Arithmetic Cohomology Groups

Arithmetic Cohomology Groups

K. Sugahara

L. Weng

Abstract

We first introduce global arithmetic cohomology groups for quasi- coherent sheaves on arithmetic varieties, adopting an adelic approach.

Then, we establish fundamental properties, such as topological duality and inductive long exact sequences, for these cohomology groups on arith- metic surfaces. Finally, we expose basic structures for ind-pro topologies on adelic spaces of arithmetic surfaces. In particular, we show that these adelic spaces are topologically self-dual.

Contents

1 Arithmetic Adelic Complexes 5

1.1 Parshin-Beilinson’s Theory . . . 5

1.1.1 Local fields for reduced flags . . . 5

1.1.2 Adelic cohomology theory . . . 6

1.1.3 An example . . . 8

1.2 Arithmetic Cohomology Groups . . . 9

1.2.1 Adelic rings for arithmetic surfaces . . . 9

1.2.2 Adelic spaces at infinity . . . 10

1.2.3 Arithmetic adelic complexes . . . 11

1.2.4 Cohomology theory for arithmetic curves . . . 13

2 Arithmetic Surfaces 15 2.1 Local Residue Pairings . . . 15

2.1.1 Residue maps for local fields . . . 15

2.1.2 Local residue maps . . . 16

2.2 Global Residue Pairing . . . 17

2.2.1 Global residue pairing . . . 17

2.2.2 Non-degeneracy . . . 17

2.3 Adelic Subspaces . . . 18

2.3.1 Level two subspaces . . . 18

2.3.2 Perpendicular subspaces . . . 19

2.4 Arithmetic Cohomology Groups . . . 25

2.4.1 Definitions . . . 25

2.4.2 Inductive long exact sequences . . . 26

2.4.3 Duality of cohomology groups . . . 28

3 Ind-Pro Topology in Dimension Two 29

3.1 Ind-pro topologies on adelic spaces . . . 29

3.1.1 Ind-pro topological spaces and their duals . . . 29

3.1.2 Adelic spaces and their ind-pro topologies . . . 32

3.1.3 Adelic spaces are complete . . . 33

3.1.4 Adelic spaces are compact oriented . . . 35

3.1.5 Double dual of adelic spaces . . . 35

3.2 Adelic spaces and their duals . . . 37

3.2.1 Continuity of scalar products . . . 37

3.2.2 Residue maps are continuous . . . 38

3.2.3 Adelic spaces are self-dual . . . 38

3.2.4 Proof of cohomological duality . . . 40

Introduction

In the study of arithmetic varieties, cohomology theory has been developed along with the line of establishing an intrinsic relation between arithmetic Euler characteristics and arithmetic intersections. For examples, for an arithmetic curve SpecOF associated to the integer ring OF of a number field F with discriminant ∆_F and a metrized vector sheaf E on it, we have the Arakelov- Riemann-Roch formula

χar(F, E) = deg_ar(E)−rankE

2 log|∆F|;

And, for a regular arithmetic surfaceπ:X →SpecOF and a metrized line sheaf Lon it, if we equip withX_∞a K¨ahler metric, and line sheavesλ(L) andλ(OX) with the Quillen metrics, namely, equip determinants of relative cohomology groups with determinants of L²-metrics modified by analytic torsions, then we have the Faltings-Deligne-Riemann-Roch isometry

λ(L)^⊗2⊗λ(OX)^⊗−2≃ ⟨L,(L ⊗K^⊗−_π ¹)⟩;

More generally, for higher dimensional arithmetic varieties, we have the works of (Bismut-)Gillet-Soul´e.

In this paper, we start to develop a genuine cohomology theory for arith- metic varieties, as a continuation of the works of Parshin ([P1,2]), Beilinson ([B]), Osipov-Parshin ([OP]), and our own study ([W]). Our aims here are to construct arithmetic cohomology groupsH_arⁱ for quasi-coherent sheaves on arith- metic varieties, and to establish topological dualities among these cohomology groups for arithmetic surfaces.

