma^{Z}(C,.)$ of global sections vanishing at a closed subscheme $Z$ which naturally appears in the comparison theorem of [TV] and is denoted $R\Gamma_{ar,V}$ in [KT] (see Rem. LABEL:SelGar for a more precise statement). Next we compare the complex of normic systems $R\Gamma^{\underline{Z}}(\underline{C},T_{p}(\cal A))$ underlying $N_{K_{\infty}}$ to the usual Selmer complex of $A/K$. We take the opportunity to give a tractable definition 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 of $C$) and write down the fundamental distinguished triangles that follow from it, using (5):

[TABLE]

Here $\phi$ is a semi-linear map such that $\phi{\bf 1}=pFrob$. By applying $R\varprojlim_{n}$, these in turn yield distinguished triangles of perfect complexes of $\Lambda(G)$-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 first distinguished triangle above is a $k$-vector space and the arrow denoted $\bf 1$ thus becomes invertible after inverting $S^{*}$. Whence an endomorphism $(\bf 1-\phi)_{S^{*}}\bf 1_{S^{*}}^{-1}$ acting on the localization $P_{K_{\infty},S^{*}}$ of

[TABLE]

In the case where $G=\Gamma$, the base change formula in crystalline cohomology together with a semi-linear argument shows that the first term in the second distinguished triangle above vanishes as well after $S^{*}$-localization. In the general case, an argument using Nakayama’s lemma ensures that it is still the case under the $\mu=0$ assumption. The endomorphism $({\bf 1}-\phi)_{S^{*}}{\bf 1}_{S^{*}}^{-1}$ is thus invertible, allowing us to define $\cal L_{A/K_{\infty}}\in K_{1}(\Lambda_{S^{*}})$ 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

[TABLE]

where $U(\rho^{\vee})$ denotes the unipotent convergent isocrystal associated to $\rho$ (Prop. LABEL:descrig). On the one hand the trace formula in rigid cohomology shows that $L(U,A/K,\rho^{\vee},1)$ coincides with the alternated product of the determinants of $1-p^{-1}Frob$ acting on the $L$-valued cohomology of the right hand side. Since we have no morphism from $\Lambda_{S^{*}}$ to $L$ we may not use directly the base change property of the $Det$ functor together with a localized version of (9) in order to relate the action of $1-p^{-1}Frob$ on the right hand side with $\rho(\cal L)$. We turn this difficulty by investigating carefully the spectral sequence of codescent along $Kk_{\infty}/K$ with the definition of the evaluation map in mind.

Acknowledgements. The first 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 group $G$ naturally gives rise to perfect complexes of $\Lambda(G)$-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 define and study briefly the category of normic systems along a profinite group $G$. The following definitions slightly generalize those given in [Va].

Definition 2.1**.**

Consider a profinite group $G$, $N$ a filtered index set, $(H_{n})_{n\in N}$ the filtered set of the normal open subgroups and denote $G_{n}=G/H_{n}$, $G_{m,n}=H_{n}/H_{m}$, $n\leq m$. Given a commutative ring $S$.

$(i)$* ${{}_{G}Mod(S)}$ the category of $S$-modules endowed with a discrete action of $G$, ie. which are the union of their fixed points by the $G_{n}$’s.*

$(ii)$* ${{}_{G_{n}}Mod(S)}$ the full subcategory of ${{}_{G}Mod(S)}$ whose objects are those on which $H_{n}$ acts trivially. For $m\geq n$, the inclusion functor ${{}_{G_{n}}Mod(S)}\rightarrow{{}_{G_{m}}Mod(S)}$ has a left and a right adjoint, described respectively as the coinvariants functor $(-)_{G_{m,n}}$ and the fixed points functor $(-)^{G_{m,n}}$.*

$(iv)$* ${{}_{\underline{G}}Mod(S)}$ the category of $Mod(S)$-valued normic systems is defined a follows:*

- an object is a triple $\underline{M}=((M_{n})_{n\in N},(j_{n,m}:M_{n}\rightarrow M_{m})_{n\leq m},(k_{m,n}:M_{m}\rightarrow M_{n})_{n\leq m})$ satisfying the following properties:

$(Norm1)$* $M_{n}$ is an object of ${{}_{G_{n}}Mod(S)}$, $j_{n,m}$ and $k_{m,n}$ are morphisms of ${{}_{G}Mod(S)}$.*

$(Norm2)$* If $n_{1}\leq n_{2}\leq n_{3}$ then $j_{n_{2},n_{3}}\circ j_{n_{1},n_{2}}=j_{n_{1},n_{3}}$ and $k_{n_{2},n_{1}}\circ k_{n_{3},n_{2}}=k_{n_{3},n_{1}}$.*

$(Norm3)$* If $n\leq m$, $j_{n,m}\circ k_{m,n}:M_{m}\rightarrow M_{m}$ coincides with the endomorphism $\sum_{g\in G_{m,n}}c_{g}$ of $M_{m}$. Note that this is not only a morphism in $Mod(S)$, but also in ${{}_{G}Mod(S)}$ since $H_{n}\lhd G$.*

- a morphism $\underline{\phi}:\underline{M}\rightarrow\underline{M}^{\prime}$ is a collection of morphisms of ${{}_{G}Mod(S)}$, $(\phi_{n}:M_{n}\rightarrow M^{\prime}_{n})_{n\in N}$ such that the following squares commute for each couple $n\leq m$:

[TABLE]

Let us make some remarks about the category ${{}_{G}Mod(S)}$ of discrete $G$-objects.

• For $M,M^{\prime}\in_{G}Mod(S)$:

[TABLE]

• The category ${}_{G}Mod(S)$ is a full subcategory of $Mod(S[G])$. The inclusion functor has a right adjoint $M\mapsto\cup_{n}M^{H_{n}}$. In particular, ${}_{G}Mod(S)$ has enough injectives since $Mod(S[G])$ has.

We now turn to some properties of the category ${}_{\underline{G}}Mod(S)$ of normic systems.

Proposition 2.2**.**

Small inductive and projective limits exist in ${}_{\underline{G}}Mod(S)$ and commute to the component functors ${}_{\underline{G}}Mod(S)\to_{\underline{G}_{n}}Mod(S)$. In particular, ${}_{\underline{G}}Mod(S)$ is an abelian category.

Proof. In order to form the inductive (resp. projective) limit indexed by a set $I$ in ${{}_{\underline{G}}\mathcal{C}}$, it suffices to form the limit of the components and endow them with the $j_{n,m}$ and $k_{m,n}$ provided by functoriality. This prove the first statement. The second one follows since being abelian is a property of limits ([KS], Def. 8.2.8 and Def. 8.3.5).

$\square$

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

Lemma 2.3**.**

$(i)$* If $G$ is in fact a finite group, then the category ${{}_{\underline{G}}Mod(S)}$ is equivalent to $Mod(\mu_{\lhd}(G,S))$ where $\mu_{\lhd}(G,S)$ is the normal Mackey algebra, defined as the quotient of the free associative algebra $S\{\{c_{g},g\in G\},\{j_{n,m},k_{m,n},n\leq m\}\}$ by the following relations:*

- $c_{g_{1}}c_{g_{2}}=c_{g_{1}g_{2}}$, $c_{g}j_{n,m}=j_{n,m}c_{g}$, $c_{g}k_{m,n}=k_{m,n}c_{g}$, for $g,g_{1},g_{2}\in G$ and $n\leq m$ in $N$.

- $j_{n_{2},n_{3}}j_{n_{1},n_{2}}=j_{n_{1},n_{3}}$, $k_{n_{2},n_{1}}k_{n_{3},n_{2}}=k_{n_{3},n_{1}}$, for $n_{1}\leq n_{2}\leq n_{3}$ in $N$.

- $c_{g}j_{n,n}=j_{n,n}$, $j_{n,n}=k_{n,n}$, for $n\in N$ and $g\in H_{n}$.

- $j_{n,m}k_{m,n}=\sum_{g\in G_{m,n}}c_{g}j_{m,m}$, for $n\leq m$ in $N$.

