A p-adic Regulator Map and Finiteness Results

for Arithmetic Schemes

Dedicated to Andrei Suslin on his 60th birthday

Shuji Saito^{1}and Kanetomo Sato^{2}

Received: January 21, 2009 Revised: March 19, 2010

Abstract. A main theme of the paper is a conjecture of Bloch-Kato on the image of p-adic regulator maps for a proper smooth variety X over an algebraic number field k. The conjecture for a regulator map of particular degree and weight is related to finiteness of two arithmetic objects: One is the p-primary torsion part of the Chow group in codimension 2 of X. Another is an unramified cohomology group of X. As an application, for a regular model 𝒳 of X over the integer ring of k, we prove an injectivity result on the torsion cycle class map of codimension 2 with values in a new p-adic cohomology of 𝒳 introduced by the second author, which is a candidate of the conjectural étale motivic cohomology with finite coefficients of Beilinson-Lichtenbaum.

2010 Mathematics Subject Classification: Primary 14C25, 14G40; Secondary 14F30, 19F27, 11G25. Keywords and Phrases: p-adic regulator, unramified cohomology, Chow groups, p-adic étale Tate twists

1 Introduction

Let k be an algebraic number field and let G_{k} be the absolute Galois group Gal(\overline{k}/k), where \overline{k} denotes a fixed algebraic closure of k. Let X be a projective smooth variety over k and put \overline{X} := X ⊗_{k} \overline{k}. Fix a prime p and integers r, m ≥ 1. A main theme of this paper is a conjecture of Bloch and Kato concerning the image of the p-adic regulator map

1supported by Grant-in-Aid for Scientific Research B-18340003 and S-19104001 2supported by JSPS Postdoctoral Fellowship for Research Abroad and EPSRC grant

Documenta Mathematica · Extra Volume Suslin (2010) 525–594

reg^{r,m} : CH^{r}(X, m) ⊗ ℚ_{p} ⟶ H_{cont}^{1}(k, H_{ét}^{2r−m−1}(\overline{X}, ℚ_{p}(r)))

from Bloch’s higher Chow group to continuous Galois cohomology of G_{k} ([BK2] Conjecture 5.3). See §3 below for the definition of this map in the case (r, m) = (2, 1). This conjecture affirms that its image agrees with the subspace

H_{g}^{1}(k, H_{ét}^{2r−m−1}(\overline{X}, ℚ_{p}(r))) ⊂ H_{cont}^{1}(k, H_{ét}^{2}(\overline{X}, ℚ_{p}(2)))

defined in loc. cit. (see §2.1 below), and plays a crucial role in the so-called Tamagawa number conjecture on special values of L-functions attached to X. In terms of Galois representations, the conjecture means that a 1-extension of continuous p-adic representations of G_{k}

0 ⟶ H_{ét}^{2r−m−1}(\overline{X}, ℚ_{p}(r)) ⟶ E ⟶ ℚ_{p} ⟶ 0

arises from a 1-extension of motives over k

0 ⟶ h^{2r−m−1}(X)(r) ⟶ M ⟶ h(Spec(k)) ⟶ 0,

if and only if E is a de Rham representation of G_{k}. There has been only very few known results on the conjecture. In this paper we consider the following condition, which is the Bloch-Kato conjecture in the special case (r, m) = (2, 1):

H1: The image of the regulator map

reg := reg^{2,1} : CH^{2}(X, 1) ⊗ ℚ_{p} ⟶ H_{cont}^{1}(k, H_{ét}^{2}(\overline{X}, ℚ_{p}(2))).

agrees with H_{g}^{1}(k, H_{ét}^{2}(\overline{X}, ℚ_{p}(2))).

We also consider a variant:

H1∗: The image of the regulator map with ℚ_{p}/ℤ_{p}-coefficients

reg_{ℚ_{p}/ℤ_{p}} : CH^{2}(X, 1) ⊗ ℚ_{p}/ℤ_{p} ⟶ H_{Gal}^{1}(k, H_{ét}^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))

agrees with H_{g}^{1}(k, H_{ét}^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div} (see §2.1 for H_{g}^{1}(k, −)). Here for an abelian group M, M_{Div} denotes its maximal divisible subgroup.

We will show that H1 always implies H1∗, which is not straight-forward. On the other hand the converse holds as well under some assumptions. See Remark 3.2.5 below for details.

Fact 1.1 The condition H1 holds in the following cases:

p-adic Regulator and Finiteness 527

(1) H^{2}(X, 𝒪_{X}) = 0 ([CTR1], [CTR2], [Sal]).

(2) X is the self-product of an elliptic curve over k = ℚ with square-free conductor and without complex multiplication, and p ≥ 5 ([Md], [Fl], [LS], [La1]).

(3) X is the elliptic modular surface of level 4 over k = ℚ and p ≥ 5 ([La2]).

(4) X is a Fermat quartic surface over k = ℚ or ℚ(\sqrt{−1}) and p ≥ 5 ([O]).

A main result of this paper relates the condition H1∗ to finiteness of two arithmetic objects. One is the p-primary torsion part of the Chow group CH^{2}(X) of algebraic cycles of codimension two on X modulo rational equivalence. Another is an unramified cohomology of X, which we are going to introduce in what follows.

Let 𝔬_{k} be the integer ring of k, and put S := Spec(𝔬_{k}). We assume the following:

Assumption 1.2 There exists a regular scheme 𝒳 which is proper and flat over S and whose generic fiber is X. Moreover, 𝒳 has good or semistable reduction at each closed point of S of characteristic p.

Let K = k(X) be the function field of X. For an integer q ≥ 0, let 𝒳 ^{q} be the set of all points x ∈ 𝒳 of codimension q. Fix an integer n ≥ 0. Roughly speaking, the unramified cohomology group H_{ur}^{n+1}(K, ℚ_{p}/ℤ_{p}(n)) is defined as the subgroup of H_{ét}^{n+1}(Spec(K), ℚ_{p}/ℤ_{p}(n)) consisting of those elements that are “unramified” along all y ∈ 𝒳 ^{1}. For a precise definition, we need the p-adic étale Tate twist 𝔗_{r}(n) = 𝔗_{r}(n)_{𝒳} introduced in [SH]. This object is defined in D^{b}(𝒳_{ét}, ℤ/p^{r}), the derived category of bounded complexes of étale sheaves of ℤ/p^{r}-modules on 𝒳 , and expected to coincide with Γ (2)_{ét}^{𝒳} ⊗^{𝕃} ℤ/p^{r}. Here Γ (2)_{ét}^{𝒳} denotes the conjectural étale motivic complex of Beilinson-Lichtenbaum [Be], [Li1]. We note that the restriction of 𝔗_{r}(n) to 𝒳 [p^{−1}] := 𝒳 ⊗_{ℤ} ℤ[p^{−1}] is isomorphic to μ_{p^{r}}^{⊗n}, where μ_{p^{r}} denotes the étale sheaf of p^{r}-th roots of unity.Then H_{ur}^{n+1}(K, ℚ_{p}/ℤ_{p}(n)) is defined as the kernel of the boundary map of étale cohomology groups

H_{ét}^{n+1}(Spec(K), ℚ_{p}/ℤ_{p}(n)) ⟶ ⨁_{x∈𝒳^{1}} H_{x}^{n+2}(Spec(𝒪_{𝒳,x} ), 𝔗_{∞}(n)),

where 𝔗_{∞}(n) denotes \underrightarrow{lim}_{r≥1} 𝔗_{r}(n). There are natural isomorphisms

H_{ur}^{1}(K, ℚ_{p}/ℤ_{p}(0)) ≃ H_{ét}^{1}(𝒳, ℚ_{p}/ℤ_{p}) and H_{ur}^{2}(K, ℚ_{p}/ℤ_{p}(1)) ≃ Br(𝒳 )_{p-tors},

where Br(𝒳 ) denotes the Grothendieck-Brauer group H_{ét}^{2}(𝒳, 𝔾_{m}), and for an abelian group M, M_{p-tors} denotes its p-primary torsion part. An intriguing question is as to whether the group H_{ur}^{n+1}(K, ℚ_{p}/ℤ_{p}(n)) is finite, which is related to several significant theorems and conjectures in arithmetic geometry (see Remark 4.3.1 below). In this paper we are concerned with the case n = 2. A crucial role will be played by the following subgroup of H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)):

H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2))

:= Im(H_{ét}^{3}(X, ℚ_{p}/ℤ_{p}(2)) → H_{ét}^{3}(Spec(K), ℚ_{p}/ℤ_{p}(2))) ∩ H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)).

It will turn out that CH^{2}(X)_{p-tors} and H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) are cofinitely generated over ℤ_{p} if Coker(reg_{ℚ_{p}/ℤ_{p}})_{Div} is cofinitely generated over ℤ_{p} (cf. Proposition 3.3.2, Lemma 5.2.3). Our main finiteness result is the following:

Theorem 1.3 Let 𝒳 be as in Assumption 1.2, and assume p ≥ 5. Then:

(1) H1∗ implies that CH^{2}(X)_{p-tors} and H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) are finite.

(2) Assume that the reduced part of every closed fiber of 𝒳 /S has simple normal crossings on 𝒳 , and that the Tate conjecture holds in codimension 1 for the irreducible components of those fibers (see the beginning of §7 for the precise contents of the last assumption). Then the finiteness of CH^{2}(X)_{p-tors} and H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) implies H1∗.

We do not need Assumption 1.2 to deduce the finiteness of CH^{2}(X)_{p-tors} from H1∗, by the alteration theorem of de Jong [dJ] (see also Remark 3.1.2 (3) below). However, we need a regular proper model 𝒳 as above crucially in our computations on H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)). The assertion (2) is a converse of (1) under the assumption of the Tate conjecture. We obtain the following result from Theorem 1.3 (1) (see also the proof of Theorem 1.6 in §5.1 below):

Corollary 1.4 H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) is finite in the four cases in Fact 1.1 (under the assumption 1.2).

We will also prove variants of Theorem 1.3 over local integer rings (see Theorems 3.1.1, 5.1.1 and 7.1.1 below). As for the finiteness of H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)) over local integer rings, Spiess proved that H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)) = 0, assuming that 𝔬_{k} is an ℓ-adic local integer ring with ℓ ≠ p and that either H^{2}(X, 𝒪_{X}) = 0 or 𝒳 is a product of two smooth elliptic curves over S ([Spi] §4). In [SSa], the authors extended his vanishing result to a more general situation that 𝔬_{k} is ℓ-adic local with ℓ ≠ p and that 𝒳 has generalized semistable reduction. Finally we have to remark that there exists a smooth projective surface X with p_{g}(X) ≠ 0 over a local field k for which the condition H1∗ does not hold and such that CH^{2}(X)_{tors} is infinite [AS].

We next explain an application of the above finiteness result to a cycle class map of arithmetic schemes. Let us recall the following fact due to Colliot-Thélène, Sansuc, Soulé and Gros:

Fact 1.5 ([CTSS], [Gr]) Let X be a proper smooth variety over a finite field of characteristic ℓ > 0. Let p be a prime number, which may be the same as ℓ. Then the cycle class map restricted to the p-primary torsion part

CH^{2}(X)_{p-tors} ⟶ H_{ét}^{4}(X, ℤ/p^{r}(2))

is injective for a sufficiently large r > 0. Here ℤ/p^{r}(2) denotes μ_{p^{r}}^{⊗2} if ℓ ≠ p. Otherwise ℤ/p^{r}(2) denotes W_{r} Ω_{X,log}^{2}[−2] with W_{r} Ω_{X,log}^{2} the étale subsheaf of the logarithmic part of the Hodge-Witt sheaf W_{r} Ω_{X}^{2} ([Bl1], [Il]).

In this paper, we study an arithmetic variant of this fact. We expect that a similar result holds for proper regular arithmetic schemes, i.e., regular schemes which are proper flat of finite type over the integer ring of a number field or a local field. To be more precise, let k, 𝔬_{k} and X be as before and let 𝒳 be as in Assumption 1.2. The p-adic étale Tate twist 𝔗_{r}(2) = 𝔗_{r}(2)_{𝒳} mentioned before replaces ℤ/p^{r}(2) in Fact 1.5, and there is a cycle class map

ϱ_{r}^{2} : CH^{2}(𝒳 )/p^{r} ⟶ H_{ét}^{4}(𝒳, 𝔗_{r}(2)).

We are concerned with the induced map

ϱ_{p-tors,r}^{2} : CH^{2}(𝒳 )_{p-tors} ⟶ H_{ét}^{4}(𝒳, 𝔗_{r}(2)).

It is shown in [SH] that the group on the right hand side is finite. So the injectivity of this map is closely related with the finiteness of CH^{2}(𝒳 )_{p-tors}. The second main result of this paper concerns the injectivity of this map:

Theorem 1.6 (§5) Assume that H^{2}(X, 𝒪_{X}) = 0. Then CH^{2}(𝒳 )_{p-tors} is finite and ϱ_{p-tors,r}^{2} is injective for a sufficiently large r > 0.

The finiteness of CH^{2}(𝒳 )_{p-tors} in this theorem is originally due to Salberger [Sal], Colliot-Thélène and Raskind [CTR1], [CTR2]. Note that there exists a projective smooth surface V over a number field with H^{2}(V, 𝒪_{V}) = 0 for which the map

CH^{2}(V)_{p-tors} ⟶ H_{ét}^{4}(V, μ_{p^{r}}^{⊗2})

is not injective for some bad prime p and any r ≥ 1 [Su] (cf. [PS]). Our result suggests that we are able to recover the injectivity of torsion cycle class maps by considering a proper regular model of V over the ring of integers in k. The fundamental ideas of Theorem 1.6 are the following. A crucial point of the proof of Fact 1.5 in [CTSS] and [Gr] is Deligne’s proof of the Weil conjecture [De2]. In the arithmetic situation, the role of the Weil conjecture is replaced by the condition H1, which implies the finiteness of CH^{2}(X)_{p-tors} and H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) by Theorem 1.3 (1). The injectivity result in Theorem 1.6 is derived from the finiteness of those objects.

This paper is organized as follows. In §2, we will review some fundamental facts on Galois cohomology groups and Selmer groups which will be used frequently in this paper. In §3, we will prove the finiteness of CH^{2}(X)_{p-tors} in Theorem 1.3 (1). In §4, we will review p-adic étale Tate twists briefly and then provide some fundamental lemmas on cycle class maps and unramified cohomology groups. In §5, we will first reduce Theorem 1.6 to Theorem 1.3 (1), and then reduce the finiteness of H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) in Theorem 1.3 (1) to Key Lemma 5.4.1. In §6, we will prove that key lemma, which will complete the proof of Theorem 1.3 (1). §7 will be devoted to the proof of Theorem 1.3 (2). In the appendix A, we will include an observation that the finiteness of H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)) is deduced from the Beilinson–Lichtenbaum conjectures on motivic complexes.

Acknowledgements. The second author expresses his gratitude to University of Southern California and The University of Nottingham for their great hospitality. The authors also express their gratitude to Professors Wayne Raskind, Thomas Geisser and Ivan Fesenko for valuable comments and discussions. Thanks are also due to the referee and Masanori Asakura, who read the manuscript of this paper carefully and gave them enlightening comments.

Notation

1.6. For an abelian group M and a positive integer n, _{n}M and M/n denote the kernel and the cokernel of the map M ⟶^{×n} M, respectively. See §2.3 below for other notation for abelian groups. For a field k, \overline{k} denotes a fixed separable closure, and G_{k} denotes the absolute Galois group Gal(\overline{k}/k). For a discrete G_{k}-module M, H^{∗}(k, M) denote the Galois cohomology groups H_{Gal}^{∗}(G_{k}, M), which are the same as the étale cohomology groups of Spec(k) with coefficients in the étale sheaf associated with M.

1.7. Unless indicated otherwise, all cohomology groups of schemes are taken over the étale topology. For a scheme X, an étale sheaf ℱ on X (or more generally an object in the derived category of sheaves on X_{ét}) and a point x ∈ X, we often write H_{x}^{∗}(X, ℱ) for H_{x}^{∗}(Spec(𝒪_{X,x}), ℱ). For a pure-dimensional scheme X and a nonnegative integer q, let X^{q} be the set of all points on X of codimension q. For a point x ∈ X, let κ(x) be its residue field. For an integer n ≥ 0 and a noetherian excellent scheme X, CH_{n}(X) denotes the Chow group of algebraic cycles on X of dimension n modulo rational equivalence. If X is pure-dimensional and regular, we will often write CH^{dim(X)−n}(X) for this group. For an integral scheme X of finite type over Spec(ℚ), Spec(ℤ) or Spec(ℤ_{ℓ}), we define CH^{2}(X, 1) as the cohomology group, at the middle, of the Gersten complex of Milnor K-groups

K_{2}^{M}(L) ⟶ ⨁_{y∈X^{1}} κ(y)^{×} ⟶ ⨁_{x∈X^{2}} ℤ,

where L denotes the function field of X. As is well-known, this group coincides with a higher Chow group ([Bl3], [Le2]) by localization sequences of higher Chow groups ([Bl4], [Le1]) and the Nesterenko-Suslin theorem [NS] (cf. [To]).

1.8. In §§4–7, we will work under the following setting. Let k be an algebraic number field or its completion at a finite place. Let 𝔬_{k} be the integer ring of k and put S := Spec(𝔬_{k}). Let p be a prime number, and let 𝒳 be a regular scheme which is proper flat of finite type over S and satisfies the following condition:

Assumption 1.8.1 If p is not invertible in 𝔬_{k}, then 𝒳 has good or semistable reduction at each closed point of S of characteristic p.

This condition is the same as Assumption 1.2 when k is a number field.

1.9. Let k be an algebraic number field, and let 𝒳 → S = Spec(𝔬_{k}) be as in 1.8. In this situation, we will often use the following notation. For a closed point v ∈ S, let 𝔬_{v} (resp. k_{v}) be the completion of 𝔬_{k} (resp. k) at v, and let 𝔽_{v} be the residue field of k_{v}. We put

𝒳_{v} := 𝒳 ⊗_{𝔬_{k}} 𝔬_{v},  X_{v} := 𝒳 ⊗_{𝔬_{k}} k_{v},  Y_{v} := 𝒳 ⊗_{𝔬_{k}} 𝔽_{v}

and write j_{v} : X_{v} ↪ 𝒳_{v} (resp. i_{v} : Y_{v} ↪ 𝒳_{v}) for the natural open (resp. closed) immersion. We put \overline{Y_{v}} := Y_{v} ×_{𝔽_{v}} \overline{𝔽_{v}}, and write Σ for the set of all closed point on S of characteristic p.

1.10. Let k be an ℓ-adic local field with ℓ a prime number, and let 𝒳 → S = Spec(𝔬_{k}) be as in 1.8. In this situation, we will often use the following notation. Let 𝔽 be the residue field of k and put

X := 𝒳 ⊗_{𝔬_{k}} k, Y := 𝒳 ⊗_{𝔬_{k}} 𝔽.

We write j : X ↪ 𝒳 (resp. i : Y ↪ 𝒳 ) for the natural open (resp. closed) immersion. Let k^{ur} be the maximal unramified extension of k, and let 𝔬^{ur} be its integer ring. We put

𝒳 ^{ur} := 𝒳 ⊗_{𝔬_{k}} 𝔬^{ur},  X^{ur} := 𝒳 ⊗_{𝔬_{k}} k^{ur},  \overline{Y} := Y ×_{𝔽} \overline{𝔽}.

