The Kernel of the Reciprocity Map of Simple Normal Crossing Varieties over Finite Fields

The Kernel of the Reciprocity Map of Simple Normal Crossing Varieties over Finite Fields

by

PatrickForré

Abstract

For a smooth and proper varietyY over a finite fieldkthe reciprocity mapρ^Y : CH0(Y)→ π^ab₁ (Y)is injective with dense image. For a proper simple normal crossing variety this is no longer true in general. In this paper we give a description of the kernel and cokernel of the reciprocity map in terms of homology groups of a complex filled with descent data using an algebraic Seifert–van Kampen theorem. Furthermore, we give a new criterion for the injectivity of the reciprocity map for proper simple normal crossing varieties over finite fields.

2010 Mathematics Subject Classification:11G25, 11G45.

Keywords:higher dimensional class field theory, reciprocity map, abelianized fundamental group, Seifert–van Kampen theorem, simple normal crossing varieties over finite fields.

§1. Introduction

The class field theory of smooth, proper varietiesY over finite fieldsk was devel- oped by S. Lang, S. Bloch, K. Kato and S. Saito. To determine the structure of the abelianized étale fundamental groupπ^ab₁ (Y)classifying the finite abelian étale coverings ofY, the following reciprocity map was considered.

Definition 1.1 (The reciprocity map). Let Y be a scheme of finite type over Z and π₁^ab(Y) =L

Y⁰∈π0(Y)π^ab₁ (Y⁰) the abelianized étale fundamental group ofY. Let y ∈ Y be a closed point. Then the residue field κ(y) is a finite field. There- fore we can consider the image of the Frobenius automorphism ϕy ∈ G_κ(y) = π^ab₁ (Spec(κ(y)))inπ^ab₁ (Y)via the push-forward of the natural map

iy: Spec(κ(y))→Y.

Communicated by A. Tamagawa. Received November 24, 2011. Revised March 6, 2012.

This is an extract from the author’s PhD thesis [For11].

P. Forré: Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, 93053 Regensburg, Germany;

e-mail:patrick.forre@mathematik.uni-regensburg.de

c 2012 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

This mapping extends linearly to the group of zero-cycles ofY: ρ⁰:Z₀(Y) = M

y∈Y(0)

Z·y→π^ab₁ (Y), X

y

n_y·y7→X

y

n_y·(i_y)_∗(ϕ_y).

ρ⁰ is called thereciprocity map ofY.

Theorem 1.2. 1. Let Y be a proper scheme over a finite fieldk. The reciprocity map ρ⁰ factors through rational equivalence to give a map from the Chow group of zero-cycles(cf. [Ful98, §1.3]):

ρ: CH₀(Y)→π^ab₁ (Y), which is also called the reciprocity map ofY.

2. Let Y be a smooth and proper variety over a finite fieldk. Then the reciprocity map ρ: CH0(Y)→π₁^ab(Y) is injective and the cokernel of ρis isomorphic to (ˆZ/Z)^π0(Y), which is a uniquely divisible group. Moreover, we have a commu- tative diagram of exact sequences

0 //A0(Y) //

 _

∼ ρ₀

CH0(Y) ^deg //

 _

ρ

Z^π0(Y _ ⁾

incl

0 //π₁^geo(Y) //π₁^ab(Y) ^deg //Zˆ^π0(Y)

where the restrictionρ0 of the reciprocity map induces an isomorphism of finite groups between the kernelsA0(Y)andπ^geo₁ (Y)of the degree maps.

Note that the degree maps for connected schemesY are defined as follows:

deg :π^ab₁ (Y)→Zˆ is just the push-forward mapπ^ab₁ (Y)→π^ab₁ (Spec(k))composed with the isomorphism π₁^ab(Spec(k)) ∼= ˆZ, sending the Frobenius automorphism of kto 1. The mapdeg : CH0(Y)→Z is given by sending a cycleP

iai·[xi]to P

iai·[κ(x) :k], which is well-defined for proper schemesY overkand coincides in this case with the proper push-forward mapCH₀(Y)→CH₀(Spec(k))(cf. [Ful98,

§1.4]).

Proof. See [Lan56], [Blo81], [KS83], and [Ras95, §5] for a summary. We also men- tion [Wie06], [Wie07], [JS03], [KS09], [KS10].

In the study of varieties over local fields (cf. e.g. [Sat05], [JS03]) one is con- fronted with the reciprocity map of simple normal crossing varieties over finite fields.

Definition 1.3 (Simple normal crossing varieties). Let k be a field and Y an equidimensional and separated scheme of finite type over k. Then Y is called

anormal crossing variety overkifY is everywhere étale locally isomorphic to Spec(k[T0, . . . , Td]/(T0T1· · ·Tr)),

withd= dimY and some 0≤r≤d.