- $\sum_{n\in N}j_{n,n}=1$.

$(ii)$* In general, ${{}_{\underline{G}}Mod(S)}$ is equivalent to the category of cocartesian sections of the cofibered category above $N^{op}$ associated to the covariant pseudo-functor on $N^{op}$ mapping $n$ to $Mod(\mu_{\lhd}(G_{n},S))$, and $m\geq n$ to the functor $Mod(\mu_{\lhd}(G_{m},S))\rightarrow Mod(\mu_{\lhd}(G_{n},S))$ given by extension of scalars through the morphism $\mu_{\lhd}(G_{m},S)\rightarrow\mu_{\lhd}(G_{n},S)$ defined by killing the $j_{a,b}$’s and $k_{b,a}$’s with $b\nleq n$.*

Proof. $(i)$ Let us describe the functor ${{}_{\underline{G}}Mod(S)}\rightarrow Mod(\mu_{\lhd}(G,S))$. A normic system $\underline{M}$ is sent to $\oplus_{n}M_{n}$, together with its obvious structure of $\mu_{\lhd}(G,S)$-module: $c_{g}$ acts on every components whereas $j_{n,m}$ (resp. $k_{m,n}$) sends the $n$-th (resp. $m$-th) component into the $m$-th (resp. $n$-th) one. By definition, a morphism of normic systems $\underline{M}\rightarrow\underline{M}^{\prime}$ consists in a collection of morphisms $\phi_{n}:M_{n}\rightarrow M_{n}^{\prime}$, compatible with the $c_{g}$’s, the $j_{n,m}$’s and the $k_{m,n}$’s. It thus gives rise to a morphism $\oplus_{n}M_{n}\rightarrow\oplus_{n}M^{\prime}_{n}$, compatible with the action of $\mu_{\lhd}(G,S)$ and this is clearly compatible to composition. We thus have defined the desired functor ${{}_{\underline{G}}Mod(S)}\rightarrow Mod(\mu_{\lhd}(G,S))$. It now remains to notice that both the full faithfulness and the essential surjectivity of this functor immediately follow from the isomorphism of algebras $\mu_{\lhd}(G,S)\simeq\prod_{n\in N}j_{n,n}\mu_{\lhd}(G,S)$.

$(ii)$ We apply $(i)$ to the case $G=G_{n}$. Letting $n$ vary, this gives an equivalence of cofibered categories. Whence the result, since ${{}_{\underline{G}}Mod(S)}$ is clearly equivalent to the category of cocartesian sections of the cofibered category associated to $n\mapsto{{}_{\underline{G_{n}}}Mod(S)}$.

$\square$

The following corollary answers a question of [Va].

Corollary 2.4**.**

The category ${{}_{\underline{G}}Mod(S)}$ has enough injectives.

Proof. Let $\mathcal{F}/N^{op}$ denote the fibered category $n\mapsto Mod(\mu_{\lhd}(G_{n},S))$ considered previously and let $Sect(\mathcal{F}/N^{op})$ (resp. $Cocart(\mathcal{F}/N^{op})$) denote its category of sections (resp. cocartesian sections). Denoting $|N|$ the discrete category underlying $N$, there are three obvious forgetful functors

[TABLE]

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. For $F_{3}$, this is easy and left to the reader. For $F_{2}$, this results from [SGA4], Vbis, 1.2.10 (note that here $\mathcal{F}/N^{op}$ is indeed bifibered). For $F_{1}$, we use the following general easy fact concerning an arbitrary cofibered category $\mathcal{F}/\mathcal{S}$ with cocleavage $f\mapsto f_{!}$:

Fact: Assume that for any $f:X\rightarrow Y$ the following properties are verified:

• The functor "composition with $f$" :$\mathcal{S}_{/X}\rightarrow\mathcal{S}_{/Y}$ is cofinal.

• $f_{!}$ commutes to projective limits indexed by $\mathcal{S}_{/X}$.

Then the inclusion functor $Cocart(\mathcal{F}/\mathcal{S})\rightarrow Sect(\mathcal{F}/\mathcal{S})$ has a right adjoint, which takes a section $X\mapsto s(X)$ to the cocartesian section $X\mapsto\displaystyle{\lim_{\longleftarrow}}_{f:Z\rightarrow X}f_{!}s(Z)$.

$\square$

Remark 2.5**.**

One may show that ${{}_{\underline{G}}Mod(S)}$ has enough projectives as well.

2.2. From sheaves to normic systems

As before let $G=\varprojlim_{n}G_{n}$ denote a profinite group and let $G_{n}=G/H_{n}$. Recall that the classifying topos $B_{G}$ is the category of discrete left $G$-sets. Consider another arbitrary topos $E$ together with its structural morphism $\pi:E\to Set$ and assume given a projective system $\underline{X}=(X_{n})$ of $G$-objects of $E$ such that $X_{n}$ is a torsor of $E$ under $\pi^{-1}G_{n}$. In this situation we have for $m\geq n$ a commutative diagram of topoi as follows:

[TABLE]

Here the horizontal arrows are by functoriality of the classifying topos with respect to the group (their inverse and direct images functors are inflation and fixed points by the adequate subgroup) and the oblique morphisms, defined by $\underline{X}$, are as follows:

• If $Y$ is in $B_{G_{n}}$ and $Y^{0}$ denotes the underlying set, the formula $g.(x,y)=(gx,gy)$ defines an action of $\pi^{-1}G_{n}$ on $X_{n}\times\pi^{-1}Y^{0}$ and $p_{n}^{-1}Y=\pi^{-1}G_{n}\backslash(X_{n}\times\pi^{-1}Y^{0})$ is the coinvariant object of this action. If $Z$ is an object of $E$, then $p_{n,*}Z$ is the set $Hom(X_{n},Z)$ endowed with the left action of $G_{n}$ induced by the inverse action on $X_{n}$: $(g.f)(x)=f(g^{-1}x)$.

• If $Y$ is in $B_{G}$ then the formula $g.(x,y)=(gx,gy)$ defines an action of $p_{\infty}^{-1}G$ on $X_{m}\times\pi^{-1}(Y^{H_{n}})^{0}$ for each $m,n$ in $N$ and $p_{\infty}^{-1}Y=\displaystyle{\lim_{\longrightarrow}}_{n}\displaystyle{\lim_{\longleftarrow}}_{m}{\pi^{-1}G}\backslash(X_{m}\times\pi^{-1}(Y^{H_{n}})^{0})$ (Note that $p_{\infty}^{-1}$ is exact, since $\displaystyle{\lim_{\longleftarrow}}_{m}$ is essentially constant while $\displaystyle{\lim_{\longrightarrow}}_{n}$ is filtered). If $Z$ is an object of $E$ then $p_{\infty,*}Z$ is the set $\varinjlim_{n}Hom(X_{n},Z)$ endowed with the left action of $G$ induced by the inverse action of $G$ on $f$.

Endowing the set $G_{n}$ with its left action by translations turns it into an object of $B_{G}$ (resp. $B_{G_{m}}$, if $m\geq n$) whose image by $p_{\infty}$ (resp. $p_{m}$) is nothing but $X_{n}$.

For the purpose of what follows we will consider a ring $S$. Endowing it with a trivial action of $G$, we may as well view it as a ring of $B_{G_{n}}$ or $B_{G}$.

Definition 2.6**.**

We define a functor

[TABLE]

by sending an $S$-module $M$ of $B_{G}$ to $(M_{n},j_{n,m},k_{m,n})$, where $M_{n}=M^{H_{n}}$, $j_{n,m}$ is the inclusion and $k_{m,n}$ the trace map.

Lemma + definition 2.7**.**

Let $G$, $S$, $E$ and $\underline{X}$ be as above. Consider furthermore a ring $R$ of $E$ and a ring homomorphism $S\to\pi_{*}R$. Letting $Mod(E,R)$ denote the category of $R$-modules of $E$, we define the functor of Normic sections along $\underline{X}$

[TABLE]

by composing $p_{\infty,*}:Mod(E,R)\to Mod(B_{G},S)$ with the functor $\Gamma(\underline{G},-):Mod(B_{G},S)\to{}{{}_{\underline{G}}Mod(S)}$ of Def. 2.6.