2 Preliminaries on Galois Cohomology

In this section, we provide some preliminary lemmas which will be frequently used in this paper. Let k be an algebraic number field (global field) or its completion at a finite place (local field). Let 𝔬_{k} be the integer ring of k, and put S := Spec(𝔬_{k}). Let p be a prime number. If k is global, we often write Σ for the set of the closed points on S of characteristic p.

2.1 Selmer Group

Let X be a proper smooth variety over Spec(k), and put \overline{X} := X ⊗_{k} \overline{k}. If k is global, we fix a non-empty open subset U_{0} ⊂ S \ Σ for which there exists a proper smooth morphism 𝒳_{U_{0}} → U_{0} with 𝒳_{U_{0}} ×_{U_{0}} k ≃ X. For v ∈ S^{1}, let k_{v} and 𝔽_{v} be as in the notation 1.9. In this section we are concerned with G_{k}-modules

V := H^{i}(\overline{X}, ℚ_{p}(n)) and A := H^{i}(\overline{X}, ℚ_{p}/ℤ_{p}(n)).

For M = V or A and a non-empty open subset U ⊂ U_{0}, let H^{∗}(U, M) denote the étale cohomology groups with coefficients in the smooth sheaf on U_{ét} associated to M.

Definition 2.1.1 (1) Assume that k is local. Let H_{f}^{1}(k, V) and H_{g}^{1}(k, V) be as defined in [BK2] (3.7). For ∗ ∈ {f, g}, we define

H_{∗}^{1}(k, A) := Im(H_{∗}^{1}(k, V) ⟶ H^{1}(k, A)).

(2) Assume that k is global. For M ∈ {V, A} and a non-empty open subset U ⊂ S, we define the subgroup H_{f,U}^{1}(k, M) ⊂ H_{cont}^{1}(k, M) as the kernel of the natural map

H_{cont}^{1}(k, M) ⟶ ∏_{v∈U^{1}} \frac{H_{cont}^{1}(k_{v}, M)}{H_{f}^{1}(k_{v}, M)} × ∏_{v∈S\U} \frac{H_{cont}^{1}(k_{v}, M)}{H_{g}^{1}(k_{v}, M)}.

If U ⊂ U_{0}, we have

H_{f,U}^{1}(k, M) = Ker(H^{1}(U, M) ⟶ ∏_{v∈S\U} H_{cont}^{1}(k_{v}, M)/H_{g}^{1}(k_{v}, M)).

We define the group H_{g}^{1}(k, M) and H_{ind}^{1}(k, M) as

H_{g}^{1}(k, M) := \underrightarrow{lim}_{U⊂U_{0}} H_{f,U}^{1}(k, M), H_{ind}^{1}(k, M) := \underrightarrow{lim}_{U⊂U_{0}} H^{1}(U, M),

where U runs through all non-empty open subsets of U_{0}. These groups are independent of the choice of U_{0} and 𝒳_{U_{0}} (cf. [EGA4] 8.8.2.5).

(3) If k is local, we define H_{ind}^{1}(k, M) to be H_{cont}^{1}(k, M) for M ∈ {V, A}.

Note that H_{ind}^{1}(k, A) = H^{1}(k, A).

2.2 p-adic Point of Motives

We provide a key lemma from p-adic Hodge theory which play crucial roles in this paper (see Corollary 2.2.3 below). Assume that k is a p-adic local field, and that there exists a regular scheme 𝒳 which is proper flat of finite type over S = Spec(𝔬_{k}) with 𝒳 ⊗_{𝔬_{k}} k ≃ X and which has semistable reduction. Let i and n be non-negative integers. Put

V^{i} := H^{i+1}(\overline{X}, ℚ_{p}), V^{i}(n) := V^{i} ⊗_{ℚ_{p}} ℚ_{p}(n),

and

H^{i+1}(𝒳, τ_{≤n}Rj_{∗}ℚ_{p}(n)) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{r≥1} H^{i+1}(𝒳, τ_{≤n}Rj_{∗}μ_{p^{r}}^{⊗n}),

where j denotes the natural open immersion X ↪ 𝒳 . There is a natural pull-back map

α : H^{i+1}(𝒳, τ_{≤n}Rj_{∗}ℚ_{p}(n)) ⟶ H^{i+1}(X, ℚ_{p}(n)).

Let H^{i+1}(𝒳, τ_{≤r}Rj_{∗}ℚ_{p}(n))^{0} be the kernel of the composite map

α′ : H^{i+1}(𝒳, τ_{≤n}Rj_{∗}ℚ_{p}(n)) ⟶^{α} H^{i+1}(X, ℚ_{p}(n)) ⟶ (V^{i+1}(n))^{G_{k}}.

For this group, there is a composite map

\overline{α} : H^{i+1}(𝒳, τ_{≤n}Rj_{∗}ℚ_{p}(n))^{0} ⟶ F^{1}H^{i+1}(X, ℚ_{p}(n)) ⟶ H_{cont}^{1}(k, V^{i}(n)).

Here the first arrow is induced by α, the second is an edge homomorphism in a Hochschild-Serre spectral sequence

E_{2}^{u,v} := H_{cont}^{u}(k, V^{v}(n)) ⟹ H_{cont}^{u+v}(X, ℚ_{p}(n))(≃ H^{u+v}(X, ℚ_{p}(n))),

and F^{•} denotes the filtration on H^{i+1}(X, ℚ_{p}(n)) resulting from this spectral sequence. To provide with Corollary 2.2.3 below concerning the image of \overline{α}, we need some strong results in p-adic Hodge theory. We first recall the following comparison theorem of log syntomic complexes and p-adic vanishing cycles due to Tsuji, which extends a comparison result of Kurihara [Ku] to semistable families. Let Y be the closed fiber of 𝒳 → S and let ι : Y ↪ 𝒳 be the natural closed immersion.

Theorem 2.2.1 ([Ts2] Theorem 5.1) For integers n, r with 0 ≤ n ≤ p − 2 and r ≥ 1, there is a canonical isomorphism

η : s_{r}^{log}(n) ^{∼} ⟶ ι_{∗}ι^{∗}(τ_{≤n}Rj_{∗}μ_{p^{r}}^{⊗n}) in D^{b}(𝒳_{ét}, ℤ/p^{r}),

where s_{r}^{log}(n) = s_{r}^{log}(n)_{𝒳} is the log syntomic complex defined by Kato [Ka2].

Put

H^{∗}(𝒳, s_{ℚ_{p}}^{log}(n)) := ℚ_{p} ⊗_{ℤ_{p}} \underleftarrow{lim}_{r≥1} H^{∗}(𝒳, s_{r}^{log}(n)),

and define H^{i+1}(𝒳, s_{ℚ_{p}}^{log}(n))^{0} as the kernel of the composite map

H^{i+1}(𝒳, s_{ℚ_{p}}^{log}(n)) ^{∼} ⟶_{η} H^{i+1}(𝒳, τ_{≤n}Rj_{∗}ℚ_{p}(n)) ⟶^{α′} (V^{i+1}(n))^{G_{k}},

where we have used the properness of 𝒳 over S. There is an induced map

\overline{η} : H^{i+1}(𝒳, s_{ℚ_{p}}^{log}(n))^{0} ^{∼} ⟶_{η} H^{i+1}(𝒳, τ_{≤n}Rj_{∗}ℚ_{p}(n))^{0} ⟶^{\overline{α}} H_{cont}^{1}(k, V^{i}(n)).

Concerning this map, we have the following fact due to Langer and Nekovář:

Theorem 2.2.2 ([La3], [Ne2] Theorem 3.1) Im(\overline{η}) agrees with H_{g}^{1}(k, V^{i}(n)).

As an immediate consequence of these facts, we obtain

Corollary 2.2.3 Assume that p ≥ n + 2. Then Im(\overline{α}) = H_{g}^{1}(k, V^{i}(n)).

Remark 2.2.4 (1) Theorem 2.2.2 is an extension of the p-adic point conjecture raised by Schneider in the good reduction case [Sch]. This conjecture was proved by Langer-Saito [LS] in a special case and by Nekovář [Ne1] in the general case.

(2) Theorem 2.2.2 holds unconditionally on p, if we define H^{i+1}(𝒳, s_{ℚ_{p}}^{log}(n)) using Tsuji’s version of log syntomic complexes 𝒮_{r} (n) (r ≥ 1) in [Ts1] §2.

2.3 Elementary Facts on ℤ_{p}-Modules

For an abelian group M, let M_{Div} be its maximal divisible subgroup. For a torsion abelian group M, let Cotor(M) be the cotorsion part M/M_{Div}.

Definition 2.3.1 Let M be a ℤ_{p}-module.

(1) We say that M is cofinitely generated over ℤ_{p} (or simply, cofinitely generated), if its Pontryagin dual Hom_{ℤ_{p}}(M, ℚ_{p}/ℤ_{p}) is a finitely generated ℤ_{p}-module.

(2) We say that M is cofinitely generated up to a finite-exponent group, if M_{Div} is cofinitely generated and Cotor(M) has a finite exponent.

(3) We say that M is divisible up to a finite-exponent group, if Cotor(M) has a finite exponent.

Lemma 2.3.2 Let 0 → L → M → N → 0 be a short exact sequence of ℤ_{p}- modules.

(1) Assume that L, M and N are cofinitely generated. Then there is a positive integer r_{0} such that for any r ≥ r_{0} we have an exact sequence of finite abelian p-groups

0 → _{p^{r}}L → _{p^{r}}M → _{p^{r}}N → Cotor(L) → Cotor(M) → Cotor(N) → 0.

Consequently, taking the projective limit of this exact sequence with respect to r ≥ r_{0} there is an exact sequence of finitely generated ℤ_{p}-modules

0 → T_{p}(L) → T_{p}(M) → T_{p}(N) → Cotor(L) → Cotor(M) → Cotor(N) → 0,

where for an abelian group A, T_{p}(A) denotes its p-adic Tate module.

(2) Assume that L is cofinitely generated up to a finite-exponent group. Assume further that M is divisible, and that N is cofinitely generated and divisible. Then L and M are cofinitely generated.

(3) Assume that L is divisible up to a finite-exponent group. Then for a divisible subgroup D ⊂ N and its inverse image D′ ⊂ M, the induced map (D′)_{Div} → D is surjective. In particular, the natural map M_{Div} → N_{Div} is surjective.

(4) If L_{Div} = N_{Div} = 0, then we have M_{Div} = 0.

Proof. (1) There is a commutative diagram with exact rows

\xymatrix{0 \ar[r] & L \ar[r] \ar[d]_{×p^{r}} & M \ar[r] \ar[d]_{×p^{r}} & N \ar[r] \ar[d]_{×p^{r}} & 0 \\ 0 \ar[r] & L \ar[r] & M \ar[r] & N \ar[r] & 0.}

One obtains the assertion by applying the snake lemma to this diagram, noting Cotor(A) ≃ A/p^{r} for a cofinitely generated ℤ_{p}-module A and a sufficiently large r ≥ 1.

(2) Our task is to show that Cotor(L) is finite. By a similar argument as for (1), there is an exact sequence for a sufficiently large r ≥ 1

0 ⟶ _{p^{r}}L ⟶ _{p^{r}}M ⟶ _{p^{r}}N ⟶ Cotor(L) ⟶ 0,

where we have used the assumptions on L and M. Hence the finiteness of Cotor(L) follows from the assumption that N is cofinitely generated.

(3) We have only to show the case D = N_{Div}. For a ℤ_{p}-module A, we have

A_{Div} = Im(Hom_{ℤ_{p}}(ℚ_{p}, A) → A)

by [J1] Lemma (4.3.a). Since Ext_{ℤ_{p}}^{1}(ℚ_{p}, L) = 0 by the assumption on L, the following natural map is surjective:

Hom_{ℤ_{p}}(ℚ_{p}, M) ⟶ Hom_{ℤ_{p}}(ℚ_{p}, N).

By these facts, the natural map M_{Div} → N_{Div} is surjective.

(4) For a ℤ_{p}-module A, we have

A_{Div} = 0 ⟺ Hom_{ℤ_{p}}(ℚ_{p}, A) = 0

by [J1] Remark (4.7). The assertion follows from this fact and the exact sequence

0 ⟶ Hom_{ℤ_{p}}(ℚ_{p}, L) ⟶ Hom_{ℤ_{p}}(ℚ_{p}, M) ⟶ Hom_{ℤ_{p}}(ℚ_{p}, N).

This completes the proof of the lemma. □

2.4 Divisible Part of H^{1}(k, A)

Let the notation be as in §2.1. We prove here the following general lemma, which will be used frequently in §§3–7:

Lemma 2.4.1 Under the notation in Definition 2.1.1 we have

Im(H_{ind}^{1}(k, V) → H^{1}(k, A)) = H^{1}(k, A)_{Div},
Im(H_{g}^{1}(k, V) → H^{1}(k, A)) = H_{g}^{1}(k, A)_{Div}.

Proof. The assertion is clear if k is local. Assume that k is global. Without loss of generality we may assume that A is divisible. We prove only the second equality and omit the first one (see Remark 2.4.9 (2) below). Let U_{0} ⊂ S be as in §2.1. We have

Im(H_{f,U}^{1}(k, V) → H^{1}(U, A)) = H_{f,U}^{1}(k, A)_{Div}  (2.4.2)

for non-empty open U ⊂ U_{0}. This follows from a commutative diagram with exact rows

\xymatrix{0 \ar[r] & H_{f,U}^{1}(k, V) \ar[r] \ar[d] & H^{1}(U, V) \ar[r] \ar[d]_{α} & ∏_{v∈S\U} H_{cont}^{1}(k_{v}, V)/H_{g}^{1}(k_{v}, V) \ar[d]_{β} \\ 0 \ar[r] & H_{f,U}^{1}(k, A) \ar[r] & H^{1}(U, A) \ar[r] & ∏_{v∈S\U} H^{1}(k_{v}, A)/H_{g}^{1}(k_{v}, A)}

and the facts that Coker(α) is finite and that Ker(β) is finitely generated over ℤ_{p}. By (2.4.2), the second equality of the lemma is reduced to the following assertion:

\underrightarrow{lim}_{U⊂U_{0}} (H_{f,U}^{1}(k, A)_{Div}) = ( \underrightarrow{lim}_{U⊂U_{0}} H_{f,U}^{1}(k, A)))_{Div}.  (2.4.3)

To show this equality, we will prove the following sublemma:

Sublemma 2.4.4 For an open subset U ⊂ U_{0}, put

C_{U} := Coker(H_{f,U_{0}}^{1}(k, A) → H_{f,U}^{1}(k, A)).

Then there exists a non-empty open subset U_{1} ⊂ U_{0} such that the quotient C_{U}/C_{U_{1}} is divisible for any open subset U ⊂ U_{1}.

We first finish our proof of (2.4.3) admitting this sublemma. Let U_{1} ⊂ U_{0} be a non-empty open subset as in Sublemma 2.4.4. Noting that H_{f,U}^{1}(k, A) is cofinitely generated, there is an exact sequence of finite groups

Cotor(H_{f,U_{1}}^{1}(k, A)) ⟶ Cotor(H_{f,U}^{1}(k, A)) ⟶ Cotor(C_{U}/C_{U_{1}}) ⟶ 0

for open U ⊂ U_{1} by Lemma 2.3.2 (1). By this exact sequence and Sublemma 2.4.4, the natural map Cotor(H_{f,U_{1}}^{1}(k, A)) → Cotor(H_{f,U}^{1}(k, A)) is surjective for any open U ⊂ U_{1}, which implies that the inductive limit

\underrightarrow{lim}_{U⊂U_{0}}Cotor(H_{f,U}^{1}(k, A))

is a finite group. The equality (2.4.3)follows easily from this.

Proof of Sublemma 2.4.4. We need the following general fact:

Sublemma 2.4.5 Let \underline{N} = \{N_{λ}\}_{λ∈Λ} be an inductive system of cofinitely generated ℤ_{p}-modules indexed by a filtered set Λ such that Coker(N_{λ} → N_{λ′}) is divisible for any two λ, λ′ ∈ Λ with λ′ ≥ λ. Let L be a cofinitely generated ℤ_{p}-module and {f_{λ} : N_{λ} → L}_{λ∈Λ} be ℤ_{p}-homomorphisms compatible with the transition maps of \underline{N}. Then there exists λ_{0} ∈ Λ such that Coker(Ker(f_{λ_{0}}) → Ker(f_{λ})) is divisible for any λ ≥ λ_{0}.

Proof of Sublemma 2.4.5. Let f_{∞} : N_{∞} → L be the limit of f_{λ}. The assumption on \underline{N} implies that for any two λ, λ′ ∈ Λ with λ′ ≥ λ, the quotient Im(f_{λ′})/Im(f_{λ}) is divisible, so that

Cotor(Im(f_{λ})) → Cotor(Im(f_{λ′})) is surjective.  (2.4.6)

By the equality Im(f_{∞}) = \underrightarrow{lim}_{λ∈Λ} Im(f_{λ}), there is a short exact sequence

0 ⟶ \underrightarrow{lim}_{λ∈Λ} (Im(f_{λ})_{Div}) ⟶ Im(f_{∞}) ⟶ \underrightarrow{lim}_{λ∈Λ} Cotor(Im(f_{λ})) ⟶ 0,

and the last term is finite by the fact (2.4.6) and the assumption that L is cofinitely generated. Hence we get

\underrightarrow{lim}_{λ∈Λ} (Im(f_{λ})_{Div}) = Im(f_{∞})_{Div}.

Since Im(f_{∞})_{Div} has finite corank, there exists an element λ_{0} ∈ Λ such that Im(f_{λ})_{Div} = Im(f_{∞})_{Div} for any λ ≥ λ_{0}. This fact and (2.4.6)imply the equality

Im(f_{λ}) = Im(f_{λ_{0}}) for any λ ≥ λ_{0}.  (2.4.7)

Now let λ ∈ Λ satisfy λ ≥ λ_{0}. Applying the snake lemma to the commutative diagram

[figure: N_{λ_{0}}; f_{λ_{0}}; N_{λ}; f_{λ}; N_{λ}/N_{λ_{0}}; 0; 0; L; L; 0,]

we get an exact sequence

Ker(f_{λ_{0}}) ⟶ Ker(f_{λ_{0}}) ⟶ N_{λ}/N_{λ_{0}} ⟶^{0} Coker(f_{λ_{0}}) ^{∼} ⟶ Coker(f_{λ}),

which proves Sublemma 2.4.5, beucase N_{λ}/N_{λ_{0}} is divisible by assumption. □

We now turn to the proof of Sublemma 2.4.4. For non-empty open U ⊂ U_{0}, there is a commutative diagram with exact rows

H^{1}(U_{0}, A) → H^{1}(U, A) → ⨁_{v∈U_{0}\U}A(−1)^{G_{𝔽v}} ⟶^{β_{U}} H^{2}(U_{0}, A)
r_{U_{0}} ↓  ^{r_{U}} ↓  ^{α_{U}} ↓
0 → ⨁_{v∈S\U_{0}} H_{/g}^{1}(k_{v}, A) → ⨁_{v∈S\U} H_{/g}^{1}(k_{v}, A) → ⨁_{v∈U_{0}\U} H_{/g}^{1}(k_{v}, A),

where we put

H_{/g}^{1}(k_{v}, A) := H^{1}(k_{v}, A)/H_{g}^{1}(k_{v}, A)

for simplicity. The upper row is obtained from a localization exact sequence of étale cohomology and the isomorphism

H_{v}^{2}(U_{0}, A) ≃ H^{1}(k_{v}, A)/H^{1}(𝔽_{v}, A) ≃ A(−1)^{G_{𝔽v}} for v ∈ U_{0} \ U, where we have used the fact that the action of G_{k} on A is unramified at v ∈ U_{0}. The map α_{U} is obtained from the facts that H_{g}^{1}(k_{v}, A) = H^{1}(k_{v}, A)_{Div} if v ∉ Σ and that H^{1}(𝔽_{v}, A) is divisible (recall that A is assumed to be divisible). It gives

Ker(α_{U}) = ⨁_{v∈U_{0}\U} (A(−1)^{G_{𝔽v}} )_{Div}.  (2.4.8)

Now let ϕ_{U} be the composite map

ϕ_{U} : Ker(α_{U}) ↪ ⨁_{v∈U_{0}\U} A(−1)^{G_{𝔽v}} ⟶^{β_{U}} H^{2}(U_{0}, A),

and let

ψ_{U} : Ker(ϕ_{U}) ⟶ Coker(r_{U_{0}})

be the map induced by the above diagram. Note that

C_{U} ≃ Ker(ψ_{U}), since H_{f,U}^{1}(k, A) = Ker(r_{U}).