A normal crossing varietyY is calledsimple if any irreducible component of Y is smooth overk.

The main results in this paper concerning the reciprocity map of simple normal crossing varieties over finite fields are summarized in the following theorem. In the proof, an algebraic Seifert–van Kampen theorem [Sti06] is used, leading to a more explicit understanding of the kernel of the reciprocity map, in contrast to [JS03]

using étale homology theory and cohomological Hasse principles. Also here the characteristic ofkposes no problem.

Theorem 1.4. Let Y =Sm

i=1Yi be a proper simple normal crossing variety over a finite field kwith irreducible components Y_i ,→Y. Let

Y^[k] := a

i0<i1<···<ik

Yi₀×Y Yi₁×Y · · · ×Y Yi_k

be the disjoint union of thek-fold intersections,k≥0, andΓY be the corresponding dual complex. Consider the complex

π^ab₁ (Y^[•]) : · · ·−→^d2 π₁^ab(Y^[1])−→^d1 π^ab₁ (Y^[0])−→^d0 π^ab₁ (Y) withd_k :=Pk

j=0(−1)^j(δ_j^k)_∗.

1. The cokernel and kernel of the reciprocity map ρ^Y : CH0(Y)→π^ab₁ (Y)

are given by

coker(ρ^Y)∼= H−1(π^ab₁ (Y^[•]))⊕(ˆZ/Z)^π0(Y), ker(ρ^Y)∼= H0(π₁^ab(Y^[•])), where the last group is finite.

2. There is also an exact sequence H₂(Γ_Y,Zˆ)→CH₀(Y) ^ρ

Y

−−→π₁^ab(Y)→(ˆZ/Z)^π0(Y)⊕H₁(Γ_Y,Zˆ)→0.

3. If, furthermore, every component ofY^[0] and ofY^[1]is geometrically connected overk andCH₀(Y^[0])is torsion-free, thenρ^Y is injective.

The last point gives a new criterion for the injectivity of the reciprocity map for simple normal crossing varieties which does not use the vanishing of the second homology groupH₂(Γ_Y,ˆ

Z)of the dual complex.

The next sections will lead to the proof of the theorem above.

§2. An algebraic Seifert–van Kampen theorem

In this section we will introduce the dual complex of a variety carrying informa- tion about how the components intersect. Furthermore, we will cite an algebraic Seifert–van Kampen theorem and compute the abelianized version. The missing information is then controlled by the first and second homology groups of the dual complex.

Definition 2.1 (The dual complex). Let(I, <)be a totally ordered set. LetY = S

i∈IY_ibe a locally noetherian scheme with closed subschemesY_i,→Y. Fork≥0 we let

Y^[k]:= a

i₀<i₁<···<ik

Y_i0,i1,...,ik

be the disjoint union of

Y_i0,i1,...,ik:=Y_i0×_Y Y_i1×_Y . . .×_Y Y_ik,

thek-fold scheme-theoretic intersection of theYi, andY^[−1]:=Y. For instance we have

Y^[0]= a

i₀∈I

Yi0 and Y^[1]= a

i₀<i₁∈I

Yi0∩Yi1.

For every integerk≥1there arek+ 1morphisms

δ_j^k:Y^[k]→Y^[k−1] forj= 0, . . . , k given by the closed immersions

δ^k_j :Yi₀,...,i_k,→Y_i

0,...,iˆj,...,ik,

where the indexi_j∈Iis omitted on the right side. Forj= 0, . . . , kwe get induced maps

∂_j^k:π₀(Y^[k])→π₀(Y^[k−1])

on the connected components. Notice that we also have a canonical maph=δ⁰₀: Y^[0]→Y =Y^[−1] and the corresponding induced maps.

ThereforeΓ := (π0(Y^[•]),(∂_j^•)j)is a semi-simplicial complex, called thedual complex to(Y,(Yi)_i∈I,(I, <)). The elements ofπ0(Y^[k])are calledk-simplices.

The homology groups of the complex

C(Γ, A) : · · ·−→^d3 A^(π0(Y[k])) −→^d2 A^(π0(Y[1]))−→^d1 A^(π0(Y[0])) withdk:=Pk

j=0(−1)^j∂_j^kwill be denoted byHk(Γ, A) := ker(dk)/im(dk+1). Notice thatA^(π0(Y[k])) is placed in degreek.

Theorem 2.2 (Algebraic Seifert–van Kampen Theorem). Let (I, <) be a finite and totally ordered set and Y = S

v∈IYv be a locally noetherian and connected scheme with closed and connected subschemesYv,→Y. Leth:Y^[0]=`

v∈IYv→Y be the canonical map.