For any $F$ in $Mod(E,R)$, the object $((M_{n}),(j_{n,m}),(k_{m,n})):=\Gamma(\underline{X},F)$ satisfies 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 of $G_{n}$ coming from the right action of $G_{n}$ on $X_{n}$: $(g,x)\mapsto g^{-1}x$.

-$j_{n,m}:M_{n}\to M_{m}$ is the restriction along $X_{m}\rightarrow X_{n}$. and induces an isomorphism $M_{n}\simeq M_{m}^{G_{m,n}}$ (ie. $F(X_{n})\mathop{\rightarrow}\limits F(X_{m})^{G_{m,n}}$).

Proof. By definition we have $M_{n}=M^{H_{n}}=q_{n,*}M$, where $M=p_{\infty,*}F$. Now, $q_{n}p_{\infty}\simeq p_{n}$, therefore $M_{n}\simeq p_{n,*}F=F(X_{n})$. The second property is clear.

$\square$

Remark 2.8**.**

If the topos $E$ is locally connected, it is possible to build trace maps along finite locally free morphisms, such as $X_{m}\to X_{n}$. One may then show that $k_{m,n}$ coincides with the trace map.

Lemma 2.9**.**

Consider another topos $E^{\prime}$ with structural morphism $\pi^{\prime}:E^{\prime}\to Set$ and a morphism of ringed topoi $f:(E^{\prime},R^{\prime})\to(E,R)$. Let us moreover denote $\underline{X}^{\prime}=(f^{-1}X_{n})$ the projective system of torsors of $E^{\prime}$ deduced from $\underline{X}$ and consider the morphism $S\to\pi^{\prime}_{*}R^{\prime}$ induced by $h:S\to\pi_{*}R$ and $R\to f_{*}R^{\prime}$. The functor $\Gamma(\underline{X}^{\prime},-)$ associated to these data is subject to a canonical isomorphism

[TABLE]

of functors $D^{+}(E^{\prime},R^{\prime})\to D^{+}({{}_{\underline{G}}Mod(S)})$.

Proof. This simply follows from the fact that the morphism of ringed topoi $(E^{\prime},R^{\prime})\to(B_{G},S)$ defined by $\underline{X}$ coincides with the one obtained by composing $f$ with the morphism $(E,R)\to(B_{G},S)$ defined by $\underline{X}$.

$\square$

The derived functor

[TABLE]

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 the $n$th component functor

[TABLE]

sends $R\Gamma(\underline{X},F)$ to $R\Gamma(X_{n},F_{|X_{n}})$.

2.3. Descent, codescent and duality

We now assume that the ring $S$ of Def. 2.6 is ${\mathbb{Z}_{p}}$ 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 ${\operatorname{\Lambda}}:={\mathbb{Z}_{p}}[[G]]:=\displaystyle{\lim_{\longleftarrow}}{\mathbb{Z}_{p}}[G_{n}]$ is Noetherian of finite global dimension $d+1$ and has an open pro-$p$-subgroup. As is well known, these condition is in particular verified if $G$ is a compact $p$-adic Lie group of dimension $d$ without $p$-torsion.

• Limits. Forgetting the $k_{m,n}$’s (resp. the $j_{n,m}$’s), inflating from $G_{n}$-modules to discrete $G$-modules (resp. abstract ${\operatorname{\Lambda}}$-modules) and then forming the limit of the resulting inductive (resp. projective) system gives rise to a functor which will abusively be denoted

[TABLE]

The first is exact and thus passes to derived categories while the second is only left exact, but right derivable. From now on, we always assume that $R\displaystyle{\lim_{\longleftarrow}}$ has finite cohomological dimension. This is e.g. the case if $N$ has a numerable cofinal subset (as is always the case in practice).

Proposition 2.10**.**

There are canonical adjunctions

[TABLE]

Here, $\underline{{\operatorname{\Lambda}}}$ denotes the natural normic system of right ${{\operatorname{\Lambda}}}$-modules $({\mathbb{Z}_{p}}[G_{n}])$ and $\underline{{\operatorname{\Lambda}}}\otimes_{{\operatorname{\Lambda}}}(-):Mod({\operatorname{\Lambda}})\rightarrow{{}_{\underline{G}}Mod({\mathbb{Z}_{p}})}$ denotes the induced right exact, left derivable functor.

Proof. The derived version are easily deduced from the obvious ordinary adjunctions using that ${{}_{G}Mod({\mathbb{Z}_{p}})}$ and ${{}_{\underline{G}}Mod({\mathbb{Z}_{p}})}$ have enough injectives (cf. Prop. 2.4). In [Va], this was not known and a finiteness assumption was thus needed to avoid deriving $\displaystyle{\lim_{\longleftarrow}}$ (cf. loc. cit. Prop. 4.2. and Rem. 4.3).

$\square$

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 ${\mathbb{Z}_{p}}$-module $I$. If $M$ (resp. $M_{n}$) is a ${\mathbb{Z}_{p}}$-module endowed with a left action of $G$ (resp. $G_{n}$) then $Hom_{\mathbb{Z}_{p}}(M,I)$ (resp. $Hom_{\mathbb{Z}_{p}}(M_{n},I)$) is endowed a left action of $G$ (resp. $G_{n}$) as well by reversing the action on $M$ (ie. $(g.f)(m)=f(g^{-1}m)$). If the action of $G$ is discrete the action of $G$ on $Hom_{\mathbb{Z}_{p}}(M,I)$ extends to an action of $\Lambda$. If now $\underline{M}=(M_{n},j_{n,m},k_{m,n})$ is a normic system then $Hom_{\mathbb{Z}_{p}}(\underline{M},I):=(Hom_{\mathbb{Z}_{p}}(M_{n},I),Hom_{\mathbb{Z}_{p}}(k_{m,n},I),Hom_{\mathbb{Z}_{p}}(j_{n,m},I))$ is a normic system as well. Using an injective resolution of ${\mathbb{Z}_{p}}$ we get duality functors

[TABLE]

It might be useful to point out the following relation to Pontryagin duality:

[TABLE]

If now $M$ is a $\Lambda$-module we view $Hom_{\Lambda}(M,\Lambda)$ as a left $\Lambda$-module via the right action of $\Lambda$ on itself and the involution $g\mapsto g^{-1}$. Using projective resolutions of $M$ we get a functor

[TABLE]

Theorem 2.11**.**

Let $E$, $X_{n}$ be as in the previous paragraph.

$(i)$* $(Descent)$ There is a canonical isomorphism*

[TABLE]

of functors $D^{+}(E,{\mathbb{Z}_{p}})\rightarrow D^{+}({{}_{\underline{G}}Mod({\mathbb{Z}_{p}})})$.

$(ii)$* Consider a bounded complex $F$ of ${\mathbb{Z}_{p}}$-modules of $E$ (resp. and assume that there exists $q_{0}$) such that $H^{q}(X_{n},F_{|X_{n}})$ is finitely generated over ${\mathbb{Z}_{p}}$ (resp. trivial) for any $n\in N$, $q\geq 0$ (resp. $q\geq q_{0}$).*

$(Codescent)$* Functorially in $F$, There is a canonical isomorphism in $D^{+}({{}_{\underline{G}}Mod({\mathbb{Z}_{p}})})$:*

[TABLE]

$(Perfectness)$* $R\displaystyle{\lim_{\longleftarrow}}R\Gamma(\underline{X},F)$ and $RHom_{\mathbb{Z}_{p}}(\displaystyle{\lim_{\longrightarrow}}R\Gamma(\underline{X},F),{\mathbb{Z}_{p}})$ are in $D^{p}({\operatorname{\Lambda}})$ (the derived category of perfect complexes).*

$(Duality)$* Functorially in $F$, There is a canonical isomorphism in $D^{p}({\operatorname{\Lambda}}{\hbox{-}mod})$:*

[TABLE]

Proof. $(i)$ We are going to check that the adjunction morphism of $(\displaystyle{\lim_{\longrightarrow}},R\Gamma(\underline{G},-))$ is an isomorphism. Replacing $F$ by an injective resolution and truncating, one reduces to the case where $F$ is an injective ${\mathbb{Z}_{p}}$-module of $E$ placed in degree [math]. Let $\underline{M}:=\Gamma(\underline{X},F)$. Then $\forall n$, $M_{n}$ is injective in ${{}_{G_{n}}Mod({\mathbb{Z}_{p}})}$ (indeed $M_{n}$ corresponds to $p_{n*}F$) and the descent map induces an isomorphism $M_{n}\rightarrow M_{m}^{G_{m,n}}$. Similarly $\varinjlim_{n}M_{n}$ is injective and the adjunction morphism of descent occurring in $D^{+}({{}_{\underline{G}}Mod({\mathbb{Z}_{p}})})$ is represented at level $n$ by the isomorphism $\Gamma(X_{n},F_{|X_{n}})\rightarrow\Gamma(H_{n},\displaystyle{\lim_{\longrightarrow}}_{m}\Gamma(X_{m},F_{|X_{m}}))$.