By (2.4.8), the inductive system {Ker(α_{U})}_{U⊂U_{0}} and the maps {ϕ_{U}}_{U⊂U_{0}} satisfy the assumptions in Sublemma 2.4.5. Hence there exists a non-empty open subset U′ ⊂ U_{0} such that Ker(ϕ_{U})/Ker(ϕ_{U′}) is divisible for any open U ⊂ U′. Then applying Sublemma 2.4.5 again to the inductive system {Ker(ϕ_{U})}_{U⊂U′} and the maps {ψ_{U}}_{U⊂U′}, we conclude that there exists a non-empty open subset U_{1} ⊂ U′ such that the quotient

Ker(ψ_{U})/Ker(ψ_{U_{1}}) = C_{U}/C_{U_{1}}

is divisible for any open subset U ⊂ U_{1}. This completes the proof of Sublemma 2.4.4 and Lemma 2.4.1. □

Remark 2.4.9 (1) By the argument after Sublemma 2.4.4, Cotor(H_{g}^{1}(k, A)) is finite if A is divisible.

(2) One obtains the first equality in Lemma 2.4.1 by replacing the local terms H_{/g}^{1}(k_{v}, A) in the above diagram with Cotor(H^{1}(k_{v}, A)).

2.5 Cotorsion Part of H^{1}(k, A)

Assume that k is global, and let the notation be as in §2.1. We investigate here the boundary map

δ_{U_{0}} : H^{1}(k, A) ⟶ ⨁_{v∈(U_{0})^{1}} A(−1)^{G_{𝔽v}}

arising from a localization exact sequence of étale cohomology and the purity for discrete valuation rings. Concerning this map, we prove the following standard lemma, which will be used in our proof of Theorem 1.3:

Lemma 2.5.1 (1) The map

δ_{U_{0},Div} : H^{1}(k, A)_{Div} ⟶ ⨁_{v∈(U_{0})^{1}} (A(−1)^{G_{𝔽v}} )_{Div}

induced by δ_{U_{0}} has cofinitely generated cokernel.

(2) The map

δ_{U_{0},Cotor} : Cotor(H^{1}(k, A)) ⟶ ⨁_{v∈(U_{0})^{1}} Cotor(A(−1)^{G_{𝔽v}} )

induced by δ_{U_{0}} has finite kernel and cofinitely generated cokernel.

We have nothing to say about the finiteness of the cokernel of these maps.

Proof. For a non-empty open U ⊂ U_{0}, there is a commutative diagram of cofinitely generated ℤ_{p}-modules

\xymatrix{ & H^{1}(U, A)_{Div} \ar[r]^{γ_{U}} \ar@{^{(}->}[d] & ⨁_{v∈U_{0}\U} ((A(−1)^{G_{𝔽v}} )_{Div} \ar@{^{(}->}[d] \\ H^{1}(U_{0}, A) \ar[r] & H^{1}(U, A) \ar[r]^{α_{U}} & ⨁_{v∈U_{0}\U} A(−1)^{G_{𝔽v}} \ar[r]^{β_{U}} & H^{2}(U_{0}, A),}

where the lower row is obtained from a localization exact sequence of étale cohomology and the purity for discrete valuation rings, and γ_{U} is induced by α_{U}. Let

f_{U} : Cotor(H^{1}(U, A)) ⟶ ⨁_{v∈U_{0}\U} Cotor(A(−1)^{G_{𝔽v}} )

be the map induced by α_{U}. By a diagram chase, we obtain an exact sequence

Ker(f_{U}) ⟶ Coker(γ_{U}) ⟶ Coker(α_{U}) ⟶ Coker(f_{U}) ⟶ 0.

Taking the inductive limit with respect to all non-empty open subsets U ⊂ U_{0}, we obtain an exact sequence

Ker(δ_{U_{0},Cotor}) → Coker(δ_{U_{0},Div}) → \underrightarrow{lim}_{U⊂U_{0}} Coker(α_{U}) → Coker(δ_{U_{0},Cotor}) → 0,

where we have used Lemma 2.4.1 to obtain the equalities

Ker(δ_{U_{0},Cotor}) = \underrightarrow{lim}_{U⊂U_{0}} Ker(f_{U}) and Coker(δ_{U_{0},Div}) = \underrightarrow{lim}_{U⊂U_{0}} Coker(γ_{U}).

Since \underrightarrow{lim}_{U⊂U_{0}} Coker(α_{U}) is a subgroup of H^{2}(U_{0}, A), it is cofinitely generated. Hence the assertions in Lemma 2.5.1 are reduced to showing that Ker(δ_{U_{0},Cotor}) is finite. We prove this finiteness assertion. The lower row of the above diagram yields exact sequences

Cotor(H^{1}(U_{0}, A)) ⟶ Cotor(H^{1}(U, A)) ⟶ Cotor(Im(α_{U})) ⟶ 0, (2.5.2)
T_{p}(Im(β_{U})) ⟶ Cotor(Im(α_{U})) ⟶ ⨁_{v∈U_{0}\U} Cotor(A(−1)^{G_{𝔽v}} ),  (2.5.3)

where the second exact sequence arises from the short exact sequence

0 ⟶ Im(α_{U}) ⟶ ⨁_{v∈U_{0}\U} A(−1)^{G_{𝔽v}} ⟶ Im(β_{U}) ⟶ 0

(cf. Lemma 2.3.2 (1)). Taking the inductive limit of (2.5.2) with respect to all nonempty open U ⊂ U_{0}, we obtain the finiteness of the kernel of the map

Cotor(H^{1}(k, A)) ⟶ \underrightarrow{lim}_{U⊂U_{0}} Cotor(Im(α_{U})).

Taking the inductive limit of (2.5.3) with respect to all non-empty open U ⊂ U_{0}, we see that the kernel of the map

\underrightarrow{lim}_{U⊂U_{0}} Cotor(Im(α_{U})) ⟶ ⨁_{v∈(U_{0})^{1}} Cotor(A(−1)^{G_{𝔽v}} ),

is finite, because we have

\underrightarrow{lim}_{U⊂U_{0}} T_{p}(Im(β_{U})) ⊂ T_{p}(H^{2}(U_{0}, A))

and the group on the right hand side is a finitely generated ℤ_{p}-module. Thus Ker(δ_{U_{0},Cotor}) is finite and we obtain Lemma 2.5.1. □

2.6 Local-Global Principle

Let the notation be as in §2.1. If k is local, then the Galois cohomological dimension cd(k) is 2 (cf. [Se] II.4.3). In the case that k is global, we have cd(k) = 2 either if p ≥ 3 or if k is totally imaginary. Otherwise, H^{q}(k, A) is finite 2-torsion for q ≥ 3 (cf. loc. cit. II.4.4 Proposition 13, II.6.3 Theorem B). As for the second Galois cohomology groups, the following local-global principle due to Jannsen [J2] plays a fundamental role in this paper (see also loc. cit. §7 Corollary 7):

Theorem 2.6.1 ([J2] §4 Theorem 4) Assume that k is global and that i ≠ 2(n−1). Let P be the set of all places of k. Then the map

H^{2}(k, H^{i}(\overline{X}, ℚ_{p}/ℤ_{p}(n))) ⟶ ⨁_{v∈P} H^{2}(k_{v}, H^{i}(\overline{X}, ℚ_{p}/ℤ_{p}(n)))

has finite kernel and cokernel, and the map

H^{2}(k, H^{i}(\overline{X}, ℚ_{p}/ℤ_{p}(n))_{Div}) ⟶ ⨁_{v∈P} H^{2}(k_{v}, H^{i}(\overline{X}, ℚ_{p}/ℤ_{p}(n))_{Div})

is bijective.

We apply these facts to the filtration F^{•} on H^{∗}(X, ℚ_{p}/ℤ_{p}(n)) resulting from the Hochschild-Serre spectral sequence

E_{2}^{u,v} = H^{u}(k, H^{v}(\overline{X}, ℚ_{p}/ℤ_{p}(n))) ⟹ H^{u+v}(X, ℚ_{p}/ℤ_{p}(n)).  (2.6.2)

Corollary 2.6.3 Assume that k is global and that i ≠ 2n. Then:

(1) F^{2}H^{i}(X, ℚ_{p}/ℤ_{p}(n)) is cofinitely generated up to a finite-exponent group.

(2) For v ∈ P, put X_{v} := X ⊗_{k} k_{v}. Then the natural maps

F^{2}H^{i}(X, ℚ_{p}/ℤ_{p}(n)) ⟶ ⨁_{v∈P} F^{2}H^{i}(X_{v}, ℚ_{p}/ℤ_{p}(n)),
F^{2}H^{i}(X, ℚ_{p}/ℤ_{p}(n))_{Div} ⟶ ⨁_{v∈P} F^{2}H^{i}(X_{v}, ℚ_{p}/ℤ_{p}(n))_{Div}

have finite kernel and cokernel (and the second map is surjective).

Proof. Let 𝔬_{k} be the integer ring of k, and put S := Spec(𝔬_{k}). Note that the set of all finite places of k agrees with S^{1}.

(1) The group H^{2}(k_{v}, H^{i−2}(\overline{X}, ℚ_{p}/ℤ_{p}(n))_{Div}) is divisible and cofinitely generated for any v ∈ S^{1}, and it is zero if p |̸ v and X has good reduction at v, by the local Poitou-Tate duality [Se] II.5.2 Théorème 2 and Deligne’s proof of the Weil conjecture [De2] (see [Sat2] Lemma 2.4 for details). The assertion follows from this fact and Theorem 2.6.1.

(2) We prove the assertion only for the first map. The assertion for the second map is similar and left to the reader. For simplicity, we assume that

(♯) p ≥ 3 or k is totally imaginary.

Otherwise one can check the assertion by repeating the same arguments as below in the category of abelian groups modulo finite abelian groups. By (♯), we have cd_{p}(k) = 2 and there is a commutative diagram

\xymatrix{H^{2}(k, H^{i−2}(\overline{X}, ℚ_{p}/ℤ_{p}(n))) \ar[r] \ar@{->>}[d] & ⨁_{v∈S^{1}} H^{2}(k_{v}, H^{i−2}(\overline{X}, ℚ_{p}/ℤ_{p}(n))) \ar@{->>}[d] \\ F^{2}H^{i}(X, ℚ_{p}/ℤ_{p}(n)) \ar[r] & ⨁_{v∈S^{1}} F^{2}H^{i}(X_{v}, ℚ_{p}/ℤ_{p}(n)),}

where the vertical arrows are edge homomorphisms of Hochschild-Serre spectral sequences and these arrows are surjective. Since

H^{2}(k_{v}, H^{i−2}(\overline{X}, ℚ_{p}/ℤ_{p}(n))) = 0 for archimedean places v

by (♯), the top horizontal arrow has finite kernel and cokernel by Theorem 2.6.1. Hence it is enough to show that the right vertical arrow has finite kernel. For any v ∈ S^{1}, the v-component of this map has finite kernel by Deligne’s criterion [De1] (see also [Sat2] Remark 1.2). If v is prime to p and X has good reduction at v, then the v-component is injective. Indeed, there is an exact sequence resulting from a Hochschild-Serre spectral sequence and the fact that cd(k_{v}) = 2:

H^{i−1}(X_{v}, ℚ_{p}/ℤ_{p}(n)) →^{d} H^{i−1}(\overline{X}, ℚ_{p}/ℤ_{p}(n)))^{G_{kv}}

→ H^{2}(k_{v}, H^{i−2}(\overline{X}, ℚ_{p}/ℤ_{p}(n))) → F^{2}H^{i}(X_{v}, ℚ_{p}/ℤ_{p}(n)).

The edge homomorphism d is surjective by the commutative square

\xymatrix{H^{i−1}(Y_{v}, ℚ_{p}/ℤ_{p}(n)) \ar@{->>}[r] \ar[d] & H^{i−1}(\overline{Y_{v}}, ℚ_{p}/ℤ_{p}(n)))^{G_{𝔽v}} \ar[d]^{≀} \\ H^{i−1}(X_{v}, ℚ_{p}/ℤ_{p}(n)) \ar[r]^{d} & H^{i−1}(\overline{X}, ℚ_{p}/ℤ_{p}(n)))^{G_{kv}} .}

Here Y_{v} denotes the reduction of X at v and \overline{Y_{v}} denotes Y_{v} ⊗_{𝔽_{v}} \overline{𝔽_{v}}. The left (resp. right) vertical arrow arises from the proper base-change theorem (resp. proper smooth base-change theorem), and the top horizontal arrow is surjective by the fact that cd(𝔽_{v}) = 1. Thus we obtain the assertion. □

3 Finiteness of Torsion in a Chow Group

Let k, S, p and Σ be as in the beginning of §2, and let X be a proper smooth geometrically integral variety over Spec(k). We introduce the following technical condition:

H0: The group H_{ét}^{3}(\overline{X}, ℚ_{p}(2))^{G_{k}} is trivial.

If k is global, H0 always holds by Deligne’s proof of the Weil conjecture [De2]. When k is local, H0 holds if dim(X) = 2 or if X has good reduction (cf. [CTR2] §6); it is in general a consequence of the monodromy-weight conjecture.

3.1 Finiteness of CH^{2}(X)_{p-tors}

The purpose of this section is to show the following result, which is a generalization of a result of Langer [La4] Proposition 3 and implies the finiteness assertion on CH^{2}(X)_{p-tors} in Theorem 1.3 (1):

Theorem 3.1.1 Assume H0, H1∗ and either p ≥ 5 or the equality

H_{g}^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div} = H^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div}.  (∗_{g})

Then CH^{2}(X)_{p-tors} is finite.

Remark 3.1.2 (1) (∗_{g}) holds if H^{2}(X, 𝒪_{X}) = 0 or if k is ℓ-adic local with ℓ ≠ p.

(2) Crucial facts to this theorem are Lemmas 3.2.2, 3.3.5 and 3.5.2 below. The short exact sequence in Lemma 3.2.2 is an important consequence of the Merkur’ev-Suslin theorem [MS].

(3) In Theorem 3.1.1, we do not need to assume that X has good or semistable reduction at any prime of k dividing p (cf. 1.8.1), because we do not need this assumption in Lemma 3.5.2 by the alteration theorem of de Jong [dJ].

3.2 Regulator Map

We recall here the definition of the regulator maps

reg_{Λ} : CH^{2}(X, 1) ⊗ Λ ⟶ H_{ind}^{1}(k, H^{2}(\overline{X}, Λ(2)))  (3.2.1)

with Λ = ℚ_{p} or ℚ_{p}/ℤ_{p}, assuming H0. The general framework on étale Chern class maps and regulator maps is due to Soulé [So1], [So2]. We include here a more elementary construction of reg_{Λ}, which will be useful in this paper. Let K := k(X) be the function field of X. Take an open subset U_{0} ⊂ S \ Σ = S[p^{−1}] and a smooth proper scheme 𝒳_{U_{0}} over U_{0} satisfying 𝒳_{U_{0}} ×_{U_{0}} Spec(k) ≃ X. For an open subset U ⊂ U_{0}, put 𝒳_{U} := 𝒳_{U_{0}} ×_{U_{0}} U and define

N^{1}H^{3}(𝒳_{U}, μ_{p^{r}}^{⊗2}) := Ker(H^{3}(𝒳_{U}, μ_{p^{r}}^{⊗2}) → H^{3}(K, μ_{p^{r}}^{⊗2})).

Lemma 3.2.2 For an open subset U ⊂ U_{0}, there is an exact sequence

0 ⟶ CH^{2}(𝒳_{U}, 1)/p^{r} ⟶ N^{1}H^{3}(𝒳_{U}, μ_{p^{r}}^{⊗2}) ⟶ _{p^{r}}CH^{2}(𝒳_{U}) ⟶ 0

See §1.7 for the definition of CH^{2}(𝒳_{U}, 1).

Proof. The following argument is due to Bloch [Bl], Lecture 5. We recall it for the convenience of the reader. There is a localization spectral sequence

E_{1}^{u,v} = ⨁_{x∈(𝒳_{U})^{u}} H_{x}^{u+v}(𝒳_{U}, μ_{p^{r}}^{⊗2}) ⟹ H^{u+v}(𝒳_{U}, μ_{p^{r}}^{⊗2}).  (3.2.3)

By the relative smooth purity, there is an isomorphism

E_{1}^{u,v} ≃ ⨁_{x∈(𝒳_{U})^{u}} H^{v−u}(x, μ_{p^{r}}^{⊗2−u}),  (3.2.4)

which implies that N^{1}H^{3}(𝒳_{U}, μ_{p^{r}}^{⊗2}) is isomorphic to the cohomology of the Bloch-Ogus complex

H^{2}(K, μ_{p^{r}}^{⊗2}) ⟶ ⨁_{y∈(𝒳_{U})^{1}} H^{1}(y, μ_{p^{r}}) ⟶ ⨁_{x∈(𝒳_{U})^{2}} ℤ/p^{r}.

By Hilbert’s theorem 90 and the Merkur’ev-Suslin theorem [MS], this complex is isomorphic to the Gersten complex

K_{2}^{M}(K)/p^{r} ⟶ ⨁_{y∈(𝒳_{U})^{1}} κ(y)^{×}/p^{r} ⟶ ⨁_{x∈(𝒳_{U})^{2}} ℤ/p^{r}.

On the other hand, there is an exact sequence obtained by a diagram chase

0 ⟶ CH^{2}(𝒳_{U}, 1) ⊗ ℤ/p^{r} ⟶ CH^{2}(𝒳_{U}, 1; ℤ/p^{r}) ⟶ _{p^{r}}CH^{2}(𝒳_{U}) ⟶ 0.

Here CH^{2}(𝒳_{U}, 1; ℤ/p^{r}) denotes the cohomology of the above Gersten complex and it is isomorphic to N^{1}H^{3}(𝒳_{U}, μ_{p^{r}}^{⊗2}). Thus we obtain the lemma. □

Put

M^{q} := H^{q}(\overline{X}, Λ(2)) with Λ ∈ \{ℚ_{p}, ℚ_{p}/ℤ_{p}\}.