The approach we take in this paper is an adelic one. Here, we use three main ideas form the literatures. Namely, the first one of adelic complexes initiated in the classical works [P1,2] and [B], (see also [H]), which is recalled in§1.1 and used in §1.2 systematically; the second one of ind-pro structures over adelic spaces from [OP], which is recalled in§1.2.1 and motivates our general constructions in

§1.2.2; and the final one on the uniformity structure between finite and infinite adeles from [W], which is recalled in§1.2.4 and plays an essential role in§1.2.3

when we construct our adelic spaces. In particular, we are able to introduce arithmetic adelic complexes (A^∗ar(X,F), d^∗) for quasi-coherent sheaves F over arithmetic varietiesX, and hence are able to define their associated arithmetic cohomology groupsH_arⁱ (X,F) :=Hⁱ(A^∗ar(X,F), d^∗). Consequently, we have the following

Theorem I. Let X be an arithmetic variety and F be a quasi-coherent sheaf on X, then there exist a natural arithmetic adelic complex (A^∗ar(X,F), d^∗)and hence arithmetic cohomology groups H_arⁱ (X,F) := Hⁱ(A^∗ar(X,F), d^∗). In par- ticular,H_arⁱ (X,F) = 0unless i= 0,1, . . . ,dimX_ar.

To understand this general cohomology theory in down-to-earth terms, in section two, we develop a much more refined cohomology theory for Weil divisors D over arithmetic surfaces X. This, in addition, is based on a basic theory for canonical ind-pro topologies over arithmetic adelic spaces. Recall that, by definition,

A^arX :=A^arX(OX)≃ lim

−→D lim

←−E E≤D

A^arX,12(D)/

A^arX,12(E).

Here A^ar_X,12(D) is one of the level two subspaces of A^ar_X introduced in §2.3.1.

Moreover, for divisorsE ≤D, A^arX,12(D)/

A^arX,12(E) are locally compact. Thus, using first projective then inductive limits, we obtain a canonical ind-pro topol- ogy onA^arX. In particular, we have the following natural generalization of topo- logical theory for one dimensional adeles (see e.g., [Iw], [T]) to dimension two.

Theorem II. Let X be an arithmetic surface. With respect to the canonical ind-pro topology on A^arX, we have

(1)A^arX is a Hausdorff, complete, and compact oriented topological group;

(2)A^arX is self-dual. That is, as topological groups, [A^arX ≃ A^arX.

With these basic topological structures exposed, next we apply them to our cohomology groups. For this, we first recall an arithmetic residue theory in

§2.1, by adopting a very precise approach of Morrow [M1,2], a special case of a general theory on residues of Grothendieck (see e.g., [L], [B] and [Y]). Then, we introduce a global pairing⟨·,·⟩ω in§2.2 on the arithmetic adelic spaceA^arX, and prove the following

Proposition A. Let X be an arithmetic surface and ω be a non-zero rational differential on X. Then, the natural residue pairing ⟨·,·⟩ω:A^arX×A^arX →S¹ is non-degenerate.

Moreover, we construct the so-called level two adelic subspacesA^arX,01,A^arX,02

and A^arX,12(D) of A^arX in §2.3.1. Accordingly, we calculate their perpendicular subspaces with respect to our global residue pairing.

Proposition B.LetX be an arithmetic surface, D be a Weil divisor andω be a non-zero rational differential on X. Denote by (ω) the canonical divisor on X associated to ω. Then we have

(1) Level two subspaces A^arX,01,A^arX,02 andA^arX,12(D)are closed in A^arX; (2) With respect to the residue pairing ⟨·,·⟩ω,

(A^ar_X,01)_⊥

=A^ar_X,01, (

A^ar_X,02)_⊥

=A^ar_X,02, and (

A^ar_X,12(D))_⊥

=A^ar_X,12((ω)−D).

Our lengthy proof for (2) is based on the residue formulas for horizontal and vertical curves on arithmetic surfaces established in [M2]. Moreover, as one can find from the proof of this theorem in §2.3.2, all the level two adelic subspaces A^arX,01,A^arX,02 andA^arX,12(D) are characterized by these perpendicular properties as well. As for (1), our proof in§3.1.3 uses a topological notion of completeness in an essential way.

With the help of these level two subspaces, now we are ready to write down the adelic complex of §1.2.3 and hence its cohomology groupsH_arⁱ (X,OX(D)) associated to the line bundleOX(D) on an arithmetic surfaceX very explicitly.