$(ii)$ Under the stated assumptions,

• $(Codescent)$ and $(Perfectness)$ follow from $(i)$, by [Va] 4.4 $(ii)$ et 4.6 $(i)$.

• $(Duality)$ follows from $(Codescent)$, by loc. cit. 4.6 $(iii)$.

$\square$

Remark 2.12**.**

*Properties $(Codescent)$ and $(Perfectness)$ are a crucial tool in this paper. It seems plausible that they hold without assuming that $G$ has finite cohomological dimension but we have not tried to check this.

2.4. Examples

Start with a projective system of surjective étale schemes $C_{n}$ over a base scheme $C$ and assume that the $C_{n}$’s are endowed with compatible actions of the $G_{n}$’s such that each $G_{n}\times C_{n}\to C_{n}\times_{C}C_{n}$, $(g,x)\mapsto(gx,x)$ is an isomorphism. (In view of our applications, let us notice that if $C$ and $C_{n}$ are proper smooth curves over a field with respective function fields $K$ and $K_{n}$ then the latter condition holds as soon as the unramified extension $K_{n}/K$ is Galois). These data represent a projective system of torsors, say $\underline{X}$ 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, say $\underline{X}^{\prime}$, of the big, say flat, topos $E^{\prime}=C_{FL}$. Note that if $\epsilon:C_{FL}\to C_{et}$ denotes the natural morphism, then $\underline{X}^{\prime}=\epsilon^{-1}\underline{X}$.

Consider now the small étale crystalline topos $E^{\prime\prime}$ of $C/{\mathbb{Z}_{p}}$ and the projection morphism $u:E^{\prime\prime}=(C/{\mathbb{Z}_{p}})_{crys,et}\to C_{et}=E$. Then we define a projective system of torsors of $E^{\prime\prime}$ by pulling back via $u$: $\underline{X}^{\prime\prime}:=u^{-1}\underline{X}$.

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

[TABLE]

whenever one is given ${\mathbb{Z}_{p}}$-algebras $R$, $R^{\prime}$, $R^{\prime\prime}$ in $E$, $E^{\prime}$, $E^{\prime\prime}$ and homomorphisms $R\to\epsilon_{*}R^{\prime}$, $R\to\epsilon_{*}R^{\prime\prime}$.

The $n$-component of the left (resp. middle, resp. right) normic section functor computes the cohomology of the localized topos $C_{FL}/X^{\prime}_{n}$, (resp. $C_{et}/X_{n}$, resp. $(C/{\mathbb{Z}_{p}})_{crys,et}/X_{n}^{\prime\prime}$), ie. of $C_{n,FL}$ (resp. $C_{n,et}$, resp. $(C_{n}/{\mathbb{Z}_{p}})_{crys,et}$). We will thus use the more suggestive notations $R\Gamma(\underline{C}_{FL},-)$ (resp. $R\Gamma(\underline{C}_{et},-)$, resp. $R\Gamma_{crys,et}(\underline{C}/{\mathbb{Z}_{p}},-)$).

3. K-theory for $\Lambda(G)$

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

3.1. Review of $K_{0}$ and $K_{1}$

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

• $K_{0}(R)$ is the abelian group defined by generators $[P]$, where $P$ is a finitely generated projective $R$-modules, and relations

1. (i)

$[P]=[Q]$ if $P$ is isomorphic to $Q$ as a $R$-module. 2. (ii)

$[P\oplus Q]=[P]+[Q]$

• $K_{1}(R)$ is the abelian group defined by generators $[P,f]$, where $P$ is a finitely generated projective $R$-module and $f$ is an automorphism of $P$ with relations (the group law is denoted multiplicatively):

1. (i)

$[P,f]=[Q,g]$ if $P$ is isomorphic to $Q$ as a $R$-module via an isomorphism which is compatible with $f$ and $g$. 2. (ii)

$[P\oplus Q,f\oplus g]=[P,f][Q,g]$ 3. (iii)

$[P,fg]=[P,f][P,g]$

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

• The functor $Hom_{R}(-,R)$ realizes an anti-equivalence between finitely generated projective left modules and finitely generated projective right modules. As a result, one gets an isomorphism $K_{i}(R)\simeq K_{i}(R^{op})$, if $R^{op}$ denotes the opposed ring. Both this isomorphism and its inverse will be denoted $(-)^{*}$ (eg. $[P,f]^{*}=[P^{*},f^{*}]$ if $f^{*}$ denotes the transpose of $f$).

• Morita equivalence. Let us fix a finitely generated projective right $R$-module $V$ and let $V^{*}=Hom_{R}(V,R)$. Then $V$ (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 finitely generated, one has a canonical isomorphism of $(R,R)$-bimodules: $V^{*}\otimes_{End_{R}(V)}V\simeq R$ (resp. of $(End_{R}(V),End_{R}(V))$-bimodules: $V\otimes_{R}V^{*}\simeq End_{R}(V)$.

The functor $V^{*}\otimes_{End_{R}(V)}(-):End_{R}(V){\hbox{-}mod}\rightarrow R{\hbox{-}mod}$ is thus an equivalence of categories, with $V\otimes_{R}(-)$ as quasi-inverse. In particular, there is a canonical isomorphism $K_{i}(End_{R}(V))\simeq K_{i}(R)$, $i=0,1$.

3.2. The determinant functor

Let $\cal P(R)$ denote the category of strictly perfect (ie. bounded with projective finitely generated objects) complexes, $K^{p}(R)$ its homotopy category and $D^{p}(R)$ its essential image in the derived category (which is naturally equivalent, and identified, 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

[TABLE]

where $\cal C_{R}$ is the Picard category (ie. a category together with an endobifunctor, 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 ${\cal C}_{R}$ is a couple $(P,Q)$ of finitely generated projective $R$-modules.

• •

$Mor((P,Q),(P^{\prime},Q^{\prime}))$ is empty if $[P]-[Q]\neq[P^{\prime}]-[Q^{\prime}]$ in $K_{0}(R)$. Else, there exists an $R$-module $M$ such that

[TABLE]

We set $I_{M}:=Isom(P\oplus Q^{\prime}\oplus M,P^{\prime}\oplus Q\oplus M)$ and $G_{M}:=Aut(P^{\prime}\oplus Q\oplus M)$ and we define the set of morphisms from $(P,Q)$ to $(P^{\prime},Q^{\prime})$ as $K_{1}(R)\times^{G_{M}}I_{M}$ where the right hand side denotes the quotient of $K_{1}(R)\times I_{M}$ by the action of $G_{M}$ given by

[TABLE]

where $x\in K_{1}(R)$, $y\in I_{M}$, $g\in G_{M}$ and $\bar{g}$ is its image in $K_{1}(R)$. As seen easily, this set does not depend on $M$, up to a canonical isomorphism, and this fact can be used to define composition in a natural way. Note that by definition, one has a canonical identification $Aut_{\cal C_{R}}((P,Q))=K_{1}(R)$ for any object $(P,Q)$.