Letsbe a geometric point ofY and for everyk= 0,1,2and everyt∈π₀(Y^[k]) lets(t)be a geometric point of t. Let Γbe the dual complex to(Y,(Yi)i∈I,(I, <)).

Fix a maximal subtree T of Γ and for every boundary map∂ :t →t⁰ inΓ_≤2 let γt,t⁰ :s(t⁰) Γ(∂)s(t)be a fixed path in the sense of algebraic paths between base points, i.e. a fixed isomorphism between the corresponding fibre functors. Then canonically with respect to all these choices we have an isomorphism

π1(Y, s)∼=

∗

whereH is the closed normal subgroup generated by the edge and cocycle relations:

edg(e, g_e) :=−→e ·π₁(δ¹₀)(g_e)·(−→e)⁻¹·π₁(δ¹₁)(g_e)⁻¹, coc(f) := (−−→

∂₂²f)·α^(f)₁₀₂(α^(f)₁₂₀)⁻¹·(−−→

∂²₀f)·α^(f)₂₁₀(α^(f)₂₀₁)⁻¹·(−−→

∂₁²f)⁻¹·α^(f)₀₂₁(α^(f)₀₁₂)⁻¹, for all parameter valuese∈Γ1,ge∈π1(e, s(e))andf ∈Γ2. Here−→e is defined to be the corresponding topological generators of the profinite group

ˆ

π₁(Γ, T) :=

e∈Γ

∗

Zˆ /(−→

e⁰ |e⁰ ∈T₁)∼=_e∈Γ

∗

1\T1

Zˆ.

The map π1(δ_i¹) uses the fixed path γ_δ1

i(e),e. Finally α^(f)_ijk is defined using the i-th vertex v = v_i(f) ∈ Γ₀ of f and the edge e = e_ij(f) ∈ Γ₁ with vertices {vi(f), v_j(f)}={∂₀¹(e), ∂₁¹(e)}, as

α^(f)_ijk:=γ_{vi(f), eij(f)}◦γ_{eij(f), f} ◦(γ_{vi(f), f})⁻¹∈π1(Yv_i, s(vi)).

Proof. This is a special case of [Sti06, Corollary 5.4] noting that closed immersions are monomorphisms ([Sti06, Definition 4.2]) and using [Sti06, Theorem 5.2(1)] to see that h—a proper, surjective morphism of finite presentation—is a universal effective descent morphism for finite étale covers ([Sti06, Definition 5.1]).

Corollary 2.3 (The abelianized fundamental group). Let the notation be as in 2.2. The abelianized fundamental group of Y is then given by

π^ab₁ (Y) = π₁^ab(Y^[0])⊕ˆπ₁^ab(Γ, T) /H, where

π^ab₁ (Y^[k]) := M

Z∈π0(Y^[k])

π₁^ab(Z),

ˆ

π₁^ab(Γ, T) := M

e∈Γ1

Zˆ

/(−→

e⁰|e⁰∈T1)∼= M

e∈Γ1\T1

Zˆ,

andH is topologically generated by the relations

d1(g) and coc(f) =β(f) +d2(f), where

d1:π^ab₁ (Y^[1])→π₁^ab(Y^[0]), g7→π^ab₁ (δ₀¹)(g)−π₁^ab(δ¹₁)(g), d₂: M

f∈Γ2

Zˆ →πˆ^ab₁ (Γ, T), X

f

n_f·f 7→X

f

n_f·(−−→

∂₀²f−−−→

∂₁²f +−−→

∂²₂f),

β : M

f∈Γ2

Zˆ →π₁^ab(Y^[0]),

X

f

nf·f 7→X

f

nf·(α^(f)₁₀₂−α^(f)₁₂₀+α^(f)₂₁₀−α^(f)₂₀₁+α^(f)₀₂₁−α^(f)₀₁₂),

coc : M

f∈Γ2

Zˆ →π₁^ab(Y^[0])⊕πˆ₁^ab(Γ, T), X

f

nf·f 7→X

f

nf· β(f) +d2(f) ,

with all the abelian groups written additively. Furthermore, for an abelian groupA consider the complex

π^ab₁ (Y^[•])⊗_ZA: π₁^ab(Y^[1])⊗_ZA−→^d1 π^ab₁ (Y^[0])⊗_ZA−→^d0 π₁^ab(Y)⊗_ZA with homology groupsH_i(π₁^ab(Y^[•]), A). We then have an isomorphism

H1(Γ, A)∼= H−1(π₁^ab(Y^[•]), A) := coker(d0),

and a surjection

H2(Γ, A)

β_|

H0(π₁^ab(Y^[•]), A) := ker(d0)/im(d1).