Indeed, according to §1.2.3, or more directly, following [P], we arrive at the following central

Definition. LetX be an arithmetic surface andDbe a Weil divisor onX. We define arithmetic cohomology groupsH_arⁱ (X,OX(D))for the line bundleOX(D) on X,i= 0,1,2, by

H_ar⁰(X,OX(D)) :=A^arX,01∩A^arX,02∩A^arX,12(D);

H_ar¹(X,OX(D)) :=((

A^arX,01+A^arX,02

)∩A^arX,12(D)

)/(A^arX,01∩A^arX,12(D) +A^arX,02∩A^arX,12(D) )

; H_ar²(X,OX(D)) :=A^arX,012

/(A^arX,01+A^arX,02+A^arX,12(D) )

.

Similar to usual cohomology theory, these cohomology groups admit a natu- ral inductive structure. For details, please refer to Propositions 17, 18 in§2.4.2.

Moreover, induced from the canonical ind-pro topology on A^arX, we obtain nat- ural topological structures for our cohomology groups, since, from Proposition B(1) above, the subspaces A^arX,01,A^arX,02 and A^arX,12(D) are all closed. Conse- quently, as one of the main results of this paper, with the use of Theorem II above, in§3.2.4, we are able to establish the following

Theorem III.LetX be an arithmetic surface with a canonical divisorKX and D be a Weil divisor on X. Then, as topological groups,

H_arⁱ (X,\OX(D))≃H_ar²⁻ⁱ(X,OX(KX−D)) i= 0,1,2.

Our theory is natural and proves to be very useful. For example, as recalled in §1.2.4, in [W], based on Tate’s thesis ([T]), for a metrized bundle E on an arithmetic curve SpecOF, we are able to prove a refined arithmetic duality:

h¹_ar( X, E)

= h⁰_ar(

X, K_X⊗E^∨) , and establish ‘the’ arithmetic Riemann-Roch theorem:

h⁰_ar( X, E)

−h¹_ar( X, E)

= deg_ar(E)−rankE

2 log|∆F|,

(wherehⁱ denotes the arithmetic count ofH_arⁱ ,) and obtain an effective version of ampleness and vanishing theorem. All this plays an essential role in our studies of non-abelian zeta functions for number fields.

1 Arithmetic Adelic Complexes

1.1 Parshin-Beilinson’s Theory

For later use, we here recall some basic constructions of adelic cohomology theory for Noetherian schemes of Parshin-Beilinson ([P1,2], [B], see also [H]).

1.1.1 Local fields for reduced flags

LetFbe a number field withOFthe ring of integers, andπ:X→SpecOFbe an integral arithmetic variety. By a flagδ= (p₀, p₁, . . . , p_n) onX, we mean a chain of integral subschemes pi satisfyingpi+1∈ {pi}=:Xi; and we callδreduced if dimpi = n−i for each i. For a reduced δ, with respect to each affine open neighborhood U = SpecB of the closed pointpn, we obtain a chain, denoted also by δ with an abuse of notation, of prime divisors on U. Consequently, through processes of localizations and completions, we can associate toδa ring

Kδ :=Cp₀S_p⁻₀¹. . . Cp_nS_p⁻_n¹B.

Here, as usual, for a ringR, anR-moduleM and a prime idealpofR, we write S_p⁻¹M for the localization ofM atS_p=R\p, andC_pM = lim_←n∈NM/pⁿM its p-adic completion.

The ringKδ is independent of the choices ofB. Indeed, following [P2], we can introduce inductively schemesX_i,α^′

i as in the following diagram X₀ ⊃ X₁ ⊃ X₂ ⊃ · · ·

↑ ↑

X₀^′ ⊃ X1,α₁ ⊃ ↑

↑

X_1,α^′ ₁ ⊃ X2,α₂

↑ ...

whereX^′denotes the normalization of a schemeX, andX_i,αidenotes an integral subscheme inX_i^′_−1,α

i−1 which dominatesX_i. In particular,

(i)X_1,α1, being an integral subvariety of the normal schemeX₀^′, defines a dis- crete valuation of the field of rational functions on X₀, whose residue field coincides with the field of rational functions on the normal schemeX_1,α^′

1. (ii) More generally, for a fixedi, 1≤i≤n,Xi,α_i, being an integral subvariety of the normal schemeX_i^′_−1,α

i−1, defines a discrete valuation of the field of rational functions on X_i^′_−1,α

i−1, whose residue field coincides with the field of rational functions on the normal schemeX_i,α^′ _i.