• •

The product is defined as

[TABLE]

and admits naturally $(0,0)$ as a unit. Every object $(P,Q)$ in ${\cal C}_{R}$ admits $(P,Q)^{-1}:=(Q,P)$ as a natural inverse.

It follows immediately from its construction that the functor $(\ref{defDet})$ is compatible with base change, ie. the diagram

[TABLE]

is naturally pseudo-commutative for any $R^{\prime}/R$. Let us quickly review the construction of $Det_{R}$ following [FK] 1.2.

• •

A strictly perfect complex $P$ is sent to $Det_{R}(P):=(P^{+},P^{-})$ where $P^{+}:=\oplus_{i\in 2\mathbb{Z}}P^{i}$ and $P^{-}:=\oplus_{i\in 1+2\mathbb{Z}}P^{i}$.

• •

Any exact sequence $0\rightarrow P^{\prime}\rightarrow P\rightarrow P^{\prime\prime}\rightarrow 0$ of $\cal P(R)$ induces a canonical isomorphism $Det_{R}(P)\simeq Det_{R}(P^{\prime}).Det_{R}(P^{\prime\prime})$. Using this, one constructs a canonical trivialization $can_{P}:Det_{R}(0)\simeq Det_{R}(P)$, for any strictly perfect complex $P$ which is acyclic.

• •

If $Cone(f)$ denotes the mapping cone of a morphism $f:P\rightarrow P^{\prime}$ of strictly perfect complexes then $Hom_{\cal C_{R}}(Det_{R}(P),Det_{R}(P^{\prime}))$ identifies with $Hom_{\cal C_{R}}(Det_{R}(0),Det_{R}(Cone(f)))$ (indeed $Cone(f)^{+}\simeq P^{-}\oplus P^{\prime+}$ and $Cone(f)^{-}\simeq P^{+}\oplus P^{\prime-}$). In particular, when $f$ is a quasi-isomorphism, then $can_{Cone(f)}:Det_{R}(0)\to Det_{R}(Cone(f))$ induces a morphism $Det_{R}(P)\rightarrow Det_{R}(P^{\prime})$, which we denote $Det_{R}(f)$. One may check that it is compatible with composition and only depends on the homotopy class of $f$ so that the functor $Det_{R}$ is finally defined.

We will make essential use of the homorphism

[TABLE]

which is induced by $(\ref{defDet})$ for any perfect complex $P$. Of course if $P$ is reduced to a (finitely generated projective) module $P^{0}$ placed in degree zero then $Det_{R}(P)=(P^{0},0)$ and $Det_{R}(f):(P^{0},0)\to(P^{0},0)$ is the class of $f^{0}:P^{0}\to P^{0}$. In that case $(\ref{Detaut})$ is thus nothing but the tautological map $Aut_{R}(P^{0})\to K_{1}(R)$.

Let us now state some multiplicative properties.

• •

Consider a morphism between exact sequences of strictly perfect complexes

[TABLE]

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

[TABLE]

• •

Consider an automorphism $f$ of some $P$ in $K^{p}(R)$. Then:

• $Det_{R}(f[1])=Det_{R}(f)^{-1}$ in $Aut_{\cal C_{R}}(Det_{R}(P))=K_{1}(R)$.

• $f$ is the homotopy class of some automorphism of the complex $P$. Let us fix such one and denote $f^{q}:P^{q}\to P^{q}$ its component of degree $q$. Then in $K_{1}(R)=Aut_{\cal C_{R}}(Det_{R}(P))=Aut_{\cal C_{R}}(Det_{R}(P^{q}))$, we have $Det_{R}(f)=\prod Det_{R}(f^{q})^{(-1)^{q}}$.

• If $P$ is cohomologically perfect (ie. each $H^{q}$ is an object of $D^{p}(R)$) then $Det_{R}(f)=\prod Det_{R}(H^{q}(f))^{(-1)^{q}}$ in $K_{1}(R)=Aut_{\cal C_{R}}(Det_{R}(P))=Aut_{\cal C_{R}}(Det_{R}(H^{q}(P)))$.

3.3. Localization

The main $K$-group of interest for Iwasawa theory is a relative one. We recall its definition ([FK] 1.3). Consider a strictly full triangulated subcategory $\Sigma$ of $D^{p}(R)$. The group $K_{1}(R,\Sigma)$ is then defined by generators and relations as follows:

• •

Generators: $[C,a]$ where $C$ is an object of $\Sigma$ and $a:Det_{R}(0)\rightarrow Det_{R}(C)$ is a trivialization of $C$.

• •

Relations: Let $C,C^{\prime},C"$ be objects of $\Sigma$.

• –

If $C\simeq 0$ then $[C,can_{C}]=1$.

• –

If $f:C\simeq C^{\prime}$, then compatible trivializations of $C$ and $C^{\prime}$ give rise to the same element in $K_{1}(R,\Sigma)$ (ie. $[C^{\prime},Det_{R}(f)\circ a]=[C,a]$ if $a:Det_{R}(0)\simeq Det_{R}(C)$).

• –

If $0\rightarrow C^{\prime}\rightarrow C\rightarrow C^{\prime\prime}\rightarrow 0$ is an exact sequence of $\cal P(R)$, and $a:1\simeq Det_{R}(C)$, $a^{\prime}:1\simeq Det_{R}(C^{\prime})$, then

[TABLE]

where $a:1\mathop{\rightarrow}\limits^{a^{\prime}.a^{\prime\prime}}Det_{R}(C^{\prime}).Det_{R}(C^{\prime\prime})\simeq Det_{R}(C)$.

There is a localization exact sequence (cf [FK] 1.3.15)

[TABLE]

(here $K_{0}(\Sigma)$ denotes the Grothendieck group of the triangulated category $\Sigma$) where:

• the first map sends $[P,f]\in K_{1}(R)$ to the complex $[P\mathop{\rightarrow}\limits^{f}P]$ placed in degrees $[-1,0]$, together with the trivialization which is represented by the identity of $P$.

• $\partial$ sends $[C,f]\in K_{1}(R,\Sigma)$ to the element $-[C]\in K_{0}(\Sigma)$.

• the last map sends the class of a strictly perfect complex $[C]$ to the alternated sum $\sum(-1)^{i}[C^{i}]$.

As checked easily, this localization sequence is functorial with respect to $(R,\Sigma)$: if $R^{\prime}$ is an $R$-algebra, and if $\Sigma^{\prime}$ is a strictly full triangulated subcategory of $D^{p}(R^{\prime})$ containing the essential image of $\Sigma$ under the functor $R^{\prime}\otimes^{L}_{R}(-):D^{p}(R)\rightarrow D^{p}(R^{\prime})$ then one has a morphism $K_{1}(R,\Sigma)\rightarrow K_{1}(R^{\prime},\Sigma^{\prime})$ which is compatible with the obvious functoriality maps.

If now $T\subset R$ is a left denominator set (cf [FK], 1.3.6) then we can apply the above constructions to the full triangulated subcategory $\Sigma_{T}$ of $D^{p}(R)$ consisting of complexes whose image under the functor $R_{T}\otimes^{L}_{R}(-):D^{p}(R)\to D^{p}(R_{T})$ become acyclic. We note that if $R$ is Noetherian and regular then $K_{0}(\Sigma_{T})$ is isomorphic to the Grothendieck group of finitely generated $T$-torsion modules, by sending $[C]$ to $\sum(-1)^{i}H^{i}(C)$. For a general $R$ we have the following result.

Proposition 3.1**.**

([FK], 1.3.7) There is a canonical isomorphism of groups

[TABLE]

sending $[C,a]$ to the isomorphism $Det_{R_{T}}(0)\mathop{\rightarrow}\limits^{a_{T}}Det_{R_{T}}C_{T}\mathop{\rightarrow}\limits^{can_{C_{T}}^{-1}}Det_{R_{T}}(0)$ viewed as an element of $K_{1}(R_{T})=Aut_{\mathcal{C}_{R_{T}}}(Det_{R_{T}}(0))$ (here $a_{T}$ and $C_{T}$ are deduced from $a$ and $C$ by localization). This isomorphism is functorial with respect to $(R,T)$.