Proof. The first statement immediately follows from 2.2 by abelianization. For clarity in the description of the homology groups we suppress the terms ⊗_ZA in every line. Nowd₀is the canonical map

d0:π^ab₁ (Y^[0])→π₁^ab(Y) = π₁^ab(Y^[0]) ⊕ ˆπ₁^ab(Γ, T) /H.

So we get

H−1(π₁^ab(Y^[•]), A) = coker(d0)

= π₁^ab(Y^[0])⊕πˆ^ab₁ (Γ, T)

/ π^ab₁ (Y^[0]) +H

= π₁^ab(Y^[0])⊕πˆ^ab₁ (Γ, T)

/ π^ab₁ (Y^[0])⊕im(d2)

∼= ˆπ₁^ab(Γ, T))/im(d2)

= H₁(Γ, T;A)∼= H1(Γ, A).

The last isomorphism holds becauseT is a maximal subtree ofΓ. Moreover, H₀(π^ab₁ (Y^[•]), A) = ker(d₀)/im(d₁) = π^ab₁ (Y^[0])∩H

/im(d₁).

Now consider the exact sequence 0→ π₁^ab(Y^[0])∩H

/im(d1)→H/im(d1)→^ω π₁^ab(Y^[0]) +H

/π^ab₁ (Y^[0])→0.

BecauseH =him(d₁),im(coc)iwe have a surjection coc : M

f∈Γ2

ZˆH/im(d1).

Furtherd2:L

f∈Γ2

Zˆ→ˆπ₁^ab(Γ, T) factors throughω:

d₂: M

f∈Γ2

Zˆ

−cocH/im(d₁)^ω π^ab₁ (Y^[0]) +H

/π₁^ab(Y^[0])−−→^pr2 πˆ^ab₁ (Γ, T).

Therefore the restriction ofcoctoker(d2), which by definition coincides with the restriction ofβ toker(d₂), induces a well-defined and surjective map

β_|= coc_|: ker(d2) π₁^ab(Y^[0])∩H

/im(d1).

BecauseT is a maximal subtree ofΓ, we haveker(d₂) = ker(d₂). Moreover β : M

f∈Γ2

Zˆ →π₁^ab(Y^[0]),

X

f

nf·f 7→X

f

nf·(α^(f)₁₀₂−α^(f)₁₂₀+α^(f)₂₁₀−α^(f)₂₀₁+α^(f)₀₂₁−α^(f)₀₁₂),

vanishes on the image of

d3: M

Z∈Γ3

Zˆ → M

f∈Γ₂

Zˆ

by definition of theα^(f)_ijk and alternating signs. Therefore we also get a surjection H2(Γ, A) = ker(d2)/im(d3)

β_|

H0(π₁^ab(Y^[•]), A), which finishes the proof.

§3. The reciprocity map for simple normal crossing varieties In this section we will determine the kernel and cokernel of the reciprocity map of simple normal crossing varieties in terms of homology groups of a complex filled with descent data. From this description we will deduce a criterion for the injec- tivity of the reciprocity map without using the vanishing of the second homology groupH2(Γ).

Lemma 3.1. 1. Let f :Y →Spec(k)be a normal scheme which is separated, of finite type and geometrically connected over a finite fieldk. The degree map

deg :Z0(Y)→Z, X

y∈Y0

ay·y7→ X

y∈Y0

ay·[κ(y) :k],

is then surjective.

2. Let Y = S

v∈IYv be a connected scheme which is proper over a finite field k such thatYv are closed normal connected subschemes ofY. Letkv :=OY_v(Yv).

The image of the degree map CH0(Y)→CH0(k)∼=Zis then given by im deg = gcd

v∈I

([k_v:k])·Z.

Proof. 1. Consider the commutative diagram Z0(Y)

deg

ρ⁰ //π^ab₁ (Y)

f∗

Z  //Galk

where Galk ∼= ˆZ and f∗ is surjective by [Sza09, Prop. 5.5.4] because Y is geo- metrically connected. Now let n be the natural number given by nZ = im deg.

By Lang’s theorem (see [Sza09, Thm. 5.8.16], [Mil80, Section VI.12])ρ⁰ has dense image and therefore so doesf_∗◦ρ⁰. ButnZis only dense inˆ

Zif and only ifn= 1.

2. By the Stein factorisation everyY_vis geometrically connected over the finite fieldkv. Now consider the commutative diagram

L

v∈ICH0(Yv) ////

⊕_vdeg_v

CH0(Y)

deg

L

v∈ICH₀(k_v) //CH₀(k)

where⊕_vdeg_vis surjective by the first point and the bottom map is componentwise onZgiven by multiplying with the degrees[k_v:k]and summing up. So the image ofdeg equals the image of the bottom map, which isgcd_v∈I([kv:k])·Z.