For an open subset U ⊂ U_{0} let H^{∗}(U, M^{q}) be the étale cohomology with coefficients in the smooth sheaf associated with M^{q}. There is a Leray spectral sequence

E_{2}^{u,v} = H^{u}(U, M^{v}) ⟹ H^{u+v}(𝒳_{U}, Λ(2)).

By Lemma 3.2.2, there is a natural map

CH^{2}(𝒳_{U}, 1) ⊗ Λ ⟶ H^{3}(𝒳_{U}, Λ(2)).

Noting that E_{2}^{0,3} is zero or finite by H0, we define the map

reg_{𝒳_{U} ,Λ} : CH^{2}(𝒳_{U}, 1) ⊗ Λ ⟶ H^{1}(U, M^{2})

as the composite of the above map with an edge homomorphism of the Leray spectral sequence. Finally we define reg_{Λ} in (3.2.1)by passing to the limit over all non-empty open U ⊂ U_{0}. Our construction of reg_{Λ} does not depend on the choice of U_{0} or 𝒳_{U_{0}}.

Remark 3.2.5 By Lemma 2.4.1, H1 always implies H1∗. If k is local, H1∗ conversely implies H1. If k is global, one can check that H1∗ implies H1, assuming that the group Ker(CH^{2}(𝒳_{U_{0}}) → CH^{2}(X)) is finitely generated up to torsion and that the Tate conjecture for divisors holds for almost all closed fibers of 𝒳_{U_{0}}/U_{0}.

3.3 Proof of Theorem 3.1.1

We start the proof of Theorem 3.1.1, which will be completed in §3.5 below. By Lemma 3.2.2, there is an exact sequence

0 ⟶ CH^{2}(X, 1) ⊗ ℚ_{p}/ℤ_{p} ⟶^{ϕ} N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶ CH^{2}(X)_{p-tors} ⟶ 0,

(3.3.1)

where we put

N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) := Ker(H^{3}(X, ℚ_{p}/ℤ_{p}(2)) → H^{3}(K, ℚ_{p}/ℤ_{p}(2))).

In view of (3.3.1), Theorem 3.1.1 is reduced to the following two propositions:

Proposition 3.3.2 (1) If k is local, then CH^{2}(X)_{p-tors} is cofinitely generated over ℤ_{p}.

(2) Assume that k is global, and that Coker(reg_{ℚ_{p}/ℤ_{p}})_{Div} is cofinitely generated over ℤ_{p}. Then CH^{2}(X)_{p-tors} is cofinitely generated over ℤ_{p}.

Proposition 3.3.3 Assume H0, H1∗ and either p ≥ 5 or (∗_{g}). Then we have

Im(ϕ) = N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))_{Div}.

We will prove Proposition 3.3.2 in §3.4 below, and Proposition 3.3.3 in §3.5 below.

Remark 3.3.4 (1) If k is local, then H^{3}(X, ℚ_{p}/ℤ_{p}(2)) is cofinitely generated. Hence Proposition 3.3.2 (1) immediately follows from the exact sequence (3.3.1).

(2) When k is global, then H^{1}(k, A)_{Div}/H_{g}^{1}(k, A)_{Div} with A := H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)) is cofinitely generated by Lemma 2.4.1. Hence H1∗ implies the second assumption of Proposition 3.3.2 (2).

Let F^{•} be the filtration on H^{∗}(X, ℚ_{p}/ℤ_{p}(2)) resulting from the Hochschild-Serre spectral sequence (2.6.2). The following fact due to Salberger will play key roles in our proof of the above two propositions:

Lemma 3.3.5 ([Sal] Main Lemma 3.9) The following group has a finite exponent:

N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ∩ F^{2}H^{3}(X, ℚ_{p}/ℤ_{p}(2)).

3.4 Proof of Proposition 3.3.2

For (1), see Remark 3.3.4 (1). We assume that k is global, and prove (2). Put

H^{3} := H^{3}(X, ℚ_{p}/ℤ_{p}(2)) and Γ := ϕ(CH^{2}(X, 1) ⊗ ℚ_{p}/ℤ_{p}) ⊂ H^{3}

(cf. (3.3.1)). Let F^{•} be the filtration on H^{3} resulting from the spectral sequence (2.6.2), and put N^{1}H^{3} := N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)). We have Γ ⊂ (F^{1}H^{3})_{Div} = (H^{3})_{Div} by H0, and there is a filtration on H^{3}

0 ⊂ Γ + (F^{2}H^{3})_{Div} ⊂ (F^{1}H^{3})_{Div} ⊂ H^{3}.

By (3.3.1), the inclusion N^{1}H^{3} ⊂ H^{3} induces an inclusion CH^{2}(X)_{p-tors} ⊂ H^{3}/Γ. We show that the image of this inclusion is cofinitely generated, using the above filtration on H^{3}. It suffices to show the following lemma:

Lemma 3.4.1 (1) The kernel of CH^{2}(X)_{p-tors} → H^{3}/(Γ + (F^{2}H^{3})_{Div}) is finite.

(2) The image of CH^{2}(X)_{p-tors} → H^{3}/(F^{1}H^{3})_{Div} is finite.

(3) The second assumption of Proposition 3.3.2 (2) implies that the group

M := (F^{1}H^{3})_{Div}/(Γ + (F^{2}H^{3})_{Div})

is cofinitely generated.

Proof. (1) There is an exact sequence

0 ⟶ \frac{N^{1}H^{3} ∩ (F^{2}H^{3})_{Div}}{Γ ∩ (F^{2}H^{3})_{Div}} ⟶ CH^{2}(X)_{p-tors} ⟶ \frac{H^{3}}{Γ + (F^{2}H^{3})_{Div}}.

Hence (1) follows from Lemma 3.3.5 and Corollary 2.6.3 (1).

(2) Let U_{0} and 𝒳_{U_{0}} → U_{0} be as in §3.2. For non-empty open U ⊂ U_{0}, there is a commutative diagram up to a sign

\xymatrix{N^{1}H^{3}(𝒳_{U}, ℚ_{p}/ℤ_{p}(2)) \ar[r] \ar@{^{(}->}[d] & CH^{2}(𝒳_{U}) ⊗ ℤ_{p} \ar[d]^{ϱ} \\ H^{3}(𝒳_{U}, ℚ_{p}/ℤ_{p}(2)) \ar[r] & H^{4}(𝒳_{U}, ℤ_{p}(2))}

by the same argument as for [CTSS], §1, Proposition 1. Here the top arrow is the composite of N^{1}H^{3}(𝒳_{U}, ℚ_{p}/ℤ_{p}(2)) → CH^{2}(𝒳_{U})_{p-tors} (cf. Lemma 3.2.2) with the natural inclusion. The bottom arrow is a Bockstein map and the right vertical arrow is the cycle class map of 𝒳_{U}. Taking the inductive limit with respect to all non-empty U ⊂ U_{0}, we see that the left square of the following diagram commutes (up to a sign):

\xymatrix{N^{1}H^{3} \ar[r] \ar@{^{(}->}[d] & CH^{2}(X) ⊗ ℤ_{p} \ar[d]^{ϱ_{ind}} \ar[dr]^{ϱ_{cont}} \\ H^{3} \ar[r] & H_{ind}^{4}(X, ℤ_{p}(2)) \ar[r] & H_{cont}^{4}(X, ℤ_{p}(2)),}

where H_{cont}^{∗}(X, ℤ_{p}(2)) denotes the continuous étale cohomology [J1] and the bottom right arrow is by definition the inductive limit, with respect to U ⊂ U_{0}, of the natural restriction map

H^{4}(𝒳_{U}, ℤ_{p}(2)) = H_{cont}^{4}(𝒳_{U}, ℤ_{p}(2)) ⟶ H_{cont}^{4}(X, ℤ_{p}(2)).

The right triangle of the diagram commutes by the definition of cycle classes in loc. cit. Theorem (3.23). This diagram and the exact sequence (3.3.1)yield a commutative diagram (up to a sign)

\xymatrix{CH^{2}(X)_{p-tors} \ar[r] \ar[d] & CH^{2}(X) ⊗ ℤ_{p} \ar[d]^{ϱ_{cont}} \\ H^{3}/(F^{1}H^{3})_{Div} \ar@{^{(}->}[r] & H_{cont}^{4}(X, ℤ_{p}(2)),}

where the bottom arrow is injective by H0 and loc. cit. Theorem (5.14). Now the assertion follows from that fact that Im(ϱ_{cont}) is finitely generated over ℤ_{p} ([Sa] Theorem (4-4)).

(3) Put

N := (F^{1}H^{3})_{Div}/{Γ + (F^{2}H^{3} ∩ (F^{1}H^{3})_{Div})} = Coker(reg_{ℚ_{p}/ℤ_{p}})_{Div},

which is cofinitely generated by assumption and fits into an exact sequence

(F^{2}H^{3} ∩ (F^{1}H^{3})_{Div})/(F^{2}H^{3})_{Div} ⟶ M ⟶ N ⟶ 0.

The first group in this sequence has a finite exponent by Corollary 2.6.3 (1), N is divisible and cofinitely generated, and M is divisible. Hence M is cofinitely generated by Lemma 2.3.2 (2). This completes the proof of Lemma 3.4.1 and Proposition 3.3.2. □

3.5 Proof of Proposition 3.3.3

We put

NF^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) := N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ∩ F^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)).

Note that N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))_{Div} = NF^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))_{Div} by H0. There is an edge homomorphism of the spectral sequence (2.6.2)

ψ : F^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶ H^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2))).  (3.5.1)

The composite of ϕ in (3.3.1)and ψ agrees with reg_{ℚ_{p}/ℤ_{p}}. Hence by Lemma 3.3.5, the assertion of Proposition 3.3.3 is reduced to the following lemma, which generalizes [LS] Lemma (5.7) and extends [La1] Lemma (3.3):

Lemma 3.5.2 Assume either p ≥ 5 or (∗_{g}) (but we do not assume H1∗). Then we have

ψ(NF^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))_{Div}) ⊂ H_{g}^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2))).

We start the proof of this lemma. The assertion is obvious under the assumption (∗_{g}). Hence we are done if k is ℓ-adic local with ℓ ≠ p (cf. Remark 3.1.2 (1)). It remains to deal with the following two cases:

(1) k is p-adic local with p ≥ 5.

(2) k is global and p ≥ 5.

Put A := H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)) for simplicity. We first reduce the case (2) to the case (1). Suppose that k is global. Then there is a commutative diagram

\xymatrix{NF^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))_{Div} \ar[r] \ar[d] & H^{1}(k, A) \ar[d] \\ ∏_{v∈S^{1}} NF^{1}H^{3}(X_{v}, ℚ_{p}/ℤ_{p}(2))_{Div} \ar[r] & ∏_{v∈S^{1}} H^{1}(k_{v}, A),}

where the vertical arrows are natural restriction maps. By this diagram and the definition of H_{g}^{1}(k, A), the case (2) is reduced to the case (1).

We prove the case (1). We first reduce the problem to the case where X has semistable reduction. By the alteration theorem of de Jong [dJ], there exists a proper flat generically finite morphism X′ → X such that X′ is projective smooth over k and has a proper flat regular model over the integral closure 𝔬′ of 𝔬_{k} in Γ (X′, 𝒪_{X′}) with semistable reduction. Put

L := Frac(𝔬′) and A′ := H^{2}(X′ ⊗_{L} \overline{k}, ℚ_{p}/ℤ_{p}(2)).

Then there is a commutative diagram whose vertical arrows are natural restriction maps

\xymatrix{NF^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))_{Div} \ar[r] \ar[d] & H^{1}(k, A) \ar[r] \ar[d] & H^{1}(k, A)/H_{g}^{1}(k, A) \ar[d] \\ NF^{1}H^{3}(X′, ℚ_{p}/ℤ_{p}(2))_{Div} \ar[r] & H^{1}(L, A′) \ar[r] & H^{1}(L, A′)/H_{g}^{1}(L, A′).}

Our task is to show that the composite of the upper row is zero. Because X′ and X are proper smooth varieties over k, the restriction map r : A → A′ has a quasi-section s : A′ → A with s ∘ r = d · id_{A}, where d denotes the extension degree of the function field of X′ ⊗_{L} \overline{k} over that of \overline{X}. Hence by the functoriality of H_{g}^{1}(k, A) in A, the right vertical arrow in the above diagram has finite kernel, and the problem is reduced to showing that the composite of the lower row is zero. Thus we are reduced to the case that X has a proper flat regular model 𝒳 over S = Spec(𝔬_{k}) with semistable reduction. We prove this case in what follows.

Let j : X ↪ 𝒳 be the natural open immersion. There is a natural injective map

[figure: α_{r} : H^{3}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}); H^{3}(X, μ_{p^{r}}^{⊗2})]

induced by the natural morphism τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2} → Rj_{∗}μ_{p^{r}}^{⊗2}. By Corollary 2.2.3, it suffices to show the following two lemmas (see also Remark 3.5.6 below):

Lemma 3.5.3 N^{1}H^{3}(X, μ_{p^{r}}^{⊗2}) ⊂ Im(α_{r}) for any r ≥ 1.

Lemma 3.5.4 Put

H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}/ℤ_{p}(2)) := \underrightarrow{lim}_{r≥1} H^{3}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}),

and define H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}/ℤ_{p}(2))^{0} as the kernel of the natural map

H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}/ℤ_{p}(2)) → H^{3}(\overline{X}, ℚ_{p}/ℤ_{p}(2)).

Then the canonical map

H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}(2))^{0} ⟶ H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}/ℤ_{p}(2))^{0}

has finite cokernel, where H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}(2))^{0} is as we defined in §2.2.

To prove Lemma 3.5.3, we need the following fact due to Hagihara, whose latter vanishing will be used later in §6:

Lemma 3.5.5 ([SH] A.2.4, A.2.6) Let n, r and c be integers with n ≥ 0 and r, c ≥ 1. Then for any q ≤ n + c and any closed subscheme Z ⊂ Y with codim_{𝒳} (Z) ≥ c, we have

H_{Z}^{q}(𝒳, τ_{≤n}Rj_{∗}μ_{p^{r}}^{⊗n}) = 0 = H_{Z}^{q+1}(𝒳, τ_{≥n+1}Rj_{∗}μ_{p^{r}}^{⊗n}).

Proof of Lemma 3.5.3. We compute the local-global spectral sequence

E_{1}^{u,v} = ⨁_{x∈𝒳^{a}} H_{x}^{u+v}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}) ⟹ H^{u+v}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}).

By the first part of Lemma 3.5.5 and the smooth purity for points on X, we have

E_{1}^{u,v} = \begin{cases}H^{v}(K, μ_{p^{r}}^{⊗2}) & (if u = 0) \\ ⨁_{x∈X^{u}} H^{v−u}(x, μ_{p^{r}}^{⊗2−u}) & (if v ≤ 2). \end{cases}

Repeating the same computation as in the proof of Lemma 3.2.2, we obtain

N^{1}H^{3}(X, μ_{p^{r}}^{⊗2}) ≃ E_{2}^{1,2} = E_{∞}^{1,2} ↪ H^{3}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}),

which implies Lemma 3.5.3. □

Remark 3.5.6 Lemma 3.5.3 extends a result of Langer-Saito ([LS] Lemma (5.4)) to regular semistable families and removes the assumption in [La1] Lemma (3.1) concerning Gersten’s conjecture for algebraic K-groups. Therefore the same assumption in loc. cit. Theorem A has been removed as well.

Proof of Lemma 3.5.4. By the Bloch-Kato-Hyodo theorem on the structure of p-adic vanishing cycles ([BK1], [Hy]), there is a distinguished triangle of the following form in D^{b}(𝒳_{ét}) (cf. [SH], (4.3.3)):

τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2} ⟶ τ_{≤2}Rj_{∗}μ_{p^{r+s}}^{⊗2} ⟶ τ_{≤2}Rj_{∗}μ_{p^{s}}^{⊗2} ⟶ (τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2})[1]

Taking étale cohomology groups, we obtain a long exact sequence

⋯ → H^{q}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}) → H^{q}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{r+s}}^{⊗2}) → H^{q}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{s}}^{⊗2})
→ H^{q+1}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}) → · · · .  (3.5.7)

We claim that H^{q}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}) is finite for any q and r. Indeed, the claim is reduced to the case r = 1 by the exactness of (3.5.7) and this case follows from the Bloch-Kato-Hyodo theorem mentioned above and the properness of 𝒳 over S. Hence taking the projective limit of (3.5.7) with respect to r and then taking the inductive limit with respect to s we obtain a long exact sequence

⋯ → H^{q}(𝒳, τ_{≤2}Rj_{∗}ℤ_{p}(2)) → H^{q}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}(2)) → H^{q}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}/ℤ_{p}(2))

→ H^{q+1}(𝒳, τ_{≤2}Rj_{∗}ℤ_{p}(2)) → ⋯,

where H^{q}(𝒳, τ_{≤2}Rj_{∗}ℤ_{p}(2)) is finitely generated over ℤ_{p} for any q. The assertion in the lemma easily follows from this exact sequence and a similar long exact sequence of étale cohomology groups of \overline{X}. The details are straight-forward and left to the reader. □

This completes the proof of Lemma 3.5.2, Proposition 3.3.3 and Theorem 3.1.1.

4 Cycle Class Map and Unramified Cohomology

Let k, S, p, 𝒳 and K be as in the notation 1.8. In particular, we always assume that 𝒳 satisfies 1.8.1. In this section we give a brief review of p-adic étale Tate twists and provide some preliminary results on cycle class maps. The main result of this section is Corollary 4.4.3 below.

4.1 p-adic Étale Tate Twist

Let n and r be positive integers. We recall here the fundamental properties (S1)– (S7) listed below of the object 𝔗_{r}(n) = 𝔗_{r}(n)_{𝒳} ∈ D^{b}(𝒳_{ét}, ℤ/p^{r}) introduced by the second author [SH]. The properties (S1), (S2), (S3) and (S4) characterizes 𝔗_{r}(n) uniquely up to a unique isomorphism in D^{b}(𝒳_{ét}, ℤ/p^{r}).

(S1) There is an isomorphism t : 𝔗_{r}(n)|_{V} ≃ μ_{p^{r}}^{⊗n} on V := 𝒳 [p^{−1}].

(S2) 𝔗_{r}(n) is concentrated in [0, n].

(S3) Let Z ⊂ 𝒳 be a locally closed regular subscheme of pure codimension c with ch(Z) = p. Let i : Z → 𝒳 be the natural immersion. Then there is a canonical Gysin isomorphism

Gys_{i}^{n} : W_{r} Ω_{Z,log}^{n−c}[−n − c] ^{∼} ⟶ τ_{≤n+c}Ri^{!}𝔗_{r}(n) in D^{b}(Z_{ét}, ℤ/p^{r}),

where W_{r} Ω_{Z,log}^{q} denotes theétale subsheaf of the logarithmic part of the Hodge-Witt sheaf W_{r} Ω_{Z}^{q} ([Bl1], [Il]).