$\square$

Let us mention an alternative characterization of this isomorphism. Consider an endomorphism $f$ of a strictly perfect complex $P$ such that $f_{T}$ is a quasi-isomorphism. Identifying $Hom_{\cal C_{R}}(Det_{R}(P),Det_{R}(P))$ with $Hom(Det_{R}(0),Det_{R}(Cone(f)))$ sends the identity of $Det_{R}(P)$ to a morphism $triv:Det_{R}(0)\rightarrow Det_{R}(Cone(f))$. Then the class $[Cone(f),triv]\in K_{1}(R,\Sigma_{T})$ corresponds to $Det_{R_{T}}(f_{T})^{-1}\in K_{1}(R_{T})$ (both are indeed described by the same endomorphism of $P_{T}^{+}\oplus P_{T}^{-}$ coming from $can_{Cone(f)_{T}}$).

Crucial to us will be the following:

Lemma 3.2**.**

Consider morphisms $a,b:C\rightarrow C^{\prime}$ in $D^{p}(R)$ whose localizations $a_{T},b_{T}$ are isomorphisms in $D^{p}(R_{T})$. In $K_{0}(\Sigma_{T})$, one has the equality

[TABLE]

as long as the following condition holds:

$(frac)$* In $D^{p}(R)$, there exists a commutative square of the form*

[TABLE]

in which $c$ and $d$ also become isomorphisms after localization by $T$.

Proof. An easy diagram chasing shows that $[Cone(c)]-[Cone(a)]=[Cone(d)]-[Cone(b)]$. Since $Det_{R_{T}}(a_{T}b_{T}^{-1})=Det_{R_{T}}(d_{T})^{-1}Det_{R_{T}}(c_{T})$ and $\partial$ is a homomorphism, it thus suffices to prove that $\partial Det_{R_{T}}(c_{T})=[Cone(c)]$ and $\partial Det_{R_{T}}(d_{T})=[Cone(d)]$. But this is clear from the alternative characterization of the isomorphism $K_{1}(R,\Sigma_{T})\simeq K_{1}(R_{T})$ mentioned above.

$\square$

We do not know whether or not the condition $(frac)$ always holds, neither if this computation of $\partial Det_{R_{T}}(a_{T}b_{T}^{-1})$ is always correct. We thus content ourselves with the following sufficient condition.

Lemma 3.3**.**

Assume $R$ is Noetherian and consider $a,b:C\rightarrow C^{\prime}$ in $D^{p}(R)$. If there exists $f$ in the center of $R$ such that $R[{1\over f}]\otimes^{L}_{R}Cone(b)=0$, then one can find $c,d:C^{\prime}\rightarrow C^{\prime}$ such that $da=cb$. In fact one can chose $d=f^{n}$ for $n$ large enough.

Proof. Since cohomology modules of $Cone(b)$ are finitely generated and almost all of them are zero, it is possible to find $n$ such that $f^{n}:Cone(b)\rightarrow Cone(b)$ is zero. But then, the long exact sequence of $Ext$’s

[TABLE]

shows that $f^{n}a$ has a trivial image in $Ext^{1}(Cone(b),C^{\prime})$ and thus comes from some $c\in Hom(C^{\prime},C^{\prime})$ as claimed.

$\square$

3.4. The evaluation map at Artin representations

Consider a profinite group $G=\displaystyle{\lim_{\longleftarrow}}_{n}G_{n}$ and a closed normal subgroup $H$ such that $\Gamma:=G/H$ is isomorphic to ${\mathbb{Z}_{p}}$. We use the following notations.

• We let ${\operatorname{\Lambda}}={\operatorname{\Lambda}}(G):=\varprojlim_{n}{\mathbb{Z}}_{p}[G_{n}]$ 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 a $p$-adic Lie extension as in the introduction).

• If $O$ is the ring of integers of a finite extension $L$ of ${\mathbb{Q}}_{p}$, then ${\operatorname{\Lambda}}_{O}(G):=\varprojlim_{n}O[G_{n}]\simeq O\otimes_{\mathbb{Z}_{p}}{\operatorname{\Lambda}}(G)$ and ${\operatorname{\Lambda}}_{L}(G):=L\otimes_{O}{\operatorname{\Lambda}}_{O}(G)$ have similar properties.

• $S$ and $S^{*}$ denote the canonical Ore sets defined in [CFKSV]. Recall that an element $f$ of ${\operatorname{\Lambda}}(G)$ is in $S$ if and only if ${\operatorname{\Lambda}}(G)/f$ is a finitely generated ${\operatorname{\Lambda}}(H)$-module whereas $S^{*}=\cup_{k}p^{k}S$. As usual, $\Lambda_{S}=\Lambda(G)_{S}$ and $\Lambda_{S^{*}}=\Lambda(G)_{S^{*}}$ denote the corresponding localizations of $\Lambda$. If $G=\Gamma$ then $Q_{O}(\Gamma):=\Lambda(\Gamma)_{S^{*}}$ is the fraction field of $\Lambda_{O}(\Gamma)$.

-$\mathfrak{M}_{H}(G)$ denotes the category of $S^{*}$-torsion finitely generated ${\operatorname{\Lambda}}(G)$-modules. We recall that a finitely generated ${\operatorname{\Lambda}}(G)$ module $M$ is $S$-torsion (resp. $S^{*}$-torsion) if and only if it is a finitely generated ${\operatorname{\Lambda}}(H)$-module (resp. modulo its $p$-torsion). In this context, the localization exact sequence reads:

[TABLE]

• 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 $g\mapsto g^{-1}$. 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 finite rank. For any $O$-algebra $R$ (eg. $R={\operatorname{\Lambda}}_{O}(G)$), we denote $V_{R}$ (resp. ${{}_{R}V}$) the right (resp. left) $R$-module $V\otimes_{O}R$ (resp. $R\otimes_{O}V$) deduced from $V$ by extending scalars from $O$ to $R$. Also, $({{}_{R}V})^{*}:=Hom_{R}({{}_{R}V},R)$ (resp. $({V_{R}})^{*}:=Hom_{R}({V_{R}},R)$) is systematically given its right (resp. left) $R$-module structure coming from the $(R,R)$-bimodule structure of $R$.

It might be useful to recall that one has a canonical isomorphism of left (resp. right) $R$-modules $(V_{R})^{*}\simeq{{}_{R}(V^{*})}$ (resp. $({{}_{R}V})^{*}\simeq(V^{*})_{R}$) and canonical isomorphisms of $O$-algebras $End_{R}((V_{R})^{*})\simeq End_{R}(V_{R})^{op}$ and $End(({{}_{R}V})^{*})\simeq End({{}_{R}V})^{op}$ ($R$-algebras). If now $R$ is a central $O$-algebra (eg. if $R=\Lambda_{O}(G)$) we have moreover canonical isomorphisms of $O$-algebras $End_{R}({{}_{R}V})\simeq R^{op}\otimes_{O}End_{O}(V)$, $End_{R}(V_{R})\simeq End_{O}(V)\otimes_{O}R$. These isomorphisms are subject to natural compatibilities, such as the commutativity of the following diagram:

[TABLE]

Consider now an $O$-Artin representation $\rho:G\rightarrow Aut_{O}(V)$. By $O$-Artin representation we mean that $V$ is as above and $\rho$ has a finite image. Consider the unique homomorphism of ${\mathbb{Z}_{p}}$-algebras

[TABLE]

sending $g\in G\subset{\operatorname{\Lambda}}_{O}(G)$ to $v\otimes\lambda\mapsto\rho(g)(v)\otimes g\lambda$.