Proposition 3.2. Let Y =S

v∈IYv be a proper scheme over a finite fieldkwith closed and connected subschemes Yv ,→ Y which are smooth over k such that Y_v0×_Y Y_v1 are also smooth over k for allv₀, v₁∈I. Let nbe an arbitrary integer and consider the complex

π₁^ab(Y^[•])/n: · · · →π^ab₁ (Y^[1])/n−→^d1 π₁^ab(Y^[0])/n−→^d0 π₁^ab(Y)/n with dk := Pk

j=0(−1)^j(δ^k_j)∗. Then the kernel and the cokernel of the reciprocity map modulon

ρn: CH0(Y)/n→π₁^ab(Y)/n are given by the homology groups of π₁^ab(Y^[•])/n:

ker(ρ_n)∼= H0(π₁^ab(Y^[•])/n), coker(ρ_n)∼= H−1(π^ab₁ (Y^[•])/n).

Furthermore, we have an exact sequence of finite abelian groups H2(Γ,Z/n)→CH0(Y)/n→π^ab₁ (Y)/n→H1(Γ,Z/n)→0, whereΓ is the dual complex to (Y,(Yi)i∈I,(I, <)).

Proof. We have a commutative diagram of complexes CH0(Y^[1])/n ^d

0 1 //

ρ¹_n o

CH0(Y^[0])/n ^d

0 0 //

ρ⁰_n o

CH0(Y)/n //

ρn

0

π^ab₁ (Y^[1])/n ^d1 //π^ab₁ (Y^[0])/n ^d0 //π₁^ab(Y)/n

where the first row is exact in analogy to [Ful98, Ex. 1.3.1, 1.8.1] and the first two vertical maps are isomorphisms by1.2, sinceY^[0]andY^[1]are smooth and proper by assumption and (ˆZ/Z)^π0(Y[k]) is uniquely divisible. By the isomorphisms ρ⁰_n andρ¹_n we havecoker(d1)∼= CH0(Y)/n, and

d₀: coker(d₁)→π₁^ab(Y)/n coincides withρ_n. Therefore we get

ker(ρ_n)∼= ker(d₀) = H₀(π₁^ab(Y^[•])/n), coker(ρn) = coker(d0) = H₋₁(π^ab₁ (Y^[•])/n).

The statement now follows from the abelianized Seifert–van Kampen theorem2.3 withA=Z/n:

H₀(π^ab₁ (Y^[•])/n)∼= H1(Γ,Z/n), H₋₁(π^ab₁ (Y^[•])/n)H₂(Γ,Z/n).

Remark 3.3. The proofs of3.2and2.3show that

ker(ρ_n)∼= im β_|: ker(d₂)→(π₁^ab(Y^[0])/n)/im(d₁) .

Therefore non-vanishing ofβ_|results in a non-trivial kernel of the reciprocity map modulon.

If we could choose the geometric points of2.2such that the pathsγt,t⁰ generate trivial α^(f)_ijk for all parameters, then we get a vanishing β_| and therefore a trivial kernel of the reciprocity map modulonfor every integern.

Notation 3.4. For an abelian group A and a set L of primes let N(L) be the monoid of all natural numbers which have prime divisors only inL. We defineA_L to be theL-completion

A_L:= lim

n∈←−N(L)

A/n,

andAˆto be the Zˆ-completion

Aˆ:= lim

n∈N←−

A/n.

Theorem 3.5 (The reciprocity map and itsL-completion). Let L be a set of prime numbers and letY =S

v∈IY_vbe a proper scheme over a finite fieldkwith a finite number of closed and connected subschemesYv,→Y which are smooth over ksuch that Yv₀×Y Yv₁ are also smooth over kfor allv0, v1∈I. LetΓ be the dual complex to (Y,(Yi)_i∈I,(I, <)). Consider the complex

π₁^ab(Y^[•])_L: · · ·−→^d2 π^ab₁ (Y^[1])_L−→^d1 π^ab₁ (Y^[0])_L−→^d0 π^ab₁ (Y)_L withdk :=Pk

j=0(−1)^j(δ_j^k)_∗ and the reciprocity maps

ρ: CH₀(Y)→π^ab₁ (Y), ρ₀:A₀(Y)→π^geo₁ (Y),

ρ_L: CH0(Y)_L→π^ab₁ (Y)_L, ρˆ:CH0ˆ(Y)→π^ab₁ˆ(Y) =π₁^ab(Y), whereA₀(Y)andπ^geo₁ (Y)are the kernels of the corresponding degree maps.

1. Then the kernel ofρ_L is a finite abelian group and a factor group of H2(Γ,ZL), and satisfies

ker(ρ_L)∼= H0(π₁^ab(Y^[•])_L).

The cokernel ofρ_L satisfies

coker(ρ_L)∼= H−1(π^ab₁ (Y^[•])_L)∼= H1(Γ,ZL).