(S4) For x ∈ 𝒳 and q ∈ ℤ_{≥0}, we define ℤ/p^{r}(q) ∈ D^{b}(x_{ét}, ℤ/p^{r}) as

ℤ/p^{r}(q) := \begin{cases}μ_{p^{r}}^{⊗q} & (if ch(x) ≠ p) \\ W_{r} Ω_{x,log}^{q}[−q] & (if ch(x) = p). \end{cases}

Then for y, x ∈ 𝒳 with c := codim(x) = codim(y)+1, there is a commutative diagram

\xymatrix{H^{n−c+1}(y, ℤ/p^{r}(n − c + 1)) \ar[r]^{−∂^{val}} \ar[d]_{_{Gys{n \atop iy}}} & H^{n−c}(x, ℤ/p^{r}(n − c)) \ar[d]^{_{Gys{n \atop ix}}} \\ H_{y}^{n+c−1}(𝒳, 𝔗_{r}(n)) \ar[r]^{δ^{loc}} & H_{x}^{n+c}(𝒳, 𝔗_{r}(n)).}

Here for z ∈ 𝒳 , Gys_{i_{z}}^{n} is induced by the Gysin map in (S3) (resp. the absolute purity [RZ], [Th], [FG]) if ch(z) = p (resp. ch(z) ≠ p). The arrow δ^{loc} denotes the boundary map of a localization exact sequence and ∂^{val} denotes the boundary map of Galois cohomology groups due to Kato [KCT] §1.

(S5) Let Y be the union of the fibers of 𝒳 /S of characteristic p. We define the étale sheaf ν_{Y,r}^{n−1} on Y as

ν_{Y,r}^{n−1} := Ker(∂^{val} : ⨁_{y∈Y^{0}} i_{y∗}W_{r} Ω_{y,log}^{n−1} ⟶ ⨁_{x∈Y^{1}} i_{x∗}W_{r} Ω_{x,log}^{n−2}),

where for y ∈ Y, i_{y} denotes the canonical map y ↪ Y. Let i and j be as follows:

[figure: V = 𝒳 [p^{−1}]; j; 𝒳; Y.; i]

Then there is a distinguished triangle in D^{b}(𝒳_{ét}, ℤ/p^{r})

i_{∗}ν_{Y,r}^{n−1}[−n − 1] ⟶^{g} 𝔗_{r}(n) ⟶^{t′} τ_{≤n}Rj_{∗}μ_{p^{r}}^{⊗n} ⟶^{σ} i_{∗}ν_{Y,r}^{n−1}[−n],

where t′ is induced by the isomorphism t in (S1) and the acyclicity property (S2). The arrow g arises from the Gysin morphisms in (S3), σ is induced by the boundary maps of Galois cohomology groups (cf. (S4)).

(S6) There is a canonical distinguished triangle of the following form in D^{b}(𝒳_{ét}):

𝔗_{r+s}(n) ⟶ 𝔗_{s}(n) ⟶^{δ_{s,r}} 𝔗_{r}(n)[1] ⟶^{\underline{p}^{s}} 𝔗_{r+s}(n)[1].

(S7) H^{i}(𝒳, 𝔗_{r}(n)) is finite for any r and i (by the properness of 𝒳 ).

When k is p-adic local with p ≥ n + 2 and 𝒳 is smooth over S, then i^{∗}𝔗_{r}(n) is isomorphic to the syntomic complex 𝒮_{r}(n) of Kato [Ka1], which is the derived image of a syntomic sheaf of Fontaine-Messing [FM]. This fact follows from a result of Kurihara [Ku] and (S5). Therefore our object 𝔗_{r}(n) extends the syntomic complexes to the global situation. Note also that i^{∗}𝔗_{r}(n) is not the log syntomic complex s_{r}^{log}(n) unless n > dim(𝒳 ), because the latter object is isomorphic to τ_{≤n}i^{∗}Rj_{∗}μ_{p^{r}}^{⊗n} by Theorem 2.2.1.

Remark 4.1.1 The above properties of 𝔗_{r}(n) deeply rely on the computation on the étale sheaf of p-adic vanishing cycles due to Bloch-Kato [BK1] and Hyodo [Hy].

Lemma 4.1.2 Put

H^{q}(𝒳, 𝔗_{ℤ_{p}}(n)) := \underleftarrow{lim}_{r≥1} H^{q}(𝒳, 𝔗_{r}(n)), H^{q}(𝒳, 𝔗_{∞}(n)) := \underrightarrow{lim}_{r≥1} H^{q}(𝒳, 𝔗_{r}(n))

and H^{q}(𝒳, 𝔗_{ℚ_{p}}(n)) := H^{q}(𝒳, 𝔗_{ℤ_{p}}(n))⊗_{ℤ_{p}} ℚ_{p}. Then there is a long exact sequence of ℤ_{p}-modules

\begin{aligned}⋯ & ⟶ & H^{q}(𝒳, 𝔗_{ℤ_{p}}(n)) & ⟶ & H^{q}(𝒳, 𝔗_{ℚ_{p}}(n)) & ⟶ & H^{q}(𝒳, 𝔗_{∞}(n)) \\ & ⟶ & H^{q+1}(𝒳, 𝔗_{ℤ_{p}}(n)) & ⟶ & ⋯, \end{aligned}

where H^{q}(𝒳, 𝔗_{ℤ_{p}}(n)) is finitely generated over ℤ_{p}, H^{q}(𝒳, 𝔗_{∞}(n)) is cofinitely generated over ℤ_{p}, and H^{q}(𝒳, 𝔗_{ℚ_{p}}(n)) is finite-dimensional over ℚ_{p}.

Proof. The assertions immediately follow from (S6) and (S7). The details are straight-forward and left to the reader. □

4.2 Cycle Class Map

Let us review the definition of the cycle map

ϱ_{r}^{n} : CH^{n}(𝒳 )/p^{r} ⟶ H^{2n}(𝒳, 𝔗_{r}(n)).

Consider the local-global spectral sequence

E_{1}^{u,v} = ⨁_{x∈𝒳^{u}} H_{x}^{u+v}(𝒳, 𝔗_{r}(n)) ⟹ H^{u+v}(𝒳, 𝔗_{r}(n)).  (4.2.1)

By (S3) and the absolute cohomological purity [FG] (cf. [RZ], [Th]), we have

E_{1}^{u,v} ≃ ⨁_{x∈𝒳^{u}} H^{v−u}(x, ℤ/p^{r}(n − u)) for v ≤ n.  (4.2.2)

This implies that there is an edge homomorphism E_{2}^{n,n} → H^{2n}(𝒳, 𝔗_{r}(n)) with

E_{2}^{n,n} ≃ Coker(∂^{val} : ⨁_{y∈𝒳^{n−1}} H^{1}(y, ℤ/p^{r}(1)) ⟶ ⨁_{x∈𝒳^{n}} H^{0}(x, ℤ/p^{r}))
= CH^{n}(𝒳 )/p^{r},

where ∂^{val} is as in (S4). We define ϱ_{r}^{n} as the composite map

ϱ_{r}^{n} : CH^{n}(𝒳 )/p^{r} ≃ E_{2}^{n,n} ⟶ H^{2n}(𝒳, 𝔗_{r}(n)).

In what follows, we restrict our attention to the case n = 2.

Lemma 4.2.3 Let Z ⊂ 𝒳 be a closed subscheme of pure codimension 1, and let K be the function field of 𝒳 . Put

N^{1}H^{i}(𝒳, 𝔗_{r}(2)) := Ker(H^{i}(𝒳, 𝔗_{r}(2)) → H^{i}(K, μ_{p^{r}}^{⊗2})),
N^{2}H_{Z}^{i}(𝒳, 𝔗_{r}(2)) := Ker(H_{Z}^{i}(𝒳, 𝔗_{r}(2)) → ⨁_{z∈Z^{0}} H_{z}^{i}(𝒳, 𝔗_{r}(2))).

(1) N^{1}H^{3}(𝒳, 𝔗_{r}(2)) is isomorphic to the cohomology of the Gersten complex modulo p^{r}

K_{2}^{M}(K)/p^{r} ⟶ ⨁_{y∈𝒳^{1}} κ(y)^{×}/p^{r} ⟶ ⨁_{x∈𝒳^{2}} ℤ/p^{r},

and there is an exact sequence

0 ⟶ CH^{2}(𝒳, 1)/p^{r} ⟶ N^{1}H^{3}(𝒳, 𝔗_{r}(2)) ⟶ _{p^{r}}CH^{2}(𝒳 ) ⟶ 0.

See §1.7 for the definition of CH^{2}(𝒳, 1).

(2) There are isomorphisms

H_{Z}^{3}(𝒳, 𝔗_{r}(2)) ≃ Ker(∂^{val} : ⨁_{z∈Z^{0}} κ(z)^{×}/p^{r} → ⨁_{x∈Z^{1}} ℤ/p^{r}),
N^{2}H_{Z}^{4}(𝒳, 𝔗_{r}(2)) ≃ Coker(∂^{val} : ⨁_{z∈Z^{0}} κ(y)^{×}/p^{r} → ⨁_{x∈Z^{1}} ℤ/p^{r})
= CH_{d−2}(Z)/p^{r},

where d denotes the Krull dimension of 𝒳 .

Proof. (1) follows from a similar argument as for the proof of Lemma 3.2.2, using the spectral sequence

E_{1}^{u,v} = ⨁_{x∈𝒳^{u}} H_{x}^{u+v}(𝒳, 𝔗_{r}(2)) ⟹ H^{u+v}(𝒳, 𝔗_{r}(2)) ((4.2.1)with n = 2)

(4.2.4)

and the purity isomorphism

E_{1}^{u,v} ≃ ⨁_{x∈𝒳^{u}} H^{v−u}(x, ℤ/p^{r}(2 − u)) for v ≤ 2 ((4.2.1)with n = 2).

(4.2.5)

More precisely, since E_{1}^{u,v} = 0 for (u, v) with u > v and v ≤ 2, we have N^{1}H^{3}(𝒳, 𝔗_{r}(2)) ≃ E_{2}^{1,2}, which is isomorphic to the cohomology of the Gersten complex in the assertion by Hilbert’s theorem 90, the Merkur’ev-Suslin theorem [MS] and (S4). One can prove (2) in the same way as for (1), using the spectral sequence

E_{1}^{u,v} = ⨁_{x∈𝒳^{u}∩Z} H_{x}^{u+v}(𝒳, 𝔗_{r}(2)) ⟹ H_{Z}^{u+v}(𝒳, 𝔗_{r}(2))

and the purity isomorphism

E_{1}^{u,v} ≃ ⨁_{x∈Z^{u−1}} H^{v−u}(x, ℤ/p^{r}(2 − u)) for v ≤ 2

instead of (4.2.4)and (4.2.5). The details are straight-forward and left to the reader. □

Corollary 4.2.6 _{p^{r}}CH^{2}(𝒳 ) is finite for any r ≥ 1, and CH^{2}(𝒳 )_{p-tors} is cofinitely generated.

Proof. The finiteness of _{p^{r}}CH^{2}(𝒳 ) follows from the exact sequence in Lemma 4.2.3 (1) and (S7) in §4.1. The second assertion follows from Lemma 4.1.2 and the facts that CH^{2}(𝒳 )_{p-tors} is a subquotient of H^{3}(𝒳, 𝔗_{∞}(2)). □

4.3 Unramified Cohomology

Let K be the function field of 𝒳 . We define the unramified cohomology groups H_{ur}^{n+1}(K, ℤ/p^{r}(n)) and H_{ur}^{n+1}(K, ℚ_{p}/ℤ_{p}(n)) as follows:

H_{ur}^{n+1}(K, ℤ/p^{r}(n)) := Ker(H^{n+1}(K, μ_{p^{r}}^{⊗n}) → ⨁_{y∈𝒳^{1}} H_{y}^{n+2}(𝒳, 𝔗_{r}(n))) ,
H_{ur}^{n+1}(K, ℚ_{p}/ℤ_{p}(n)) := \underrightarrow{lim}_{r≥1} H_{ur}^{n+1}(K, ℤ/p^{r}(n)).

We mention some remarks on these groups:

Remark 4.3.1 (1) For n = 0, we have

H_{ur}^{1}(K, ℤ/p^{r}(0)) = H^{1}(𝒳, ℤ/p^{r}) and H_{ur}^{1}(K, ℚ_{p}/ℤ_{p}(0)) = H^{1}(𝒳, ℚ_{p}/ℤ_{p}).

If k is global, then H_{ur}^{1}(K, ℚ_{p}/ℤ_{p}(0)) is finite by a theorem of Katz-Lang [KL].

(2) For n = 1, we have

H_{ur}^{2}(K, ℤ/p^{r}(1)) = _{p^{r}}Br(𝒳 ) and H_{ur}^{2}(K, ℚ_{p}/ℤ_{p}(1)) = Br(𝒳 )_{p-tors}.

If k is global, the finiteness of H_{ur}^{2}(K, ℚ_{p}/ℤ_{p}(1)) is equivalent to the finiteness of the Tate-Shafarevich group of the Picard variety of X (cf. [G] III, [Ta1]).

(3) For n = d := dim(𝒳 ), H_{ur}^{d+1}(K, ℚ_{p}/ℤ_{p}(d)) agrees with a group considered by Kato [KCT], who conjectures that

H_{ur}^{d+1}(K, ℚ_{p}/ℤ_{p}(d)) = 0 if p ≠ 2 or k has no embedding into ℝ.

His conjecture is a generalization, to higher-dimensional proper arithmetic schemes, of the corresponding classical fact on the Brauer groups of local and global integer rings. The d = 2 case is proved in [KCT] and the d = 3 case is proved in [JS].

We restrict our atttention to the case n = 2 in what follows. The following standard proposition relates H_{ur}^{3}(K, ℤ/p^{r}(2)) with the cycle class map ϱ_{r}^{2}, which will be useful later.

Proposition 4.3.2 For a positive integer r, there is an exact sequence

\begin{aligned}0 & ⟶ & N^{1}H^{3}(𝒳, 𝔗_{r}(2)) & ⟶ & H^{3}(𝒳, 𝔗_{r}(2)) & ⟶ & H_{ur}^{3}(K, ℤ/p^{r}(2)) \\ & ⟶ & CH^{2}(𝒳 )/p^{r} & ⟶^{ϱ_{r}^{2}} & H^{4}(𝒳, 𝔗_{r}(2)) . \end{aligned}

Consequently, taking the inductive limit on r ≥ 1, we get an exact sequence

\begin{aligned}0 & ⟶ & N^{1}H^{3}(𝒳, 𝔗_{∞}(2)) & ⟶ & H^{3}(𝒳, 𝔗_{∞}(2)) & ⟶ & H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)) \\ & ⟶ & CH^{2}(𝒳 ) ⊗ ℚ_{p}/ℤ_{p} & ⟶^{ϱ_{ℚp/ℤp}^{2}} & H^{4}(𝒳, 𝔗_{∞}(2)) . \end{aligned}

(4.3.3)

Proof. Consider the spectral sequence (4.2.4). Since E_{1}^{u,v} = 0 for (u, v) with u > v and v ≤ 2 by (4.2.5), there is an exact sequnece

0 → E_{2}^{1,2} → H^{3}(𝒳, 𝔗_{r}(2)) → E_{2}^{0,3} → E_{2}^{2,2} → H^{4}(𝒳, 𝔗_{r}(2)).

One obtains the assertion by rewriting these E_{2}-terms by similar arguments as for the proof of Lemma 4.2.3 (1). □

Remark 4.3.4 Because the groups H^{∗}(𝒳, 𝔗_{∞}(2)) are cofinitely generated by Lemma 4.1.2, the sequence (4.3.3)implies that H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)) is cofinitely generated if and only if CH^{2}(𝒳 ) ⊗ ℚ_{p}/ℤ_{p} is cofinitely generated.

We next prove that H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)) is related with the torsion part of the cokernel of a cycle class map, assuming its finiteness. This result will not be used in the rest of this paper, but shows an arithmetic meaning of H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)). See also Appendix B below for a zeta value formula for threefolds over finite fields using unramified cohomology.

Proposition 4.3.5 Assume that H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)) is finite. Then the order of

Coker(ϱ_{ℤ_{p}}^{2} : CH^{2}(𝒳 ) ⊗ ℤ_{p} ⟶ H^{4}(𝒳, 𝔗_{ℤ_{p}}(2)))_{p-tors}

agrees with that of H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)).

Proof. Note that H^{4}(𝒳, 𝔗_{ℤ_{p}}(2)) is finitely generated over ℤ_{p} by Lemma 4.1.2, so that Coker(ϱ_{ℤ_{p}}^{2})_{p-tors} is finite. Consider the following commutative diagram with exact rows (cf. Lemma 4.1.2):

0 →  CH^{2}(𝒳)_{p-tors}  → CH^{2}(𝒳) ⊗ ℤ_{p}  →^{b} CH^{2}(𝒳) ⊗ ℚ_{p} → CH^{2}(𝒳) ⊗ ℚ_{p}/ℤ_{p} → 0
a↓  ^{ϱ{2 \atop ℤp}} ↓  ^{ϱ{2 \atop ℚp}} ↓  ^{ϱ{2 \atop ℚp/ℤp}} ↓
0 → Cotor(H^{3}(𝒳, 𝔗_{∞}(2))) → H^{4}(𝒳, 𝔗_{ℤp} (2)) →^{c} H^{4}(𝒳, 𝔗_{ℚp} (2)) → H^{4}(𝒳, 𝔗_{∞}(2)),

where the arrow a denotes the map obtained from the short exact sequence in Lemma 4.2.3 (1) and the arrows b and c are natural maps. See Lemma 4.4.2 below for the commutativity of the left square. By the finiteness of H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)), we see that

Coker(a) ≃ gr_{N}^{0}H^{3}(𝒳, 𝔗_{∞}(2)) := H^{3}(𝒳, 𝔗_{∞}(2))/N^{1}H^{3}(𝒳, 𝔗_{∞}(2))

(cf. Lemma 4.2.3 (1)) and that the natural map Ker(ϱ_{ℚ_{p}}^{2}) → Ker(ϱ_{ℚ_{p}/ℤ_{p}}^{2}) is zero (cf. (4.3.3)). The latter conclusion further implies that the kernel of the induced map Im(b) → Im(c) is divisible. Noting these facts, we obtain a short exact sequence

0 ⟶ gr_{N}^{0}H^{3}(𝒳, 𝔗_{∞}(2)) ⟶ Coker(ϱ_{ℤ_{p}}^{2})_{p-tors} ⟶ Ker(ϱ_{ℚ_{p}/ℤ_{p}}^{2}) ⟶ 0

by a diagram chase on the above diagram. Comparing this sequence with (4.3.3), we obtain the assertion. □

4.4 Torsion Cycle Class Map of Codimension Two

We define H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) as the following subgroup of H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)):

Im(H^{3}(X, ℚ_{p}/ℤ_{p}(2)) → H^{3}(K, ℚ_{p}/ℤ_{p}(2))) ∩ H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)).

In this subsection we relate the finiteness of this group with the injectivity of torsion cycle class maps of codimension two (see Corollary 4.4.3 below), which will be used in the proof of Theorem 1.6. We start with the following proposition.

Proposition 4.4.1 Assume that the quotient

gr_{N}^{0}H^{3}(𝒳, 𝔗_{∞}(2)) := H^{3}(𝒳, 𝔗_{∞}(2))/N^{1}H^{3}(𝒳, 𝔗_{∞}(2))

is finite. Then there exists a positive integer r_{0} such that the kernel of the map

ϱ_{p-tors,r}^{2} : CH^{2}(𝒳 )_{p-tors} ⟶ H^{4}(𝒳, 𝔗_{r}(2))

agrees with (CH^{2}(𝒳 )_{p-tors})_{Div} for any r ≥ r_{0}.