Accordingly, for each collection (α1, α2, . . . , αn) of indices, the chain of field of rational functions K0, K1,α₁, . . . , Kn,α_n defines an n-dimensional local field K_{(α1,...,αn)}and hence an Artin ring

Kδ:=⊕(α1,...,αn)∈ΛδK_{(α1,...,αn)}.

Theorem 1. ([P1,2], [Y]) Let δ= (p0, p1, . . . , pn)be a reduced flag on X, and K_{(α1,...,αn)}be then-dimensional local field associated to the collection of indices (α1, α2, . . . , αn)above. Then, we have

(1) The n-dimensional local field K_{(α1,...,αn)} is, up to finite extension, isomor- phic to

either F_v^′((tn−1))· · ·((t1)), or F_v^′{{tn}} · · · {tm+2}}((tm)). . .((t1)) where F_v^′ denotes a certain finite extension of some v-adic non-Archimedean local field Fv;

(2) The ring Kδ is isomorphic to Kδ. In particular, it is independent of the choices of U.

For example, ifX is an arithmetic surface, and p₁ is a vertical curve, then, up to finite extension, Kδ =F_π(p2){{u}},¹ where udenotes a local parameter of the curvep1at the point p2, andF_π(p2)denotes theπ(p2)-adic number field associated to the closed point π(p2) on SpecOF; on the other hand, if p1 is a horizontal curve, thenKδ =L((t)), wheretis a local parameter ofp1atp2, and L/F is a finite field extension. Indeed,p1 corresponds to an algebraic point on the generic fiberXF ofπ, and Lis simply the corresponding defining field.

1.1.2 Adelic cohomology theory

LetX be a Noetherian scheme, and letP(X) be the set of (integral) points of X (in the scheme theoretic sense). Forp, q∈P(X), ifq∈ {p}, we writep≥q.

LetS(X) be the simplicial set induced by (P(X),≥), i.e., the set ofm-simplices ofS(X) is defined by

S(X)_m:={

(p₀, . . . , p_m)|p_i∈P(X), p_i≥p_i+1} ,

the natural boundary mapsδ_iⁿ are defined by deleting the i-th point, and the degeneracy mapsσⁿ_i are defined by duplicating thei-th point:

δ_i^m:S(X)m→S(X)m−1, (p0, . . . , pi, . . . , pm)7→(p0, . . . ,pˇi, . . . , pm), σ^m_i :S(X)_m→S(X)_m+1, (p₀, . . . , p_i, . . . , p_m)7→(p₀, . . . , p_i, p_i, . . . , p_m).

Denote also by S(X)^red_m the subset of S(X)_m consisting of all non-degenerate simplifies, i.e.,

S(X)^red_m ={

(p0, . . . , pm)∈S(X)m dimpi̸= dimpj ∀i̸=j} .

Forp∈P(X) andManOp-module, set [M]p:= (ip)_∗M, whereip: Spec(Op),→ X denotes the natural induced morphism. Moreover, for K ⊂ S(X)m and a pointp∈P(X), introducepK⊂S(X)m−1 by

pK:={

(p1, . . . , pm)∈S(X)m−1|(p, p1, . . . , pm)∈K} . Then, we have the following

Proposition 2. ([P1,2], [B], see also [H, Prop 2.1.1]) There exists a unique system of functors{A(K,∗)}_K⊂S(X) from the category of quasi-coherent sheaves on X to the category of abelian groups, such that

(i)A(K,·)commutes with direct limits.

1Definition ofF_π(p2){{u}}will be recalled in§2.1.1.

(ii) For a coherent sheaf F on X,

A(K,F) =











∏

p∈K

lim←−lFp

/m^l_pFp, m= 0,

∏

p∈P(X)

lim←−lA(

pK,[Fp

/m^l_pFp]_p)

, m >0.

Here mp denotes the prime ideal associated top.

Consequently, for any quasi-coherent sheafF onX, there exist well-defined adelic spaces

A^mX(F) :=A(

S(X)^red_m ,F) . Clearly, if we introduce Ki₀,...,i_m = {

(p0, . . . , pm) ∈ S(X)m|codim( {pr})

= i_r ∀0≤r≤m}

, and defineAX;i0,...,im(F) :=AX

(K_i0,...,im,F) , then A^mX(F) = ⊕

0≤i₀<···<i_m≤dimX

AX;i₀,...,i_m(F).