Therefore, we have an exact sequence of finitely generated ZL-modules:

H₂(Γ,ZL)→CH₀(Y)_L−→^ρL π₁^ab(Y)_L→H₁(Γ,ZL)→0.

2. For every set Lof primes with#A0(Y)∈N(L)we have

ker(ρ_L) = ker(ρ0) = ker(ρ) = ker( ˆρ)∼= H0(π^ab₁ (Y^[•])).

3. The cokernels of ρandρˆsatisfy coker( ˆρ)∼= H−1(π^ab₁ (Y^[•]))∼= H1(Γ,ˆ

Z), coker(ρ)∼= (ˆZ/Z)^π0(Y)⊕H1(Γ,ˆ Z).

Therefore, we have an exact sequence of abelian groups

H2(Γ,Zˆ)→CH0(Y)−→^ρ π^ab₁ (Y)→(ˆZ/Z)^π0(Y)⊕H1(Γ,Zˆ)→0.

Proof. For theL-completion we have the analogous results from3.2by taking the inverse limit with the additional information that for a finitely generated abelian groupA we haveA_L=A⊗_ZZL, so that the universal coefficient theorem

0→H_i(Γ,Z)⊗_ZZL→H_i(Γ,ZL)→Tor^Z₁(H_i−1(Γ,Z),ZL)→0

gives that Hi(Γ,ZL) = Hi(Γ,Z)_L, becauseZL is torsion-free and therefore we get Tor^Z₁(H_i−1(Γ,Z),ZL) = 0.ker(ρ_L)is finite because it lies in the kernel of the degree map (see diagram below), which is finite (cf. [Blo81, Thm. 4.2] and [KS86, Thm.

6.1]). With3.1we have the commutative and exact diagram 0 //ker(ρ0)

 _

∼ //ker(ρ)

 _

//0

Z^π0(Y _ ⁾

incl

0 //A0(Y) //

ρ₀

CH0(Y) ^deg //

ρ

im deg _

//'

55

0 Zˆ^π0(Y)

0 //π^geo₁ (Y)

//π^ab₁ (Y)

deg⁰ //im deg⁰' //

44

0

0 //coker(ρ0) //coker(ρ) //L

W∈π0(Y)Zˆ/mWZ //0 wherem_W := gcd_Y

v⊆W([k_v:k_W])andk_v:=OYv(Y_v)andk_W :=OW(W).

Taking the Zˆ-completion of this diagram, we see that the two middle hori- zontal lines stay exact and that ker(ρ0)∼= ker( ˆρ). Note that A0(Y) and π₁^geo(Y) are profinite groups and do not change underZˆ-completion, i.e. we haveρˆ0=ρ0. Comparing with the original bottom line sequence we get a commutative diagram

of exact sequences 0

//ker(f)

 _

//ker(g)

 _

//

 _

0

0 //coker(ρ0) //coker(ρ)

f

//L

W∈π0(Y)Zˆ/mWZ //

g

0

0 //coker(ρ0) //coker( ˆρ) //L

W∈π0(Y)Zˆ/mWZˆ //0 from which it follows thatf is surjective, andker(f)∼= ker(g)∼= (ˆZ/Z)^π0(Y), which is a divisible group. We therefore get an isomorphism

coker(ρ)∼= (ˆZ/Z)^π0(Y)⊕coker( ˆρ).

Now letm∈N(L)be an integer withm·A0(Y) = 0andm·π₁^geo(Y)_L-tors= 0, which exists by assumption and since π₁^geo(Y)is a finitely generated abelian pro- finite group. Thenρ0 factors as

ρ0:A0(Y)→π₁^geo(Y)_L-tors,→π₁^geo(Y).

Sinceπ^geo₁ (Y)/π^geo₁ (Y)_L-tors isL-torsion-free andm∈N(L)we get an injection π^geo₁ (Y)_L-tors/m ,→π^geo₁ (Y)/m,

and therefore a factorisation modulom:

ρ0,m:A0(Y)/m→π^geo₁ (Y)_L-tors/m ,→π^geo₁ (Y)/m.

BecauseA₀(Y)/m=A₀(Y)andπ₁^geo(Y)_L-tors/m=π₁^geo(Y)_L-tors we have ker(ρ0,m) = ker(ρ0) = ker(ρ).

Now taking the limit overN(L)shows

ker(ρ_L) = ker(ρ0) = ker(ρ)

for such setsL and also for the set of all prime numbers, which gives the results forρ.ˆ

Corollary 3.6. Let the setting be as in3.5. Furthermore, assume that all compo- nents ofY,Y^[1] andY^[0] are geometrically connected overk and Y^[0] consists of

“geometrically simply connected” components, i.e.