We need the following lemma to prove this proposition (cf. [CTSS] Proposition 1):

Lemma 4.4.2 For integers r, s > 0, there is a commutative diagram up to a sign

\xymatrix{N^{1}H^{3}(𝒳, 𝔗_{s}(2)) \ar[r]^{α_{s}} \ar@{^{(}->}[d] & _{p^{s}}CH^{2}(𝒳 ) \ar[d]^{ϱ_{s,r}^{2}} \\ H^{3}(𝒳, 𝔗_{s}(2)) \ar[r]^{δ_{s,r}} & H^{4}(𝒳, 𝔗_{r}(2)),}

where α_{s} denotes the boundary map in the short exact sequence of Lemma 4.2.3 (1) and ϱ_{s,r}^{2} denotes the cycle class map ϱ_{r}^{2} restricted to _{p^{s}}CH^{2}(𝒳 ). The arrow δ_{s,r} is the connecting morphism of the distinguished triangle in (S6):

𝔗_{r+s}(2) ⟶ 𝔗_{s}(2) ⟶^{δ_{s,r}} 𝔗_{r}(2)[1] ⟶^{\underline{p}^{s}} 𝔗_{r+s}(2)[1].

Proof of Lemma 4.4.2. Note that N^{1}H^{3}(𝒳, 𝔗_{s}(2)) is generated by the image of H_{Z}^{3}(𝒳, 𝔗_{s}(2)) for closed subsets Z ⊂ 𝒳 of pure codimension 1. We fix such a Z, and endow it with the reduced subscheme structure. There is a diagram which commutes obviously

\xymatrix{H_{Z}^{3}(𝒳, 𝔗_{s}(2)) \ar[r]^{δ_{s,r}} \ar[d]_{canonical} & H_{Z}^{4}(𝒳, 𝔗_{r}(2)) \ar[d]^{canonical} \\ H^{3}(𝒳, 𝔗_{s}(2)) \ar[r]^{δ_{s,r}} & H^{4}(𝒳, 𝔗_{r}(2)).}

We show that the image of the upper δ_{s,r} lies in the subgroup

N^{2}H_{Z}^{4}(𝒳, 𝔗_{r}(2)) = Ker(H_{Z}^{4}(𝒳, 𝔗_{r}(2)) → ⨁_{z∈Z^{0}} H_{z}^{4}(𝒳, 𝔗_{r}(2)))
≃ Coker(∂^{val} : ⨁_{z∈Z^{0}} κ(z)^{×}/p^{r} → ⨁_{x∈Z^{1}} ℤ/p^{r})
= CH_{d−2}(Z)/p^{r} (d := dim(𝒳 ))

(cf. Lemma 4.2.3). Indeed, there is a commutative diagram with exact bottom row

\xymatrix{ & H_{Z}^{3}(𝒳, 𝔗_{s}(2)) \ar[r]^{δ_{s,r}} \ar[d] & H_{Z}^{4}(𝒳, 𝔗_{r}(2)) \ar[d] \\ ⨁_{z∈Z^{0}} H_{z}^{3}(𝒳, 𝔗_{r+s}(2)) \ar@{->>}[r] & ⨁_{z∈Z^{0}} H_{z}^{3}(𝒳, 𝔗_{s}(2)) \ar[r]^{δ_{s,r}} & ⨁_{z∈Z^{0}} H_{z}^{4}(𝒳, 𝔗_{r}(2)),}

whose bottom left arrow is surjective by the purity in (S3) and Hilbert’s theorem 90:

H_{z}^{3}(𝒳, 𝔗_{t}(2)) ≃ H^{1}(z, ℤ/p^{t}(1)) ≃ κ(z)^{×}/p^{t} for t = r + s, s.

Hence the lower δ_{s,r} is the zero map and the image of the upper δ_{s,r} is contained in N^{2}H_{Z}^{4}(𝒳, 𝔗_{r}(2)). Now the composite map

H_{Z}^{3}(𝒳, 𝔗_{s}(2)) ⟶^{δ_{s,r}} N^{2}H_{Z}^{4}(𝒳, 𝔗_{r}(2)) ≃ CH_{d−2}(Z)/p^{r} ⟶ CH^{2}(𝒳 )/p^{r}

agrees, up to a sign, with the composite map

H_{Z}^{3}(𝒳, 𝔗_{s}(2)) → N^{1}H^{3}(𝒳, 𝔗_{s}(2)) ⟶^{α_{s}} _{p^{s}}CH^{2}(𝒳 ) ↪ CH^{2}(𝒳 ) ↠ CH^{2}(𝒳 )/p^{r}

by (S4) and computations on boudary maps (see [CTSS] Proof of Proposition 1, Step 6). We obtain the commutativity in question from these facts, because the cycle class map ϱ_{r}^{2} for cycles on Z is given by the composite map

CH_{d−2}(Z)/p^{r} ≃ N^{2}H_{Z}^{4}(𝒳, 𝔗_{r}(2)) ↪ H_{Z}^{4}(𝒳, 𝔗_{r}(2)) → H^{4}(𝒳, 𝔗_{r}(2))

by definition. □

Proof of Proposition 4.4.1. The following argument is essentially the same as the proof of [CTSS] Corollaire 3. Taking the inductive limit of the diagram in Lemma 4.4.2 with respect to s ≥ 1, we obtain a diagram whose square commutes up to a sign and whose bottom row is exact

\xymatrix{ & N^{1}H^{3}(𝒳, 𝔗_{∞}(2)) \ar[r]^{α_{∞}} \ar@{^{(}->}[d] & CH^{2}(𝒳 )_{p-tors} \ar[d]^{ϱ_{p-tors,r}^{2}} \\ H^{3}(𝒳, 𝔗_{∞}(2)) \ar[r]^{×p^{r}} & H^{3}(𝒳, 𝔗_{∞}(2)) \ar[r]^{δ_{∞,r}} & H^{4}(𝒳, 𝔗_{r}(2)).}

Since Ker(α_{∞}) is divisible by Lemma 4.2.3 (1), this diagram induces the following commutative diagram up to a sign:

\xymatrix{ & Cotor(N^{1}H^{3}(𝒳, 𝔗_{∞}(2))) \ar[r]^{\overline{α_{∞}}} \ar@{^{(}->}[d]^{∼} & Cotor(CH^{2}(𝒳 )_{p-tors}) \ar[d]^{\overline{ϱ_{p-tors,r}^{2}}} \\ Cotor(H^{3}(𝒳, 𝔗_{∞}(2))) \ar[r]^{×p^{r}} & Cotor(H^{3}(𝒳, 𝔗_{∞}(2))) \ar[r]^{\overline{δ_{∞,r}}} & H^{4}(𝒳, 𝔗_{r}(2)),}

where the bottom row remains exact, and the injectivity of the central vertical arrow follows from the finiteness of gr_{N}^{0}H^{3}(𝒳, 𝔗_{∞}(2)). Because Cotor(H^{3}(𝒳, 𝔗_{∞}(2))) is finite by (S7) and Lemma 4.1.2, the map \overline{δ_{∞,r}} is injective for any r for which p^{r} annihilates Cotor(H^{3}(𝒳, 𝔗_{∞}(2))). Thus we obtain Proposition 4.4.1. □

Corollary 4.4.3 If H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) is finite, then there is a positive integer r_{0} such that Ker(ϱ_{p-tors,r}^{2}) = (CH^{2}(𝒳 )_{p-tors})_{Div} for any r ≥ r_{0}.

Proof. Since gr_{N}^{0}H^{3}(𝒳, 𝔗_{∞}(2)) is a subgroup of H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) (cf. (4.3.3)), the assumption implies that gr_{N}^{0}H^{3}(𝒳, 𝔗_{∞}(2)) is finite. Hence the assertion follows from Proposition 4.4.1. □

Remark 4.4.4 If k is ℓ-adic local with ℓ ≠ p, then we have 𝔗_{∞}(2) = ℚ_{p}/ℤ_{p}(2) by definition and

H^{3}(𝒳, 𝔗_{∞}(2)) = H^{3}(𝒳, ℚ_{p}/ℤ_{p}(2)) ≃ H^{3}(Y, ℚ_{p}/ℤ_{p}(2))

by the proper base-change theorem, where Y denotes the closed fiber of 𝒳 /S. The last group is finite by Deligne’s proof of the Weil conjecture [De2]. Hence ϱ_{p-tors,r}^{2} for 𝒳 is injective for a sufficiently large r ≥ 1 by Proposition 4.4.1. On the other hand, if k is global or p-adic local, then H^{3}(𝒳, 𝔗_{∞}(2)) is not in general finite. Therefore we need to consider the finiteness of the group H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) to investigate the injectivity of ϱ_{p-tors,r}^{2}.

5 Finiteness of an Unramified Cohomology Group

Let k, S, p, 𝒳 and K be as in the notation 1.8. We always assume 1.8.1 throughout this and the next section.

5.1 Finiteness of H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2))

In this and the next section, we prove the following result, which implies the finiteness of H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) in Theorem 1.3 (1). See the beginning of §3 for H0.

Theorem 5.1.1 Assume H0, H1∗ and either p ≥ 5 or the equality

H_{g}^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div} = H^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div}.  (∗_{g})

Then H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) is finite.

In this section we reduce Theorem 5.1.1 to Key Lemma 5.4.1 stated in §5.4 below. We will prove the key lemma in §6. We first prove Theorem 1.6 admitting Theorem 5.1.1.

Proof of Theorem 1.6. The assumption H^{2}(X, 𝒪_{X}) = 0 implies H1∗ and (∗_{g}) (cf. Fact 1.1, Remark 3.2.5, Remark 3.1.2 (1)). Hence H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) is finite by Theorem 5.1.1. By Corollary 4.4.3, there is a positive integer r_{0} such that

Ker(ϱ_{p-tors,r}^{2}) = (CH^{2}(𝒳 )_{p-tors})_{Div} for any r ≥ r_{0}.

Thus it remains to check that CH^{2}(𝒳 )_{p-tors} is finite, which follows from the finiteness of CH^{2}(X)_{p-tors} (cf. Theorem 3.1.1) and [CTR2] Lemma 3.3. This completes the proof. □

5.2 Proof of Theorem 5.1.1, Step 1

We reduce Theorem 5.1.1 to Proposition 5.2.2 below. Let N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) (resp. gr_{N}^{0}H^{3}(X, ℚ_{p}/ℤ_{p}(2))) be the kernel (resp. the image) of the natural map

H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶ H^{3}(K, ℚ_{p}/ℤ_{p}(2)).

In view of Lemma 4.2.3, there is a commutative diagram with exact rows

N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ↪ H^{3}(X, ℚ_{p}/ℤ_{p}(2)) → gr_{N}^{0}H^{3}(X, ℚ_{p}/ℤ_{p}(2))
δ_{1}↓  ^{δ_{2}}↓  \overline{𝔡}↓
⨁_{v∈S^{1}} N^{2}H_{Y_{v}}^{4}(𝒳, 𝔗_{∞}(2)) ↪ ⨁_{v∈S^{1}} H_{Y_{v}}^{4}(𝒳, 𝔗_{∞}(2)) → ⨁_{v∈S^{1}} ⨁_{y∈Y_{v}^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2)),

(5.2.1)

where the arrows δ_{2} and \overline{𝔡} arise from boundary maps of localization exact sequences and δ_{1} is induced by the right square. Note that we have

Ker(\overline{𝔡}) = H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)).

Proposition 5.2.2 Assume H0, H1∗ and either p ≥ 5 or (∗_{g}). Then we have

Ker(\overline{𝔡})_{Div} = 0.

The proof of this proposition will be started in §5.3 below and finished in the next section. We first finish the proof of Theorem 5.1.1, admitting Proposition 5.2.2. It suffices to show the following lemma (see also Remark 3.3.4 (2)):

Lemma 5.2.3 (1) If k is local, then Ker(\overline{𝔡}) is cofinitely generated over ℤ_{p}.

(2) Assume that k is global, and that Coker(reg_{ℚ_{p}/ℤ_{p}})_{Div} is cofinitely generated over ℤ_{p}, where reg_{ℚ_{p}/ℤ_{p}} denotes the regulator map (3.2.1). Then Ker(\overline{𝔡}) is cofinitely generated over ℤ_{p}.

Proof. (1) is obvious, because H^{3}(X, ℚ_{p}/ℤ_{p}(2)) is cofinitely generated. We prove (2). We use the notation fixed in 1.9. By Lemma 4.1.2, H^{3}(𝒳, 𝔗_{∞}(2)) is cofinitely generated. Hence it suffices to show Coker(δ_{1}) is cofinitely generated, where δ_{1} is as in (5.2.1). There is a commutative diagram

[figure: CH^{2}(X, 1) ⊗ ℚ_{p}/ℤ_{p}; ∂; N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)); δ_{1}; ⨁_{v∈S^{1}} CH_{d−2}(Y_{v}) ⊗ ℚ_{p}/ℤ_{p}; ∼; ⨁_{v∈S^{1}} N^{2}H_{Y_{v}}^{4}(𝒳, 𝔗_{∞}(2)),]

where the bottom isomorphism follows from Lemma 4.2.3 (2) and ∂ is the boundary map of the localization sequence of higher Chow groups. See (3.3.1) for the top arrow. Since N^{2}H_{Y_{v}}^{4}(𝒳, 𝔗_{∞}(2)) is cofinitely generated for any v ∈ S^{1}, it suffices to show that for a sufficiently small non-empty open subset U ⊂ S, the cokernel of the boundary map

∂_{U} : CH^{2}(X, 1) ⊗ ℚ_{p}/ℤ_{p} ⟶ ⨁_{v∈(U)^{1}} CH_{d−2}(Y_{v}) ⊗ ℚ_{p}/ℤ_{p}

is cofinitely generated. Note that CH_{d−2}(Y_{v}) = CH^{1}(Y_{v}) if Y_{v} is smooth. Now let U be a non-empty open subset of S \ Σ for which 𝒳 ×_{S} U → U is smooth. Put A := H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)), viewed as a smooth sheaf on U_{ét}. There is a commutative diagram up to a sign

\xymatrix{CH^{2}(X, 1) ⊗ ℚ_{p}/ℤ_{p} \ar[d]_{∂_{U}} \ar[r]^{^{reg^{ℚp/ℤp}}} & H^{1}(k, A) \ar[d]^{δ_{U}} \\ ⨁_{v∈U^{1}} CH^{1}(Y_{v}) ⊗ ℚ_{p}/ℤ_{p} \ar[r]^{τ_{U}} & ⨁_{v∈U^{1}} A(−1)^{G_{𝔽v}} .}

See §2.5 for δ_{U}. The bottom arrow τ_{U} is defined as the composite map

CH^{1}(Y_{v}) ⊗ ℚ_{p}/ℤ_{p} ↪ H^{2}(Y_{v}, ℚ_{p}/ℤ_{p}(1)) →^{ϵ} H^{2}(\overline{Y_{v}}, ℚ_{p}/ℤ_{p}(1))^{G_{𝔽v}} = A(−1)^{G_{𝔽v}} ,

where the first injective map is the cycle class map for divisors on Y_{v}. Note that Coker(∂_{U}) is divisible and that Ker(τ_{U}) has a finite exponent by the isomorphisms

Ker(ϵ) ≃ H^{1}(𝔽_{v}, H^{1}(\overline{Y_{v}}, ℚ_{p}/ℤ_{p}(1))) ≃ H^{1}(𝔽_{v}, Cotor(H^{1}(\overline{X}, ℚ_{p}/ℤ_{p}(1))))

for v ∈ U^{1}, where the first isomorphism follows from the Hochschild-Serre spectral sequence for Y_{v}, and the second follows from Deligne’s proof of the Weil conjecture [De2] and the proper smooth base-change theorem H^{1}(\overline{Y_{v}}, ℚ_{p}/ℤ_{p}(1)) ≃ H^{1}(\overline{X}, ℚ_{p}/ℤ_{p}(1)). Hence to prove that Coker(∂_{U}) is cofinitely generated, it suffices to show that the map

∂′ := τ_{U} ∘ ∂_{U} : CH^{2}(X, 1) ⊗ ℚ_{p}/ℤ_{p} ⟶ ⨁_{v∈U^{1}} (A(−1)^{G_{𝔽v}} )_{Div}

has cofinitely generated cokernel (cf. Lemma 2.3.2 (2)). Finally Coker(reg_{ℚ_{p}/ℤ_{p}})_{Div} is cofinitely generated by assumption, which implies that ∂′ has cofinitely generated cokernel by Lemma 2.5.1 (1). Thus we obtain Lemma 5.2.3. □

5.3 Proof of Theorem 5.1.1, Step 2

We construct a key commutative diagram (5.3.3)below and prove Lemma 5.3.5, which play key roles in our proof of Proposition 5.2.2. We need some preliminaries. We suppose that k is global until the end of Lemma 5.3.1. Let Σ ⊂ S be the set of the closed points on S of characteristic p. For non-empty open U ⊂ S, put

𝒳_{U} := 𝒳 ×_{S} U and 𝒳_{U}[p^{−1}] := 𝒳_{U} ×_{S} (S \ Σ).

Let j_{U} : 𝒳_{U}[p^{−1}] → 𝒳_{U} be the natural open immersion. There is a natural injective map

[figure: α_{U,r} : H^{3}(𝒳_{U}, τ_{≤2}Rj_{U∗}μ_{p^{r}}^{⊗2}); H^{3}(𝒳_{U}[p^{−1}], μ_{p^{r}}^{⊗2})]

induced by the canonical morphism τ_{≤2}Rj_{U∗}μ_{p^{r}}^{⊗2} → Rj_{U∗}μ_{p^{r}}^{⊗2}.

Lemma 5.3.1 We have N^{1}H^{3}(𝒳_{U}[p^{−1}], μ_{p^{r}}^{⊗2}) ⊂ Im(α_{U,r}).

Proof. We compute the local-global spectral sequence

E_{1}^{u,v} = ⨁_{x∈(𝒳_{U})^{u}} H_{x}^{u+v}(𝒳_{U}, τ_{≤2}Rj_{U∗}μ_{p^{r}}^{⊗2}) ⟹ H^{u+v}(𝒳_{U}, τ_{≤2}Rj_{U∗}μ_{p^{r}}^{⊗2}).