Moreover, since A(K,F) ⊂ ∏

(p₀,...,p_m)∈KA(

(p0, . . . , pm),F)

, we sometimes write an element f of A(K,F) as f = (fp₀,...,p_m) or f = (fX₀,...,X_m), where Xi={pi} andfp₀,...,p_m =fX₀,...,X_m ∈A(

(p0, . . . , pm),F) .

To get an adelic complex associated to X, we next introduce boundary maps d^m:A^mX⁻¹(F)→A^mX(F) as in [H, Def 2.2.2]. For K ⊂ S(X)_m and L⊂S(X)_m−1 such thatδ^m_i K⊂L for a certaini, we define a boundary map

d^m_i (K, L,F) : A(L,F)−→A(K,F) as follows.

(a) For coherent sheavesF,

(i) Wheni= 0, forp∈P(X), induced from the morphismF →[Fp/m^l_pFp]pand the inclusion pK ⊂ L, we have the morphisms A(L,F) → A(

L,[Fp/m^l_pFp]p

) and A(

L,[Fp/m^l_pFp]p

) → A(

pK,[Fp/m^l_pFp]p

). Their compositions form a projective system φ^l_p : A(L,F) → A(

pK,[Fp/m^l_pFp]p

). Accordingly, we set d^m₀(K, L,F) :=∏

p∈P(X)lim_←−lφ^l_p;

(ii) Wheni=m= 1, we obtain a projective system induced from the standard morphismsπ^l_p: Γ(

X,[Fp/m^l_pFp]_p)

→A(

pK,[Fp/m^l_pFp]_p)

. Accordingly, we set d¹₁(K, L,F) :=∏

p∈P(X)lim_←−lπ_p^l;

(iii) When i >0, m >0, we use an induction on (i, m). That is to say, we set d^m_i (K, L,F) :=∏

p∈P(X)lim_←−ld^m_i−⁻₁¹(

pK,pL,[Fp/m^l_pFp]p

).

(b) For quasi-coherent sheaves F, first we write F as an inductive limit of coherent sheaves, then we use (a) to get boundary maps for the later, finally we use the fact that in the definition of (a), all constructions commute with inductive limits. One checks (see e.g. [H]) that the resulting boundary map is well-defined.

With this, set d^m:=

∑m i=0

(−1)ⁱd^m_i (

S(X)^red_m , S(X)^red_m−1;F) .

Then we have the following

Theorem 3. ([P1,2], [B], see also [H, Thm 4.2.3]) Let X be a Noetherian scheme. Then, for any quasi-coherent sheaf F overX, we have

(1)(

A^∗X(F), d^∗)

forms a cohomological complex of abelian groups;

(2) Cohomology groups of the complex(

A^∗X(F), d^∗)

coincide with Grothendieck’s sheaf theoretic cohomology groupsHⁱ(

X,F)

. That is to say, we have, for alli, Hⁱ(

A^∗X(F), d^∗)

≃Hⁱ( X,F)

.

1.1.3 An example

LetX be an integral regular projective curve defined over a fieldk. Denote its generic point by η and its field of rational functions byk(X). For a divisorD onX, letOX(D) be the associated invertible sheaf. Then, from definition, the associated adelic spaces can be calculated as follows:

AX;0(OX(D)) =A(

{η},OX(D))

= lim

←−lOX(D)_η/

m^l_ηOX(D)_η= lim

←−l

k(X)/{0}=k(X), AX;1(OX(D)) =AX

({p} |p∈X: closed point},OX(D))

=∏

p∈X

←−limlOX(D)p

/m^l_pOX(D)p= ∏

p∈X

←−limlm⁻_p^ordp(D)/

m⁻_p^ordp(D)+l

=∏

p∈X

m⁻_p^ordp(D)= {

(ap)∈ ∏

p∈X

k(X)pordp(ap) + ordp(D)≥0 }

,

and AX;01(OX(D)) =AX

({η, p|p∈X: closed point},OX(D))

= lim

←−lA(

{p} |p∈X : closed point},[OX(D)_η/

m^l_ηOX(D)_η]_η)

=A(

{p} |p∈X : closed point},[k(X)]η

)

=A(

{p} |p∈X : closed point}, lim

−→EOX(E))

= lim

−→EAX;1(OX(E)) =∪

E

AX;1(OX(E))

= {

(a_p)∈ ∏

p∈X

k(X)_pa_p ∈ Op∀^′p }

.