π^geo₁ (Y^[0]) := ker π^ab₁ (Y^[0])−−→^deg Gal^π_k^0(Y[0])∼= ˆZ^π0(Y

[0])

vanishes.(Note that by1.2this assumption is equivalent to saying that CH0(Y^[0]) is torsion-free.) The kernel of the reciprocity map then vanishes and also modulo nfor every integer n.

Proof. We have a commutative diagram of complexes π^geo₁ (Y^[1]) //

 _

π₁^geo(Y^[0]) //

 _

π₁^geo(Y)

 _

π^ab₁ (Y^[1]) //

deg

π₁^ab(Y^[0]) //

deg

π^ab₁ (Y)

deg

Gal^π_k^{0(Y[1]) α} //Gal^π_k^0(Y[0]) //Gal^π_k^0(Y)

By geometrical connectedness the degree maps are surjective. Therefore we have a short exact sequence of complexes

0→π^geo₁ (Y^[•])→π₁^ab(Y^[•])→Gal^π_k^0(Y[•])→0.

And because Galk ∼= ˆZ is torsion-free, we get short exact sequences of complexes for every integern:

0→π₁^geo(Y^[•])/n→π₁^ab(Y^[•])/n→(Z/n)^π0(Y[•])→0.

From the long exact sequence follows the exact sequence

H0(π^geo₁ (Y^[•])/n)→H0(π₁^ab(Y^[•])/n)→H0((Z/n)^π0(Y[•])),

where the first term vanishes by assumption. For the last term we mention that (cf. [Liu02, §2.4, Ex. 4.4])

coker(α)∼= H0(Γ,Zˆ)∼= ˆZ^π0(Γ)→^∼ Zˆ^π0(Y).

The isomorphism above also holds withZ/n-coefficients. HenceH0((Z/n)^π0(Y[•]))

= 0. So H0(π₁^ab(Y^[•])/n)vanishes. By3.2 the last term is isomorphic to ker(ρn).

So the claim follows. The same way one shows thatker(ρ) = 0.

§4. Examples

Here we will give some examples of commonly used varieties and make some essen- tial observations about the interaction ofH2(ΓY)with the kernel of the reciprocity map and their torsion parts withCH0(Y).

Example 4.1. Let k be a finite field and Y = V+(T0· T1 ·T2· T3) ⊆ P³k = Proj(k[T0, . . . , T3])be the surface of the projective tetrahedron. Then the reciprocity map

ρ: CH0(Y)→π₁^ab(Y)

has trivial kernel and so also do the reciprocity maps modulonfor every integern.

But

H2(ΓY,Z/n)∼=Z/n.

SoH2(ΓY,Z/n)surjects ontoker(ρn), but e.g. does not inject intoCH0(Y)/n.

Proof. The calculation ofH2(ΓY,Z/n)is clear. The rest follows from 3.6and the fact thatπ1(P^mk , xi)∼= Galk, i.e.P^mk is “geometrically simply connected”. Note that every intersection of irreducible components is isomorphic to aP^mk .

This example can be generalized to the following:

Lemma 4.2. Let k be a field and let i:W ,→Z be a closed immersion between proper smooth geometrically connected k-varieties such thatd:= dimZ ≥3, and the complement U =Z\W is affine. Further assume one of the following proper- ties:

• The natural map π1(Z, z)→ Galk has a section for a geometric pointz on Z (which is the case ifZ has a k-rational point).

• The cohomology group H²(Galk,Q/Z)vanishes.

Then the push-forward mapi_∗:π^ab₁ (W)→π₁^ab(Z)is an isomorphism.

Proof. For a separably closed fieldk this is due to [Sat05]. The proof there uses Poincaré duality [Mil80, VI.11.1], the affine Lefschetz theorem [Mil80, VI.7.2] and the assumption dim ≥ 3 to show that Hⁱ_c(U,Q/Z)[`] vanishes for i = 1,2 and

`6=p= char(k). For `=pone needs duality results from [JSS09, Thm. 1.6, 1.7]

(cf. [Mos99], [Mil86, §1]), and the corresponding vanishing results from [Suw95, 2.1]

for the cohomology of the logarithmic part of de Rham–Witt sheaves (cf. [Ill79]).

For an arbitrary fieldk, one base changes with a separable closurekofkand uses the homotopy exact sequence from [Gro71, IX, Thm. 6.1] to get a commutative diagram of exact sequences

0 //π1(W) //

π1(W) //

Galk //0

0 //π1(Z) //π1(Z)rr //Galk //

0

suppressing the geometric points. This induces a commutative diagram of exact sequences

π₁^ab(W)_Galk //

o

π^ab₁ (W) //

Gal^ab_k //0

0 //π₁^ab(Z)Gal_k //π₁^ab(Z)rr //Gal^ab_k //

0

where the injectivity on the left at the bottom is induced by the section given by assumption or via the Pontryagin dual of the Hochschild–Serre 4-term sequence (withG= Galk for brevity)

0→H¹(G,Q/Z)→H¹(Z,Q/Z)→H¹(Z,Q/Z)^G→H²(G,Q/Z) = 0.