By the absolute cohomological purity [FG] and Lemma 3.5.5 (1), we have

E_{1}^{u,v} ≃ \begin{cases}H^{v}(K, μ_{p^{r}}^{⊗2}) & (if u = 0) \\ ⨁_{x∈(𝒳_{U} [p^{−1}])^{u}} H^{v−u}(x, μ_{p^{r}}^{⊗2−u}) & (if v ≤ 2). \end{cases}

Repeating the same computation as in the proof of Lemma 3.2.2, we obtain

N^{1}H^{3}(𝒳_{U}[p^{−1}], μ_{p^{r}}^{⊗2}) ≃ E_{2}^{1,2} = E_{∞}^{1,2} ↪ H^{3}(𝒳_{U}, τ_{≤2}Rj_{U∗}μ_{p^{r}}^{⊗2}), which completes the proof of Lemma 5.3.1. □

Now we suppose that k is either local or global, and put

𝒲 := \begin{cases}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) & (if k is ℓ-adic local with ℓ ≠ p) \\ H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}/ℤ_{p}(2)) & (if k is p-adic local) \\ \underrightarrow{lim}_{Σ⊂U⊂S} H^{3}(𝒳_{U}, τ_{≤2}Rj_{U∗}ℚ_{p}/ℤ_{p}(2)) & (if k is global), \end{cases}

(5.3.2)

where j in the second case denotes the natural open immersion X ↪ 𝒳 , and the limit in the last case is taken over all non-empty open subsets U ⊂ S which contain Σ. By Lemma 3.5.3 and Lemma 5.3.1, there are inclusions

N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⊂ 𝒲 ⊂ H^{3}(X, ℚ_{p}/ℤ_{p}(2))

and a commutative diagram

[figure: NF^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))_{Div}; ν; (𝒲 ^{0})_{Div}; ω; H^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2))).]  (5.3.3)

Here NF^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) is as we defined in §3.5, and we put

𝒲 ^{0} := Ker(𝒲 ⟶ H^{3}(\overline{X}, ℚ_{p}/ℤ_{p}(2))).  (5.3.4)

The arrows ω and ν are induced by the edge homomorphism (3.5.1). We show here the following lemma, which extends Lemma 3.5.2 under Assumption 1.8.1:

Lemma 5.3.5 Assume either p ≥ 5 or (∗_{g}). Then we have

Im(ω) ⊂ H_{g}^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2))).

Remark 5.3.6 We will prove the equality Im(ω) = H_{g}^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div} under the same assumptions, later in Lemma 7.2.2.

The following corollary of Lemma 5.3.5 will be used later in §5.4:

Corollary 5.3.7 Assume H0, H1∗ and either p ≥ 5 or (∗_{g}). Then we have

Im(ν) = Im(ω) = H_{g}^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div}.

Proof of Lemma 5.3.5. The assertion under the second condition is rather obvious. In particular, we are done if k is ℓ-adic local with ℓ ≠ p (cf. Remark 3.1.2 (1)). If k is p-adic local with p ≥ 5, the assertion follows from Corollary 2.2.3 and Lemma 3.5.4. Before proving the global case, we show the following sublemma:

Sublemma 5.3.8 Let k be an ℓ-adic local field with ℓ ≠ p. Let 𝒳 be a proper smooth scheme over S := Spec(𝔬_{k}). Put A := H^{i}(\overline{X}, ℚ_{p}/ℤ_{p}(n)) and

H_{ur}^{i+1}(X, ℚ_{p}/ℤ_{p}(n)) := Im(H^{i+1}(𝒳, ℚ_{p}/ℤ_{p}(n)) → H^{i+1}(X, ℚ_{p}/ℤ_{p}(n))).

Then we have

H_{f}^{1}(k, A) ⊂ Im(F^{1}H^{i+1}(X, ℚ_{p}/ℤ_{p}(n)) ∩ H_{ur}^{i+1}(X, ℚ_{p}/ℤ_{p}(n)) → H^{1}(k, A))

and the quotient is annihilated by #(A/A_{Div}), where F^{•} denotes the filtration induced by the Hochschild-Serre spectral sequence (2.6.2).

Proof. Put Λ := ℚ_{p}/ℤ_{p}, and let 𝔽 be the residue field of k. By the proper smooth base-change theorem, G_{k} acts on A through the quotient G_{𝔽}. It suffices to show the following two claims:

(i) We have

Im(F^{1}H^{i+1}(X, Λ(n)) ∩ H_{ur}^{i+1}(X, Λ(n)) → H^{1}(k, A)) = H^{1}(𝔽, A),

where H^{1}(𝔽, A) is regarded as a subgroup of H^{1}(k, A) by an inflation map.

(ii) We have

H_{f}^{1}(k, A) ⊂ H^{1}(𝔽, A)

and the quotient is annihilated by #(A/A_{Div}).

We show these claims. Let Y be the closed fiber of 𝒳 /S, and consider a commutative diagram with exact rows

\xymatrix{0 \ar[r] & H^{1}(𝔽, H^{i}(\overline{Y} , Λ(n))) \ar[r] \ar@{^{(}->}[d]_{σ_{1}} & H^{i+1}(Y, Λ(n)) \ar[r] \ar[d]_{σ_{2}} & H^{i+1}(\overline{Y} , Λ(n)))^{G_{𝔽}} \ar[d]_{≀}^{σ_{3}} \\ 0 \ar[r] & H^{1}(k, A) \ar[r] & H^{i+1}(X, Λ(n))/F^{2} \ar[r] & H^{i+1}(\overline{X}, Λ(n)))^{G_{k}},}

where the exactness of the upper (resp. lower) row follows from the fact that cd(G_{𝔽}) = 1 (resp. cd(G_{k}) = 2). The arrows σ_{1} and σ_{3} are induced by the isomorphism

H^{∗}(\overline{Y} , Λ(n)) ≃ H^{∗}(\overline{X}, Λ(n)) (proper smooth base-change theorem).

The arrow σ_{2} is induced by

σ′_{2} : H^{i+1}(Y, Λ(n)) ≃ H^{i+1}(𝒳, Λ(n)) ⟶ H^{i+1}(X, Λ(n)).

Since Im(σ′_{2}) = H_{ur}^{i+1}(X, Λ(n)) by definition, the claim (i) follows from the above diagram. The second assertion immediately follows from the fact that H_{f}^{1}(k, A) = Im(H^{1}(𝔽, A)_{Div} → H^{1}(k, A)). This completes the proof of Sublemma 5.3.8. □

We prove Lemma 5.3.5 in the case that k is global with p ≥ 5. Let 𝒲 and 𝒲 ^{0} be as in (5.3.2)and (5.3.4), respectively, and put

A := H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)).

Note that (𝒲 ^{0})_{Div} = 𝒲_{Div} by H0. By a similar argument as for Lemma 2.4.1, we have

𝒲_{Div} = \underrightarrow{lim}_{Σ⊂U⊂S} H^{3}(𝒳_{U}, τ_{≤2}Rj_{U∗}ℚ_{p}/ℤ_{p}(2))_{Div}.

Here the limit is taken over all non-empty open subsets U ⊂ S which contain Σ, and j_{U} denotes the natural open immersion 𝒳_{U}[p^{−1}] ↪ 𝒳_{U}. By this equality and the definition of H_{g}^{1}(k, A) (cf. Definition 2.1.1), it suffices to show the following sublemma:

Sublemma 5.3.9 Let U be an open subset of S containing Σ, and fix an open subset U′ of U \ Σ for which 𝒳_{U′} → U′ is smooth (and proper). Put 𝒲_{U} := H^{3}(𝒳_{U}, τ_{≤2}Rj_{U∗}ℚ_{p}/ℤ_{p}(2)). Then for any x ∈ (𝒲_{U})_{Div}, its diagonal image

\overline{x} = (\overline{x}_{v})_{v∈S^{1}} ∈ ∏_{v∈(U′)^{1}} \frac{H^{1}(k_{v}, A)}{H_{f}^{1}(k_{v}, A)} × ∏_{v∈S\U′} \frac{H^{1}(k_{v}, A)}{H_{g}^{1}(k_{v}, A)}

is zero.

Proof. Since (𝒲_{U})_{Div} is divisible, it suffices to show that \overline{x} is killed by a positive integer independent of x. By Lemma 5.3.8, \overline{x}_{v} with v ∈ (U′)^{1} is killed by #(A/A_{Div}). Next we compute \overline{x}_{v} with v ∈ Σ. Let 𝒳_{v} and j_{v} : X_{v} ↪ 𝒳_{v} be as in 1.9, and put

𝒲_{v} := H^{3}(𝒳_{v}, τ_{≤2}Rj_{v∗}ℚ_{p}/ℤ_{p}(2)).

By H0 over k, we have

Im((𝒲_{U})_{Div} → 𝒲_{v}) ⊂ Ker(𝒲_{v} → H^{3}(\overline{X}, ℚ_{p}/ℤ_{p}(2))_{Div}.

Hence Corollary 2.2.3 and Lemma 3.5.4 imply that \overline{x}_{v} = 0 for v ∈ Σ. Finally, because the product of the other components

∏_{v∈S\(U′∪Σ)} \frac{H^{1}(k_{v}, A)}{H_{g}^{1}(k_{v}, A)}

is a finite group, we see that all local components of \overline{x} are annihilated by a positive integer independent of x. Thus we obtain the sublemma and Lemma 5.3.5. □

5.4 Proof of Theorem 5.1.1, Step 3

We reduce Proposition 5.2.2 to Key Lemma 5.4.1 below. We replace the conditions in Proposition 5.2.2 with another condition N1: We have Im(ω) = Im(ν) in (5.3.3), and Coker(reg_{ℚ_{p}/ℤ_{p}})_{Div} is cofinitely generated over ℤ_{p}. Here reg_{ℚ_{p}/ℤ_{p}} denotes the regulator map (3.2.1).

Indeed, assuming H0, H1∗ and either p ≥ 5 or (∗_{g}), we see that N1 holds by Corollary 5.3.7 and the fact that the quotient H(k, A)_{Div}/H_{g}^{1}(k, A)_{Div}, with A = H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)), is cofinitely generated over ℤ_{p} for (cf. Lemma 2.4.1). Thus Proposition 5.2.2 is reduced to the following:

Key Lemma 5.4.1 Assume H0 and N1. Then we have Ker(\overline{𝔡})_{Div} = 0.

This lemma will be proved in the next section.

6 Proof of the Key Lemma

The notation remains as in the previous section. We always assume 1.8.1 throughout this section. The aim of this section is to prove Key Lemma 5.4.1.

6.1 Proof of Key Lemma 5.4.1

Let

𝔡 : H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶ ⨁_{v∈S^{1}} ⨁_{y∈(Y_{v})^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2))

be the map induced by \overline{𝔡} in (5.2.1). Put

Θ := H^{3}(X, ℚ_{p}/ℤ_{p}(2)) /(N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))_{Div})

and let Θ̃ ⊂ Θ be the image of Ker(𝔡). Note that we have

Θ̃ = Ker(Θ → gr_{N}^{0}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) →^{\overline{𝔡}} ⨁_{v∈S^{1}} ⨁_{y∈(Y_{v})^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2)))

and a short exact sequence

0 ⟶ Cotor(N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))) ⟶ Θ̃ ⟶ Ker(\overline{𝔡}) ⟶ 0.

If k is global, the assumption of Proposition 3.3.2 (2) is satisfied by the condition N1. Hence Cotor(N^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))) is finite in both cases k is local and global (cf. Proposition 3.3.2, (3.3.1)). By the above short exact sequence and Lemma 2.3.2 (3), our task is to show

Θ̃_{Div} = 0, assuming H0 and N1.

Let F^{•} be the filtration on H^{3}(X, ℚ_{p}/ℤ_{p}(2)) resulting from the Hochschild-Serre spectral sequence (2.6.2). We define the filtration F^{•} on Θ as that induced by F^{•}H^{3}(X, ℚ_{p}/ℤ_{p}(2)), and define the filtration F^{•}Θ̃ ⊂ Θ̃ as the pull-back of F^{•}Θ. Since H0 implies the finiteness of gr_{F}^{0} Θ̃, it suffices to show

(F^{1}Θ̃)_{Div} = 0, assuming N1.  (6.1.1)

The following lemma will play key roles:

Lemma 6.1.2 Suppose that k is local. Then the following composite map has finite kernel:

𝔡_{2} : H^{2}(k, H^{1}(\overline{X}, ℚ_{p}/ℤ_{p}(2)) ⟶ H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶^{𝔡} ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2)).

Here the first map is obtained by the Hochschild-Serre spectral sequence (2.6.2) and the fact that cd(k) = 2 (cf. §2.6). Consequently, the group F^{2}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ∩ Ker(𝔡) is finite.

Admitting this lemma, we will prove (6.1.1)in §§6.2–6.3. We will prove Lemma 6.1.2 in §6.4.

6.2 Proof of (6.1.1) in the Local Case

We prove (6.1.1) assuming that k is local and that Lemma 6.1.2 holds. Let 𝔽 be the residue field of k. By Lemma 6.1.2, F^{2}Θ̃ is finite. We prove that Im(F^{1}Θ̃ → gr_{F}^{1} Θ) is finite, which is exactly the finiteness of gr_{F}^{1} Θ̃ and implies (6.1.1). Let 𝒲 and 𝒲 ^{0} be as in (5.3.2)and (5.3.4), respectively. N1 implies

gr_{F}^{1} Θ ≃ F^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))/((𝒲 ^{0})_{Div} + F^{2}H^{3}(X, ℚ_{p}/ℤ_{p}(2))).  (6.2.1)

If p ≠ ch(𝔽), then the group on the right hand side is clearly finite. If p = ch(𝔽), then Lemma 6.2.2 below implies that the image of F^{1}Θ̃ → gr_{F}^{1} Θ is a subquotient of Cotor(𝒲 ^{0}), which is finite by the proof of Lemma 3.5.4. Thus we are reduced to

Lemma 6.2.2 If p = ch(𝔽), then Ker(𝔡) ⊂ 𝒲 .

We do not need to assume H0 or N1 in this lemma.

Proof. Let the notation be as in 1.10. Note that 𝒲 = H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}/ℤ_{p}(2)) by definition. There is a commutative diagram with distinguished rows in D^{b}(𝒳, ℤ/p^{r})

[figure: 𝔗_{r}(2); t; Rj_{∗}μ_{p^{r}}^{⊗2}; Ri_{∗}Ri^{!}𝔗_{r}(2)[1]; Ri_{∗}Ri^{!}(t)[1]; 𝔗_{r}(2)[1]; t[1]; τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}; Rj_{∗}μ_{p^{r}}^{⊗2}; Ri_{∗}Ri^{!}(τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2})[1]; (τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2})[1],]

where t is as in (S5) in §4.1. The central square of this diagram gives rise to the left square of the following commutative diagram (whose rows are not exact):

H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶  H_{Y}^{4} (𝒳, 𝔗_{∞}(2))  ⟶  ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2))
∥  ↓  ↓
H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶^{ϵ_{1}} H^{0}(Y, i^{∗}R^{3}j_{∗}ℚ_{p}/ℤ_{p}(2)) ⟶^{ϵ_{2}} ⨁_{y∈Y^{0}} H^{0}(y, i^{∗}R^{3}j_{∗}ℚ_{p}/ℤ_{p}(2)).

(6.2.3)

Here the middle and the right vertical arrow is obtained from the composite morphism

Ri_{∗}Ri^{!}𝔗_{r}(2)[1] ⟶^{Ri_{∗}Ri^{!}(t)[1]} Ri_{∗}Ri^{!}τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}[1] ^{∼} ⟵ τ_{≥3}Rj_{∗}μ_{p^{r}}^{⊗2},

and the right square commutes by the functoriality of restriction maps. The composite of the upper row is 𝔡. We have Ker(ϵ_{1}) = 𝒲 obviously, and ϵ_{2} is injective by the second assertion of Lemma 3.5.5. Hence we have Ker(𝔡) ⊂ Ker(ϵ_{2} ∘ϵ_{1}) = Ker(ϵ_{1}) = 𝒲 . □

6.3 Proof of (6.1.1) in the Global Case

We prove (6.1.1) assuming that k is global and that Lemma 6.1.2 holds. Let 𝒲 and 𝒲 ^{0} be as in (5.3.2)and (5.3.4), respectively. N1 implies

gr_{F}^{1} Θ ≃ F^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2))/((𝒲 ^{0})_{Div} + F^{2}H^{3}(X, ℚ_{p}/ℤ_{p}(2))).  (6.3.1)

We first prove the following lemma, where we do not assume H0 or N1:

Lemma 6.3.2 Ker(𝔡) ⊂ 𝒲 .

Proof. We use the notation in 1.9. By the same argument as for the proof of Lemma 6.2.2, we obtain a commutative diagram analogous to (6.2.3)

H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶  ⨁_{v∈S1} H_{Yv}^{4} (𝒳, 𝔗_{∞}(2))  ⟶  ⨁_{v∈S1} ⨁_{y∈(Yv)0} H_{y}^{4}(𝒳, 𝔗_{∞}(2))
∥  ↓  ↓
H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶^{ϵ1} ⨁_{v∈Σ} H^{0}(Y_{v}, i_{v}^{∗}R^{3}j_{v∗}ℚ_{p}/ℤ_{p}(2)) ⟶^{ϵ2} ⨁_{v∈Σ} ⨁_{y∈(Yv)0} H^{0}(y, i_{v}^{∗}R^{3}j_{v∗}ℚ_{p}/ℤ_{p}(2))

The composite of the upper row is 𝔡. The assertion follows from the facts that Ker(ϵ_{1}) = 𝒲 and that ϵ_{2} is injective (cf. Lemma 3.5.5). □

We prove (6.1.1). By Lemma 2.3.2 (4), it suffcies to show that

(F^{2}Θ̃)_{Div} = 0 = (gr_{F}^{1} Θ̃)_{Div}.

Since F^{2}H^{3}(X, ℚ_{p}/ℤ_{p}(2))∩Ker(𝔡) has a finite exponent (Corollary 2.6.3 (2), Lemma 6.1.2), we have (F^{2}Θ̃)_{Div} = 0. We show (gr_{F}^{1} Θ̃)_{Div} = 0. By (6.3.1)and Lemma 6.3.2, we have

gr_{F}^{1} Θ̃ ⊂ Ξ := 𝒲 ^{0}/((𝒲 ^{0})_{Div} + Z) with Z := 𝒲 ^{0} ∩ F^{2}H^{3}(X, ℚ_{p}/ℤ_{p}(2)).

By Corollary 2.6.3 (1), Cotor(Z) has a finite exponent, which implies

(gr_{F}^{1} Θ̃)_{Div} ⊂ Ξ_{Div} = Cotor(𝒲 ^{0})_{Div} = 0

(cf. Lemma 2.3.2 (3)). Thus we obtain (6.1.1).

6.4 Proof of Lemma 6.1.2

The case that k is p-adic local follows from [Sat1] Theorem 3.1, Lemma 3.2 (1) (cf. [Ts3]). More precisely, 𝒳 is assumed in [Sat1] §3 to have strict semistable reduction, but one can remove the strictness assumption easily. The details are left to the reader.