Remark. To calculate AX;01(OX(D)), when dealing with the constant sheaf [k(X)]η, we cannot use Proposition 2(ii) directly, since [k(X)]η is not coherent.

Instead, above, we first expressed it as an inductive limit of coherent sheaves OX(E) associated to divisorsE, then get the result from the inductive limit of adelic spaces forOX(E)’s. Indeed, if we had used Proposition 2 (ii) directly, then we would have obtained simplyA(

{p} |p∈X : closed point},[k(X)]η

)={0}, a wrong claim.

Clearly,AX;01(OX(D)) is independent ofD. We will write it asAX;01, or simplyAX. Consequently, the associated adelic complex

0−→AX;0(OX(D))⊕AX;1(OX(D)) ^d

−→1 AX;01(OX(D))−→0 is given by

0−→k(X)⊕AX;1(OX(D)) ^d

−→1 AX −→0

whered¹: (a₀, a₁)7→a₁−a₀. Therefore, H⁰(

AX(OX(D)))

=k(X) ∩ AX:1(OX(D)), H¹(

AX(OX(D)))

=AX

/(k(X) +AX;1(OX(D))) .

Note that AX(OX(D)) is simply AX(D) of [S, Ch. 2], or better, [Iw, §4].

We have proved the following

Proposition 4. (See e.g., [S, Ch. 2], [Iw,§4]) For a divisorDover an integral regular projective curve defined over a field k, we have

H⁰(

AX(OX(D)))

=H⁰(

X,OX(D))

, H¹(

AX(OX(D)))

=H¹(

X,OX(D)) .

1.2 Arithmetic Cohomology Groups

LetF be a number field withOF the ring of integers. Denote bySfin, resp. S_∞, the collection of finite, resp. infinite, places of F. Write S =Sfin∪S_∞. Let π:X→SpecOF be an integral arithmetic variety of pure dimensionn+1. That is, an integral Noetherian schemeX, a flat and proper morphismπwith generic fiberXF a projective variety of dimensionnoverF. For each v∈S, we write F_v thev-completion ofF, and for eachσ∈S_∞, we writeX_v:=X×OF SpecF_v and write φ_σ : X_σ → X_F for the map induced from the natural embedding F ,→F_σ. In particular, an arithmetic varietyX consists of two parts, the finite one, which we also denote by X, and an infinite one, which we denote by X_∞. These two parts are closely interconnected.

1.2.1 Adelic rings for arithmetic surfaces

The part of our theory on arithmetic adelic complexes for finite places now be- comes very simple. Indeed, our arithmetic variety X is assumed to be Noethe- rian, so we can apply the theory recalled in §1.1 directly. In particular, for a quasi-coherent sheafF onX, we have well-defined adelic spaces

A^finX;i₀,...,i_m(F) :=AX(Ki₀,...,i_m,F).

So to defineA^arX;i0,...,im(F), we need to understand what happens on X_∞. For this purpose, we next recall Osipov-Parshin’s construction of arithmetic adelic ringA^arX for an arithmetic surface X.

Definition 5. ([OP]) Arithmetic adelic ring of an arithmetic surface Let π:X →SpecOF be an arithmetic surface, i.e., a 2-dimensional arithmetic variety, with generic fiberXF.

(i) Finite adelic ring: From the Parshin-Beilinson theory for the Noetherian schemeX, we define

A^finX :=AX;012(OX) = lim

−→D₁

lim←−

D₂:D₂≤D₁

AX;12(D1)/

AX;12(D1).

Here D_∗’s are divisors onX andAX;12(D) :=AX;12(OX(D_∗))for∗= 1,2;

(ii)∞-adelic ring: Associated to the regular integral curveXF overF, we obtain the adelic ring

AX_F :=AX_F;01(OX_F) = lim

−→D₁

lim←−

D₂:D₂≤D₁

AX_F;1(D1)/

AX_F;1(D1).

Here D_∗’s are divisors onX_F andAXF;1(D) :=AXF;1(OXF(D_∗))for∗= 1,2.

By definition,

A^∞X :=AX_F c⊗_QR:= lim

−→D₁

lim←−

D₂:D₂≤D₁

((AX_F;1(D1)/

AX_F;1(D1))⊗b_QR) .

(iii) Arithmetic adelic ring: The arithmetic adelic ring of an arithmetic surface X is defined by

A^arX:=A^arX;012:=A^finX

⊕A^∞X.