With the previous result over separably closed fields and the snake lemma one gets the claim.

Example 4.3. Let k be a finite field and P^d+1_k be the projective space. Let f₁, . . . , f_n be homogeneous irreducible polynomials defining smooth and geomet- rically connected hypersurfaces in P^d+1k such that V₊(f_i) and V₊(f_j) intersect smoothly and every connected component of V+(fi, fj) is geometrically connected (e.g. contains a k-rational point). Then the reciprocity map ρ^Y is injective for Y :=V+(f1· · ·fn),→P^d+1_k .

Proof. Ifd < 2, then Y is a union of points or curves, so that H₂(Γ_Y) = 0. For d≥2we use4.2together withπ^ab₁ (P^d+1k ) = Gal^ab_k and refer to3.6.

Note that this example can be used to construct a huge H2(ΓY) and never- theless a vanishing kernel of the reciprocity map.

Example 4.4 (cf. [MSA99, Example 4.1], [Sug09, Example 3.2]). Let k be a fi- nite field and n > 1 an integer such that gcd(n,6 char(k)) = 1 and k contains a primitive n-th root of unity ζ. Let P³k = Proj(k[T0, T1, T2, T3]) be the projective space and

V :=V₊(T₀ⁿ+T₁ⁿ+T₂ⁿ+T₃ⁿ),→P³k

a Fermat surface and consider the free action onV given by τ: (T₀:T₁:T₂:T₃)7→(T₀:ζT₁:ζ²T₂:ζ³T₃).

ThenX :=V /hτiis a smooth and projective surface. Let

L=V+(T0+T1, T2+T3) and L⁰=V+(T0+T1, T2+ζT3)

be two lines onV andC,C⁰ be their images in X. ThenD:=C∪C⁰ is a simple normal crossing divisor onX, andC andC⁰ meet in two k-rational points. Set

Y := (X×_kO)∪(X×_k∞)∪(D×_kP¹k)⊆X×_kP¹k,

whereO = (0 : 1)and∞= (1 : 0)are rational points onP¹k. ThenY is a simple normal crossing surface inX×kP¹k which is projective and geometrically connected overk, and the reciprocity map

ρY : CH0(Y)→π₁^ab(Y)

has ker(ρ_Y) ∼= Z/n. Moreover, ker(ρ_Y⊗F) ∼= Z/n for every finite field exten- sionF|k.

Now letC∩C⁰ ={c1, c2} and let ΓY be the dual complex toY associated to a numbering of the irreducible components. Then

H₀(Γ_Y,Z) =Z, H₁(Γ_Y,Z) = 0, H₂(Γ_Y,Z)∼=Z, and therefore for every integer m,

H0(ΓY,Z/m) =Z/m, H1(ΓY,Z/m) = 0, H2(ΓY,Z/m)∼=Z/m.

Proof. For the first statements see [Sug09, Example 3.2]. For the homology groups we mention the following:Y,Y^[0]resp., has four irreducible components:

Y1=X×kO, Y2=C×kP¹k, Y3=C⁰×kP¹k, Y4=X×k∞.

There are six connected components inY^[1]:

Y12=Y1∩Y2=C×kO, Y13=Y1∩Y3=C⁰×kO, Y₂₃¹ in Y₂∩Y₃:c₁×kP¹k, Y₂₃² in Y₂∩Y₃:c₂×kP¹k, Y24=Y2∩Y4=C×k∞, Y34=Y3∩Y4=C⁰×k∞.

And there are four connected components inY^[2]:

Y₁₂₃¹ in Y1∩Y2∩Y3:c1×kO, Y₂₃₄¹ inY2∩Y3∩Y4:c1×k∞, Y₁₂₃² in Y₁∩Y₂∩Y₃:c₂×_kO, Y₂₃₄² inY₂∩Y₃∩Y₄:c₂×_k∞.

The homology groupsHi(ΓY,Z)can then be computed combinatorially. And the homology groups with coefficients in Z/mcan be computed from the homology groups with coefficients inZ by the universal coefficient theorem, observing that allHi(ΓY,Z)are torsion-free.

4.4shows that the groupsH2(ΓY,Z)andH2(ΓY,Zˆ)are torsion-free, but the reciprocity map and the reciprocity map modulonhave kernelZ/n. Therefore the kernel is not given by the torsion part ofH2(ΓY,Z)or H2(ΓY,ˆ

Z).

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