By functoriality of $K_{1}$, one has a commutative diagram:

[TABLE]

where $tw_{\rho}$ is the map sending $[P,f]\in K_{1}({\operatorname{\Lambda}}(G))$ to $[(V_{{\operatorname{\Lambda}}_{O}(G)})^{*}\otimes_{{\operatorname{\Lambda}}(G)}P,id\otimes f]\in K_{1}({\operatorname{\Lambda}}_{O}(G))$ where $(V_{{\operatorname{\Lambda}}_{O}(G)})^{*}$ is viewed as a $({\operatorname{\Lambda}}_{O}(G),{\operatorname{\Lambda}}(G))$-bimodule, the left action of ${\operatorname{\Lambda}}_{O}(G)$ being the obvious one and the right action of $\Lambda(G)$ being deduced via $\Phi$ from the natural right action of $End_{{\operatorname{\Lambda}}_{O}(G)}(V_{{\operatorname{\Lambda}}_{O}(G)})$. Furthermore the image of this element in $K_{1}(O)$ is $[V^{*}\otimes_{{\operatorname{\Lambda}}(G)}P,id_{V^{*}}\otimes f]$ where the right ${\operatorname{\Lambda}}(G)$-module structure of $V^{*}$ is deduced from the right action of $G$ on $V^{*}$: $g\mapsto\rho(g)^{*}$. The composed homomorphism

[TABLE]

where $det_{O}$ denotes the usual (commutative) determinant, will simply be denoted $\rho$.

Remark 3.4**.**

$(i)$* The element $tw_{\rho}([P,f])$ is also equal to $[V^{*}\otimes_{\mathbb{Z}_{p}}P,id\otimes f]$. Indeed, there are isomorphisms of left $\Lambda_{O}(G)$ modules*

[TABLE]

where the $(\Lambda_{O}(G),\Lambda(G))$-bimodule structure of ${}_{{\operatorname{\Lambda}}_{O}(G)}(V^{*})$ is given by $h.(\lambda\otimes\phi)=h\lambda\otimes\phi$ and $(\lambda\otimes\phi).h=\lambda h\otimes\phi\circ(\rho(h))$), the left action of $\Lambda_{O}(G)$ on $V^{*}\otimes_{\mathbb{Z}_{p}}P$ is given by $h.(\phi\otimes p)=\phi\circ(\rho(h^{-1}))\otimes hp$, the first isomorphism is induced by ${}_{{\operatorname{\Lambda}}_{O}(G)}(V^{*})\to(V_{\Lambda_{O}(G)})^{*}$, $\lambda\otimes\phi\mapsto(v\otimes\mu\mapsto\lambda\phi(v)\mu)$ and the second is given by $g\otimes\phi\otimes p\mapsto\phi\circ\rho(g^{-1})\otimes gp$.

$(ii)$* Let us examine the simple case where $P$ is free with basis $e_{1},\dots,e_{n}$ to fix the ideas. The element $[V^{*}\otimes_{\mathbb{Z}_{p}}P,id\otimes f]$ occurring in $(i)$ above has then a more convenient description as follows. Sending $\phi\otimes ge_{i}$ to $\phi\circ\rho(g)\otimes ge_{i}$ realises an isomorphism of $V^{*}\otimes_{\mathbb{Z}_{p}}P$ with $V^{*}\otimes_{\mathbb{Z}_{p}}P$ viewed as a left $\Lambda_{O}(G)$-module via $h.(\phi\otimes\lambda)=\phi\otimes h\lambda$. Through this isomorphism $id\otimes f$ translates as the automorphism sending $\phi\otimes\lambda_{j}e_{j}$ to $\iota(\lambda_{j}f(e_{j}))$ where $\iota:V^{*}\otimes_{\mathbb{Z}_{p}}P\to V^{*}\otimes_{\mathbb{Z}_{p}}P$ sends $\phi\otimes ge_{i}$ to $\phi\circ\rho(g)\otimes ge_{i}$. In particular we find that via the determinant isomorphism $K_{1}(O)\simeq O^{\times}$, $\rho([P,f])$ is nothing but the determinant of the automorphism of $O\otimes_{\Lambda_{O}(G)}(V^{*}\otimes_{\mathbb{Z}_{p}}P)\simeq\oplus_{i}V^{*}\otimes e_{i}$ sending $\phi\otimes e_{j}$ to $\sum_{i}\phi\circ\rho(f_{i}(e_{j}))$ if $f(e_{j})=\sum f_{i}(e_{j})e_{i}$.*

Next, following [CFKSV], we consider the extension of $\rho$ to a map

[TABLE]

defined as $\epsilon\rho_{Q_{O}(\Gamma)}$ where:

• the map $\epsilon:Q_{O}(\Gamma)^{\times}\rightarrow L\cup\{\infty\}$ coincides with $\Lambda_{O}(\Gamma)_{I}\to L$, the localization of the augmentation map at the augmentation ideal $I$, and takes the value $\infty$ elsewhere (ie. at the elements of $Q_{O}(\Gamma)$ which are not integral at $I$).

• the map $\rho_{Q_{O}(\Gamma)}:K_{1}({\operatorname{\Lambda}}(G)_{S^{*}})\to Q_{O}(\Gamma)^{\times}$ is defined by composing the obvious localized version of $tw_{\rho}$, $K_{1}(\Lambda(G)_{S^{*}})\to K_{1}(\Lambda_{O}(G)_{S^{*}})$, the fonctioriality map $K_{1}(\Lambda_{O}(G)_{S^{*}})\to K_{1}(Q_{O}(\Gamma))$ and the isomorphism $det_{Q_{O}(\Gamma)}:K_{1}(Q_{O}(\Gamma))\simeq Q_{O}(\Gamma)^{\times}$.

The map $\rho$ is multiplicative in the sense of the usual partial multiplication

[TABLE]

This means that for $x,y\in K_{1}({\operatorname{\Lambda}}(G)_{S^{*}})$:

$(i)$ $\rho(x^{-1})=\rho(x)^{-1}$.

$(ii)$ the formula $\rho(x)\rho(y)=\rho(xy)$ is true as soon as it makes sense, ie. whenever $(\rho(x),\rho(y))\neq(0,\infty),(\infty,0)$.

Lemma 3.5**.**

Let $[P,f]$ in $K_{1}(\Lambda(G)_{S^{*}})$ and consider its dual $[P^{\vee},f^{\vee}]$ where $(-)^{\vee}$ stands for $Hom_{\Lambda(G)_{S^{*}}}(-,\Lambda(G)_{S^{*}})$ viewed as a left $\Lambda(G)_{S^{*}}$-module via the involution $g\mapsto g^{-1}$. If $\rho$ is an Artin representation with contragredient $\rho^{\vee}$ then in $K_{1}(L)$:

[TABLE]

Proof. One easily reduces to the analogous statement for $\Lambda=\Lambda(G)$ in place of $\Lambda(G)_{S^{*}}$. In virtue of Rem. 3.4 $(i)$, it is then sufficient to prove that for a finitely generated projective $P$ over $\Lambda(G)$ we have in $L^{\times}$:

[TABLE]

the action of $\Lambda_{O}$ on $V^{*}\otimes_{\mathbb{Z}_{p}}P$ (resp. $(V^{\vee})^{*}\otimes_{\mathbb{Z}_{p}}Hom_{\Lambda}(P,\Lambda)=V\otimes_{\mathbb{Z}_{p}}Hom_{\Lambda}(P,\Lambda)$) being given by $g.(\phi\otimes p)=\phi\circ\rho(g^{-1})\otimes gp$ (resp. $g.(v\otimes\phi)=\rho(g)v\otimes(p\mapsto\phi(p)g^{-1})$). Since the determinant of an endomorphism and its transpose are equal it is sufficient to build an isomorphism

[TABLE]

identifying $Hom_{L}(id\otimes id\otimes f,L)$ to $id\otimes id\otimes Hom_{\Lambda}(f,\Lambda)$. Let us examine the left term of (13). Let us chose $n$ such that $\rho$ factors through $G_{n}=G/H_{n}$. Denoting $V_{L}=L\otimes_{O}V$ we have the following series of natural isomorphisms

[TABLE]

where the invariants or coinvariants occurring respectively in the second, third, third and fourth term are taken with respect to the following left actions of $G$ or $G_{n}$: $g.(\phi\otimes p)=\phi\circ\rho(g^{-1})\otimes gp$, $g.\psi=(\phi\otimes p\mapsto\psi(\phi\circ\rho(g)\otimes g^{-1}p))$, $g.(v\otimes\phi)=\rho(g)v\otimes(p\mapsto\phi(g^{-1}p))$, $g.(v\otimes\phi)=\rho(g)v\otimes(p\mapsto\phi(g^{-1}p))$. Regarding the second term of (13) we have

[TABLE]

where the coinvariants occurring respectively in the second, third and fourth term are taken with respect to the following left action of $G$, $H_{n}$, $G_{n}$ and $G_{n}$: $g.(v\otimes\phi)=\rho(g)v\otimes(p\mapsto\phi(p)g^{-1})$, $h.\phi=(p\mapsto\phi(p)h^{-1})$, $g.(v\otimes\phi)=\rho(g)v\otimes(p\mapsto\phi(p)g^{-1})$, $g.(v\otimes\phi)=\rho(g)v\otimes(p\mapsto\phi(p)g^{-1})$. We finally obtain (13) by noticing the isomorphism

[TABLE]

The reader may easily check by himself that (13) is compatible with the automorphisms induced by $f$ as desired.