The essential point here is, for divisorsD_i, i= 1,2,over the curveX_F, when D₂≤D₁, the quotientAX;1(D₁)/

AX;1(D₁) is a finite dimensionalF- and hence Q-vector space.

To help the reader understand this formal definition in concrete terms, we add following examples.

Example 1. OnX =P¹_Z We haveX_Q=P¹_Q andQ(

P¹_Q)

=Q(t). Easily, Q((t))⊗_QR̸=R((t)).

However, sinceQ((t)) = lim_−→

n

lim_←−

m:m≤n

t⁻ⁿQ[[t]]/

t^−mQ[[t]] and theQ-vector spaces t⁻ⁿQ[[t]]/

t^−mQ[[t]] are finite dimensional, we have Q((t))c⊗_QR= lim

−→n

lim←−

m:m≤n

(t⁻ⁿQ[[t]]/

t^−mQ[[t]])

⊗_QR

= lim

−→n

lim←−

m:m≤n

(t⁻ⁿR[[t]]/

t^−mR[[t]])

=R((t)).

Example 2. Over an arithmetic surfaceX

For a complete flag (X, C, x) onX (with Can irreducible curve on X and xa close point onC), letk(X)_C,xits associated local ring. By Theorem 1,k(X)_C,x is a direct sum of two dimensional local fields. Denote byπ_Cthe local parameter defined byC in X. Then

A^finX =AX,012= ∏

x∈C

′k(X)C,x:=∏

C

′(∏′

x:x∈Ck(X)C,x

)

:={( ∑^∞

i_C=−∞

hC(aiC)πⁱ_C^C)

C∈∏

C

( ∏

x:x∈C

k(X)C,x

):

a_iC ∈AC,01, a_iC = 0 (i_C≪0); min{i_C:a_iC ̸= 0} ≥0 (∀^′C) }

, wherehC is a lifting defined in [MZ], which we call the Madunts-Zhukov lifting.

For details, please see§3.1.2.

1.2.2 Adelic spaces at infinity

Now we are ready to treat adelic spaces at infinite places for general arithmetic varieties. Motivated by the discussion above, we make the following

Definition 6. Let π : X →SpecOF be an arithmetic variety. Let S(X_F) be the simplicial set associated to its generic fiberX_F and K⊂S(X_F)_m, m≥0, a subset.

(i) LetG be a coherent sheaf on XF. We define the associated adelic spaces by

A^∞(K,G) :=











∏

p∈K

lim←−l

(Gp/m^l_pGp⊗b_QR)

, m= 0

∏

p∈P(X)

lim←−lA^∞(

pK,[Gp/m^l_pGp]p

), m >0.

(ii) Let{Gi}ibe an inductive system of coherent sheaves onXF andF = lim

−→iGi. Then we define

A^∞(K,F) := lim

−→iA^∞(K,Gi).

Clearly, the essential part of this definition is the one for m = 0. More- over, if F = lim

−→iGi^′ is another inductive limit of coherent sheaves, we have

−→limiA^∞(K,Gi^′)≃lim

−→iA^∞(K,Gi), by the universal property of inductive limits since Gp/m^l_pGp’s are Q-vector spaces. Therefore, A^∞(K,F) is well-defined for all quasi-coherent sheaves F on XF. Moreover, as a functor from the cate- gory of coherent sheaves on XF to that of Q-vector spaces, A^∞(K,∗) is ad- ditive and exact. Hence, by [H, §1.2], A^∞(K,∗) commutes with the direct limits, even in general, for an inductive system{Fi}i of quasi-coherent sheaves

−→limiA^∞(K,Fi)̸=A^∞(K,lim

−→iFi).

1.2.3 Arithmetic adelic complexes

As mentioned at the beginning of this section, for arithmetic varieties, the finite and infinite parts are closely interconnected. Therefore, when developing an arithmetic cohomology theory, we will treat them as an unify one using an uniformity condition.

Let X be an arithmetic variety with generic fiber XF. For a point P of XF, denote its associated Zariski closure in X by EP. We call a flag δ = (p0,p1, . . . ,pk)∈S(X)horizontal, if there exists a flagδF = (P0, P1, . . . , Pk)∈ S(XF) such that (p₀,p₁, . . . ,p_k) = (EP₀, EP₁, . . . , EP_k). Accordingly, for K ⊂ S(X), we denoteK^h the collection of all horizontal flags inK andK^nh=K∖ K^h. Simply put, our uniformity condition is a constrain on adelic components associated to horizontal flags.

LetFbe a quasi-coherent sheaf onX, denote its induced sheaf on the generic fiberX_F byFF. It is well-known thatFF is quasi-coherent as well. Motivated by [W], we introduce the following

Definition 7. LetX be an arithmetic variety of dimensionn+1andFa quasi- coherent sheaf onX. Fix an index tuple(i0, . . . , im)satisfying i0≤ · · · ≤im. (i) The finite, resp. infinite, adelic space of type (i0, . . . , im) associated toF is defined by

A^finX;i₀,...,i_m(F) :=AX

(KX;i₀,...,i_m,F)

=A^finX

(K_X;^nh_i0,...,im,F)

⊕A^finX

(K_X;^h _i0,...,im,F) , resp. A^∞X;i0,...,im(F) :=A^∞X

(KX_F;i₀,...,i_m,FF

).

Here, forZ ⊂X or X_F, we set K_Z;i0,...,im :={

(p₀, . . . , p_m)∈S(Z)_mcodim_Z{p_r}=i_r ∀0≤t≤m}

; (ii) The arithmetic adelic space of type (i0, . . . , im) associated to F is defined by

A^arX;i0,...,im(F) =:





A^finX;i₀,...,i_m(F)⊕

A^∞X;i₀,...,i_m−1(FF), im=n+ 1;

A^finX

(K_X;^nh_i

0,...,i_m,F)

⊕A^fin,inf_X ( K_X;^h _i

0,...,i_m,F)

, im̸=n+ 1 where

A^fin,infX

(K_X;^h _i0,...,im,F)

⊂A^finX

(K_X;^h _i0,...,im,F) ⊕

A^∞X;i₀,...,i_m(FF) consisting of adeles satisfying, for all flags(pi₀,pi₁, . . . ,pi_m)∈KX_F;i₀,...,i_m,

fE_pi

0,E_pi

1,...,E_p

im =fp_i0,p_i1,...,p_im;

(iii) Form≥0, define them-th reduced arithmetic adelic space A^arX;m(F)of F by

A^mar,red(X,F) := ⊕

(i₀,...,i_m) 0≤i₀<i₁<···<i_m≤n+1

A^arX;i₀,...,i_m(F).

Remarks. (i) For any p ∈ P(XF), OX,E_p = OX_F,p and k(X)E_p = k(XF)p. Consequently, for any (p0, . . . ,pm)∈S(XF)m, we have a natural morphism

A(

(Ep₀, . . . , Ep_m),F)

=A(

(p0, . . . ,pm),FF

).

sinceFis quasi-coherent. It is in this sense we use the relationfE_p0,E_p1,...,E_pm = fp₀,p₁,...,p_m above. (In particular, if pi’s are vertical, there are no conditions on the corresponding components.) Clearly, this uniformity condition is an essential one, since it characters the natural interconnection between finite and infinite components of arithmetic adelic elements.

(ii) In part (ii) of the definition, we need the spaceA^∞_X;∅(FF). Here, to complete our definition, for an arithmetic varietyX, we viewA^∞_X;∅(FF) as the (−1)-level of the adelic complex for its generic fiber XF. That is to say, we define it as follows. By [Y, p. 63], we have the (-1)-simplex 1_U for open U ⊂X. Set then S(XF)₋1={1_U |U ⊂X : open}, and, forK⊂S(XF)₋1, let

A^∞X;∅(K,FF) :=

{FF(U_K,F)⊗QR, dimX ≥2 {sF ∈ FF(UK,F)⊗R|s∈ F(UK)}, dimX = 1 where UK := ∪1_U∈KU and sF denotes the section induced by s. The reason for separation of arithmetic curves with others in this latest definition is that arithmetic varieties are relative over arithmetic curves.

Moreover, from standard homotopy theory, if we introduce the boundary morphisms by

d^m_i : ⊕

A^arX;l0,...,l_m−1(F) −→ ⊕

A^arX;k0,...,km(F) (al₀,...,l_m−1) 7→ (a_k

0,...,ˆk_i,...,k_m);

and dm=∑m

i=0(−1)ⁱd^m_i : ⊕

A^arX;k₀,...,l_m−1(F) −→ ⊕

A^arX;k₀,...,k_m(F),we have