$\square$

The next results concerning the computation of (12) composed with (10) will be useful to us.

Lemma 3.6**.**

Consider an automorphism $f$ of an object $C$ of $D^{p}({\operatorname{\Lambda}}(G)_{S^{*}})$ as well as its determinant $Det_{{\operatorname{\Lambda}}(G)_{S^{*}}}(f)\in K_{1}({\operatorname{\Lambda}}(G)_{S^{*}})$ (cf. (10)). In $Q_{O}(\Gamma)^{\times}$ we have:

[TABLE]

where the tensor product occurring on the right hand side is taken with respect to the right ${\operatorname{\Lambda}}(G)_{S^{*}}$-module structure of $(V_{Q_{O}(\Gamma)})^{*}$ defined using $\Phi$ in the obvious way.

Proof. We can always assume that $C$ is strictly perfect. In this case we have by definition of $\rho_{Q_{O}(\Gamma)}$ and compatibility of $Det$ with scalar extension:

[TABLE]

where $id\otimes f$ is the automorphism of $(V_{Q_{O}(\Gamma)})^{*}\otimes_{{\operatorname{\Lambda}}(G)_{S}^{*}}C\in D^{p}(Q_{O}(\Gamma))$ deduced from $f$. Now $Q_{O}(\Gamma)$ being a field, the complex $(V_{Q_{O}(\Gamma)})^{*}\otimes_{{\operatorname{\Lambda}}(G)_{S^{*}}}C$ is cohomologically perfect, and thus

[TABLE]

$\square$

Lemma 3.7**.**

Let $\epsilon$, $I$, ${\operatorname{\Lambda}}_{O}(\Gamma)_{I}$ as in (12). Consider a finitely generated ${\operatorname{\Lambda}}_{O}(\Gamma)_{I}$-module $M$ together with an endomorphism $f:M\rightarrow M$ such that $Q_{O}(\Gamma)\otimes_{{\operatorname{\Lambda}}_{O}(\Gamma)_{I}}f$ is invertible. The formula

[TABLE]

is true whenever the right hand term makes sense in $L^{\times}\cup\{0,\infty\}$.

Proof. Let $M^{\prime}\subset M$ denote the ${\operatorname{\Lambda}}_{O}(\Gamma)_{I}$-torsion submodule of $M$, and $M^{\prime\prime}:=M/M^{\prime}$. Let us denote $f^{\prime}:M^{\prime}\rightarrow M^{\prime}$, $f^{\prime\prime}:M^{\prime\prime}\rightarrow M^{\prime\prime}$ the endomorphisms induced by $f$.

On the one hand, $Q_{O}(\Gamma)\otimes_{{\operatorname{\Lambda}}_{O}(\Gamma)_{I}}M^{\prime}=0$, and thus

[TABLE]

On the other hand, $M^{\prime\prime}$ is free since ${\operatorname{\Lambda}}_{O}(\Gamma)_{I}$ is a discrete valuation ring. We thus have a short exact sequence and an isomorphism

[TABLE]

both of which are compatible with $f$. Whence equalities:

[TABLE]

[TABLE]

Now $M^{\prime}$ is of finite length, and is thus subject to Koszul duality:

[TABLE]

Since $det_{L}$ isn’t affected by $L$-linear duality, this shows that

[TABLE]

Now, in the formula to be proven, we see that the right hand term makes sense if and only if $det_{L}(Tor_{0}^{{\operatorname{\Lambda}}_{O}(\Gamma)_{I}}(id,f)|Tor_{0}^{{\operatorname{\Lambda}}_{O}(\Gamma)_{I}}(L,M^{\prime}))$ is non zero (ie. if and only if $det_{L}(Tor_{1}^{{\operatorname{\Lambda}}_{O}(\Gamma)_{I}}(id,f)|Tor_{0}^{{\operatorname{\Lambda}}_{O}(\Gamma)_{I}}(L,M))$ is non zero) in which case the factors $det_{L}(Tor_{q}^{{\operatorname{\Lambda}}_{O}(\Gamma)_{I}}(id,f)|Tor_{0}^{{\operatorname{\Lambda}}_{O}(\Gamma)_{I}}(L,M^{\prime}))$, $q=0,1$, cancel each other and the desired formula reduces to the obvious equality

[TABLE]

$\square$

4. Selmer complexes and crystalline cohomology

In this section we define Selmer complexes for abelian varieties over a one variable function field. We begin with their basic properties and a duality theorem. In the semistable case we pursue with a review of the main result of [TV], which will be the cornerstone of our proof for the Iwasawa main conjecture.

4.1. Preliminaries

Let us begin with some technical facts regarding derived categories.

4.1.1**.**

We will frequently use derived categories of the type $D(\cal C^{\cal I})$, where $\cal C^{\cal I}$ denotes the category of contravariant functors $\cal I\rightarrow\cal C$, $\cal C$ being an abelian category. Note that $\cal CI$ is an abelian category. If $\cal C$ has enough injectives and it has products indexed by the subsets of $Ob(\cal I)$, then the same holds for $\cal CI$. If $\cal C$ is the category of modules on a ringed topos $(E,A)$, $Mod(E,A)I$ naturally identifies with the category of modules on the topos $(EI,A)$. This point of view offers the possibility to consider the more general category $Mod(E^{\cal I},A_{.})$, with $A_{.}$ a projective system of rings of $E$ indexed by $\cal I$.

These categories are especially useful for some particular choices of $\cal I$ which we explain now.

• Let $\cal I=\mathbb{N}$ viewed as the category where $Hom(k,k^{\prime})$ has one element if $k\leq k^{\prime}$ and is empty otherwise (resp. $\cal I={\mathbb{N}}^{op}$, resp. $\cal I=N_{2}:={\mathbb{N}}\sqcup_{|{\mathbb{N}}|}{\mathbb{N}}^{op}$). In that case, the objects of the category $\cal CI$ are the projective systems of $\cal C$ indexed by integers (resp. the inductive systems of $\mathcal{C}$ indexed by integers, resp. the triples $((A_{k})_{k},(p_{k^{\prime},k})_{k^{\prime}\geq k},(i_{k,k^{\prime}})_{k\leq k^{\prime}})$ where $((A_{k})_{k},(p_{k^{\prime},k})_{k^{\prime}\geq k})$ is in $\cal C^{\mathbb{N}}$ and $((A_{k})_{k},(i_{k,k^{\prime}})_{k\leq k^{\prime}})$ is in $\cal C^{{\mathbb{N}}^{op}}$).

• Let $\cal I=[1]^{a}$ for some integer $a\geq 1$, $[1]$ denoting the category $\{0,1,\leq\}$. In that case, we think of objects of $D(\cal C^{\cal I})$ as $(a+1)$-uple naive complexes which are zero outside a specified range of the form $]-\infty,+\infty[\times[i_{1},i_{1}+1]\times\dots,\times[i_{a},i_{a}+1]$. The interest of this category lies in the fact that forming total complexes (with an appropriate sign convention which the interested reader is invited to specify) give rise to a triangulated functor $Tot_{a+1}:D(\cal C^{[1]^{a}})\rightarrow D(\cal C)$.