We prove Lemma 6.1.2 assuming that k is ℓ-adic local with ℓ ≠ p. Note that in this case 𝒳 /S may not have semistable reduction. If 𝒳 /S has strict semistable reduction, then the assertion is proved in [Sat1] Theorem 2.1. We prove the general case. Put

Λ := ℚ_{p}/ℤ_{p}

for simplicity. By the alteration theorem of de Jong [dJ], we take a proper generically finite morphism f : 𝒳 ′ → X such that 𝒳 ′ has strict semistable reduction over the normalization S′ = Spec(𝔬_{k′}) of S in 𝒳 ′. sNote that 𝔡_{2} is the composite of a composite map

𝔡_{3} : H^{2}(k, H^{1}(\overline{X}, Λ(2)) ⟶ H^{3}(X, Λ(2)) ⟶^{δ^{loc}} H_{Y}^{4} (𝒳, Λ(2))

with a pull-back map

H_{Y}^{4} (𝒳, Λ(2)) ⟶ ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, Λ(2)).  (6.4.1)

Here the arrow δ^{loc} is the boundary map of a localization exact sequence. There is a commutative diagram

\xymatrix{H^{2}(k, H^{1}(\overline{X}, Λ(2))) \ar[r]^{𝔡_{3}} \ar[d]_{f^{∗}} & H_{Y}^{4} (𝒳, Λ(2)) \ar[d]^{f^{∗}} \\ H^{2}(k′, H^{1}(\overline{X}′, Λ(2))) \ar[r]^{𝔡′_{3}} & H_{Y′}^{4}(𝒳 ′, Λ(2)),}

where \overline{X}′ := 𝒳 ′ ⊗_{𝔬_{k′}} \overline{k} and Y′ denotes the closed fiber of 𝒳 ′/S′. We have already shown that Ker(𝔡′_{3}) is finite, and a standard norm argument shows that the left vertical arrow has finite kernel. Thus Ker(𝔡_{3}) is finite as well. It remains to show

Lemma 6.4.2 Im(𝔡_{3}) ∩ N^{2}H_{Y}^{4} (𝒳, Λ(2)) is finite, where N^{2}H_{Y}^{4} (𝒳, Λ(2)) denotes the kernel of the map (6.4.1).

Proof. First we note that

Im(𝔡_{3}) ⊂ Im(H^{1}(𝔽, H3 ; \overline{Y} (𝒳 ^{ur}, Λ(2))) → H_{Y}^{4} (𝒳, Λ(2))).

Indeed, this follows from the fact that 𝔡_{3} factors as follows:

H^{2}(k, H^{1}(\overline{X}, Λ(2))) ≃ H^{1}(𝔽, H^{1}(k^{ur}, H^{1}(\overline{X}, Λ(2))))

⟶ H^{1}(𝔽, H^{2}(X^{ur}, Λ(2))) ⟶ H^{1}(𝔽, H3 ; \overline{Y} (𝒳 ^{ur}, Λ(2))) ⟶ H_{Y}^{4} (𝒳, Λ(2)).

Hence it suffices to show the finiteness of the kernel of the composite map

υ : H^{1}(𝔽, H3 ; \overline{Y} (𝒳 ^{ur}, Λ(2))) ⟶ H_{Y}^{4} (𝒳, Λ(2)) ⟶ ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, Λ(2)).

There is a commutative diagram with exact rows and columns

[figure: CH_{d−2}(Y) ⊗ Λ; ι; CH_{d−2}(\overline{Y}) ⊗ Λ; H^{1}(𝔽, H3 ; \overline{Y} (𝒳 ^{ur}, Λ(2))); υ; H_{Y}^{4} (𝒳, Λ(2)); H4 ; \overline{Y} (𝒳 ^{ur}, Λ(2)); ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, Λ(2)); ⨁_{η∈(\overline{Y} )^{0}} H_{η}^{4}(𝒳 ^{ur}, Λ(2)),]

whose columns arise from the isomorphisms

N^{2}H_{Y}^{4} (𝒳, Λ(2)) ≃ CH_{d−2}(Y) ⊗ Λ, N^{2}H4 ; \overline{Y} (𝒳 ^{ur}, Λ(2)) ≃ CH_{d−2}(\overline{Y}) ⊗ Λ

with d := dim(𝒳 ) (see Lemma 4.2.3 (2), noting that 𝔗_{∞}(2) = Λ(2) in this situation). The middle exact row arises from the Hochschild-Serre spectral sequence for the covering 𝒳 ^{ur} → 𝒳 . A diagram chase shows that Ker(υ) ≃ Ker(ι), and we are reduced to showing the finiteness of Ker(ι). Because the natural restriction map

CH_{d−2}(Y)/CH_{d−2}(Y)_{tors} → CH_{d−2}(\overline{Y})/CH_{d−2}(\overline{Y})_{tors}

is injective by the standard norm argument, the finiteness of Ker(ι) follows from the following general lemma:

Lemma 6.4.3 Let e be a positive integer and let Z be a scheme which is separated of finite type over F := \overline{𝔽} with dim(Z) ≤ e. Then the group CH_{e−1}(Z)/CH_{e−1}(Z)_{tors} is a finitely generated abelian group.

Proof of Lemma 6.4.3. Obviously we may suppose that Z is reduced. We first reduced the problem to the case where Z is proper. Take a dense open immersion Z ↪ Z′ with \overline{Z}′ is proper. Writing Z′′ for Z′ \ Z, there is an exact sequence

CH_{e−1}(Z′′) ⟶ CH_{e−1}(Z′) ⟶ CH_{e−1}(Z) ⟶ 0,

where CH_{e−1}(Z′′) is finitely generated free abelian group because dim(Z′′) ≤ e − 1. Let f : Z̃ → Z be the normalization. Since f is birational and finite, one easily sees that the cokernel of f_{∗} : CH_{e−1}(Z̃) → CH_{e−1}(Z) is finite. Thus we may assume Z is a proper normal variety of dimension e over F. Since F is algebraically closed, Z has an F -rational point. Now the theory of Picard functor (cf. [Mu] §5) implies the functorial isomorphisms CH_{e−1}(Z) ≃ Pic_{Z/F} (F), where Pic_{Z/F} denotes the Picard functor for Z/F. This functor is representable by a group scheme and fits into the exact sequence of group schemes

0 ⟶ Pic_{Z/F}^{τ} ⟶ Pic_{Z/F} ⟶ NS_{Z/F} ⟶ 0,

where Pic_{Z/F}^{τ} is quasi-projective over F and the reduce part of NS_{Z/F} is associated with a finitely generated abelian group. Since F is the algebraic closure of a finite field, the group Pic_{Z/F}^{τ} (F) is torsion. Lemma 6.4.3 follows immediately from these facts. □

This completes the proof of Lemma 6.4.2, Lemma 6.1.2 and the key lemma 5.4.1. □

7 Cokernel of the Regulator Map

Let k, S, p, 𝒳 and K be as in the notation 1.8. We always assume 1.8.1 throughout this section. For a proper smooth geometrically integral variety Z over a finite field 𝔽. we say that the Tate conjecture holds in codimension 1 for Z, if the étale cycle class map

CH^{1}(Z) ⊗ ℚ_{ℓ} ⟶ H^{2}(Z ⊗_{𝔽} \overline{𝔽}, ℚ_{ℓ}(1))^{G_{𝔽}}

is surjective for a prime number ℓ ≠ ch(𝔽) ([Ta1], [Ta2]). By [Mi1] Theorems 4.1 and 6.1, this condition is independent of ℓ ≠ ch(𝔽) and equivalent to that the Grothendieck-Brauer group Br(Z) = H^{2}(Z, 𝔾_{m}) is finite.

7.1 Statement of the Result

Let 𝒞 be the category of ℤ_{p}-modules modulo the Serre subcategory consisting of p-primary torsion abelian groups of finite-exponent. In this section, we prove the following result, which implies Theorem 1.3 (2) (see the beginning of §3 for H0):

Theorem 7.1.1 Assume H0 and either p ≥ 5 or the equality

H_{g}^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div} = H^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div}.  (∗_{g})

Assume further the following conditions:

T: The reduced part of every closed fiber of 𝒳 /S has simple normal crossings on 𝒳 , and the Tate conjecture holds in codimension 1 for the irreducible components of those fibers.

F: Cotor(CH^{2}(X)_{p-tors}) has a finite exponent.

Then there exists a short exact sequence in 𝒞

0 ⟶ CH^{2}(X)_{p-tors} ⟶ \frac{H_{g}^{1}(k, H_{ét}^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))}{Im(reg_{ℚ_{p}/ℤ_{p}})} ⟶ H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)).

Moreover the image of the last map contains H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2))_{Div}.

This result is a generalization of Theorems 3.1.1 and 5.1.1 under the assumption T. The formulation of Theorem 7.1.1 in this final version was much inspired by discussions with Masanori Asakura.

7.2 Proof of Theorem 7.1.1

Let

𝔡 : H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶ ⨁_{v∈S} ⨁_{y∈(Y_{v})^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2)).

be the map induced by \overline{𝔡} in (5.2.1). Let 𝒲 be as in (5.3.2)and let 𝒲 ^{0} be as in (5.3.4). We need the following two lemmas:

Lemma 7.2.1 Assume that T holds. Then we have

𝒲_{Div} ⊂ Ker(𝔡) + F^{2}H^{3}(X, ℚ_{p}/ℤ_{p}(2)).

Lemma 7.2.2 Assume either p ≥ 5 or (∗_{g}). Then we have

Im(ω) = H_{g}^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div},

where ω is as in (5.3.3).

Lemma 7.2.1 will be proved in §§7.4–7.5 below, and Lemma 7.2.2 will be proved in §7.3 below. We prove Theorem 7.1.1 in this subsection, admitting these lemmas. Let

H^{3} := H^{3}(X, ℚ_{p}/ℤ_{p}(2)), N^{1}H^{3} ⊂ H^{3} and F^{i}H^{3} ⊂ H^{3}

be as in §3.4, and put

NF^{1}H^{3} := N^{1}H^{3} ∩ F^{1}H^{3} and A := H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)).

We see that

Cotor(N^{1}H^{3}) has a finite exponent  (7.2.3)

by the exact sequence (3.3.1)and the assumption F, and moreover that

Cotor(NF^{1}H^{3}) has a finite exponent  (7.2.4)

by H0.

We work in 𝒞 for a while to simplify the arguments. By H0, (3.3.1)and Lemma 3.3.5, there is a short exact sequence in 𝒞

0 ⟶ CH^{2}(X)_{p-tors} ⟶ H^{1}(k, A)/Im(reg_{ℚ_{p}/ℤ_{p}}) ⟶ H^{1}(k, A)/NF ⟶ 0,

where NF denotes the image of NF^{1}H^{3} in H^{1}(k, A). By the assumption ‘p ≥ 5 or (∗_{g})’ and Lemma 3.5.2, the composite map (NF^{1}H^{3})_{Div} → F^{1}H^{3} → H^{1}(k, A) factors through H_{g}^{1}(k, A). Therefore we obtain a short exact sequence in 𝒞

0 ⟶ CH^{2}(X)_{p-tors} ⟶ H_{g}^{1}(k, A)/Im(reg_{ℚ_{p}/ℤ_{p}}) ⟶ H_{g}^{1}(k, A)/NF_{Div} ⟶ 0,

whose exactness follows from (7.2.4)and the fact that the natural map (NF^{1}H^{3})_{Div} → NF_{Div} is surjective (cf. Corollary 2.6.3 (1) and Lemma 2.3.2 (3)). It remains to construct an injective map

α : H_{g}^{1}(k, A)/NF_{Div} ↪ H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) in 𝒞

whose image contains H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2))_{Div}.

We work in the usual category ℤ_{p}-modules in what follows. There are ℤ_{p}- homomorphisms

\xymatrix{H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) \ar@{^{(}->}[r]^{ι} & H^{3}/N^{1}H^{3} \ar@{->>}[d]_{π}^{(finite-exponent kernel)} \\  & H^{3}/(N^{1}H^{3} + F^{2}H^{3}),}  (7.2.5)

where ι is induced by the definition of H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) and Ker(π) has a finite exponent by Lemma 3.3.5 (i.e., π is an isomorphism in 𝒞). On the other hand, there is a diagram of ℤ_{p}-submodules of H^{3}

\xymatrix{(N^{1}H^{3})_{Div} \ar@{^{(}->}[r] \ar@{^{(}->}[d] & N^{1}H^{3} ∩ 𝒲_{Div} \ar@{^{(}->}[r] \ar@{^{(}->}[dl] & 𝒲_{Div} \ar@{^{(}->}[r]^{7.2.1} \ar@{^{(}->}[d] & Ker(𝔡) + F^{2}H^{3} \\ N^{1}H^{3} \ar@{^{(}->}[r] & Ker(𝔡) \ar@{^{(}->}[r]^{6.2.2 and 6.3.2} & 𝒲 ,}

where the inclusion Ker(𝔡) ⊂ 𝒲 is obvious when k is ℓ-adic local with ℓ ≠ p. Since N^{1}H^{3} ∩ 𝒲_{Div} is divisible up to a finite-exponent group by (7.2.3), we have

𝒲_{Div}/(N^{1}H^{3} ∩ 𝒲_{Div}) ≃ (𝒲 /N^{1}H^{3})_{Div}  (7.2.6)

by Lemma 2.3.2 (3). Now for a subgroup M ⊂ H^{3}, let \overline{M} be its image into H^{3}/(N^{1}H^{3} + F^{2}H^{3}). Then we have \overline{𝒲_{Div}} ⊂ \overline{Ker(𝔡)} ⊂ \overline{𝒲} by the above diagram, and moreover

\overline{𝒲_{Div}} = (\overline{Ker(𝔡)})_{Div} = (\overline{𝒲} )_{Div}  (7.2.7)

by (7.2.6)and the fact that Ker(π) has a finite exponent (cf. Lemma 2.3.2 (3)). Noting that Coker(H_{g}^{1}(k, A)) has a finite exponent (Remark 2.4.9 (1)), we define the desired map α in 𝒞 as that induced by the following diagram:

\xymatrix{H_{g}^{1}(k, A)/NF_{Div} \ar@{.>}[rr]^{α} &  & H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) \ar@{->>}[d]^{π∘ι} \\ H_{g}^{1}(k, A)_{Div}/NF_{Div} \ar@{^{(}->}[u] \ar@{->>}[r]^{τ} & (\overline{𝒲} )_{Div} \ar@{^{(}->}[r] & \overline{Ker(𝔡)},}

where the right vertical map is surjective by the definition of H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2)) and has finite-exponent kernel by the diagram (7.2.5). The arrow τ is obtained from Lemma 7.2.2, which is surjective by H0 and has finite-exponent kernel by (7.2.4). By the last fact, α is injective in 𝒞. Finally Im(α) contains H_{ur}^{3}(K, X; ℚ_{p}/ℤ_{p}(2))_{Div} by the surjectivity of τ and (7.2.7). Thus we obtain Theorem 7.1.1 assuming Lemmas 7.2.1 and 7.2.2.

7.3 Proof of Lemma 7.2.2

If k is local, then the assertion follows from Corollary 2.2.3 and Lemma 3.5.4. We show the inclusion Im(ω) ⊃ H_{g}^{1}(k, H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)))_{Div}, assuming that k is global (the inclusion in the other direction has been proved in Lemma 5.3.5). Put A := H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2)). By Lemma 2.3.2 (3), it is enough to show the following:

(i) The image of the composite map

\xymatrix{𝒲 ^{0} \ar@{^{(}->}[r] & F^{1}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) \ar[r]^{ψ} & H^{1}(k, A)}

contains H_{g}^{1}(k, A)_{Div}, where the arrow ψ is as in (3.5.1).

(ii) The kernel of this composite map is cofinitely generated up to a finite-exponent group.

(ii) follows from Corollary 2.6.3 (1). We prove (i) in what follows. We use the notation fixed in 1.9. Let U ⊂ S be a non-empty open subset which contains Σ and for which 𝒳_{U} → U is smooth outside of Σ. Let j_{U} : 𝒳_{U}[p^{−1}] → 𝒳_{U} be the natural open immersion. Put U′ := U \ Σ and Λ := ℚ_{p}/ℤ_{p}. For v ∈ Σ, put

M_{v} := F^{1}H^{3}(X_{v}, Λ(2))/(H^{3}(𝒳_{v}, τ_{≤2}R(j_{v})_{∗}Λ(2))^{0})_{Div},

where the superscript 0 means the subgroup of elements which vanishes in

H^{3}(\overline{X}, Λ(2)) ≃ H^{3}(X_{v} ⊗_{k_{v}} \overline{k_{v}}, Λ(2)).

We construct a commutative diagram with exact rows

\xymatrix{0 \ar[r] & Ker(r_{U}) \ar[r] \ar[d]_{c_{U}} & F^{1}H^{3}(𝒳_{U}[p^{−1}], Λ(2)) \ar[r]^{r_{U}} \ar[d]_{ψ_{U}} & ⨁_{v∈Σ} M_{v} \ar[d]_{b_{Σ}} \\ 0 \ar[r] & Ker(a_{U}) \ar[r] & H^{1}(U, A) \ar[r]^{a_{U}} & ⨁_{v∈Σ} H_{/g}^{1}(k_{v}, A),}

where F^{1} on H^{3}(𝒳_{U}[p^{−1}], Λ(2)) means the filtration resulting from the Hochschild-Serre spectral sequence (3.2.3) for 𝒳_{U}[p^{−1}], and ψ_{U} is an edge homomorphism of that spectral sequence. The arrows r_{U} and a_{U} are natural pull-back maps, and we put

H_{/g}^{1}(k_{v}, A) := H^{1}(k_{v}, A)/H_{g}^{1}(k_{v}, A).

The existence of b_{Σ} follows from the local case of Lemma 7.2.2, and c_{U} denotes the map induced by the right square. Note that Ker(a_{U}) contains H_{f,U}^{1}(k, A). Now let

c : 𝒲 ^{†} := \underrightarrow{lim}_{Σ⊂U⊂S} Ker(r_{U}) ⟶ \underrightarrow{lim}_{Σ⊂U⊂S} Ker(a_{U})

be the inductive limit of c_{U}, where U runs through all non-empty open subsets of S which contains Σ and for which 𝒳_{U} → U is smooth outside of Σ. Because the group on the right hand side contains H_{g}^{1}(k, A), it remains to show that (iii) Coker(c) has a finite exponent.

(iv) 𝒲 ^{†} is contained in 𝒲 ^{0}.

(iv) is rather straight-forward and left to the reader. We prove (iii). For U ⊂ S as above, applying the snake lemma to the above diagram, we see that the kernel of the natural map

Coker(c_{U}) ⟶ Coker(ψ_{U})

is a subquotient of Ker(b_{Σ}). By the local case of Lemma 7.2.2, we have

Ker(b_{Σ}) ≃ ⨁_{v∈Σ} Im(F^{2}H^{3}(X_{v}, Λ(2)) → M_{v})

and the group on the right hand side is finite by Lemma 7.5.1 below. On the other hand Coker(ψ_{U}) is zero if p ≥ 3, and killed by 2 if p = 2. Hence passing to the limit, we see that Coker(c) has a finite exponent. This completes the proof of Lemma 7.2.2.

7.4 Proof of Lemma 7.2.1, Step 1

We start the proof of Lemma 7.2.1. Our task is to show the inclusion

𝔡(𝒲_{Div}) ⊂ 𝔡(F^{2}H^{3}(X, ℚ_{p}/ℤ_{p}(2))).  (7.4.1)

If k is global, then the assertion is reduced to the local case, because the natural map

F^{2}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶ ⨁_{v∈S^{1}} F^{2}H^{3}(X_{v}, ℚ_{p}/ℤ_{p}(2))

has finite cokernel by Corollary 2.6.3 (2).

Assume now that k is local. In this subsection, we treat the case that k is ℓ-adic local with ℓ ≠ p. We use the notation fixed in 1.10. Recall that Y has simple normal crossings on 𝒳 by the assumption T. Note that 𝔡 factors as

H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶ H_{Y}^{4} (𝒳, ℚ_{p}/ℤ_{p}(2)) ⟶^{ι} ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, ℚ_{p}/ℤ_{p}(2)),

and that Im(𝔡) ⊂ Im(ι). There is a short exact sequence

0 → H^{1}(𝔽, H3 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))) → H_{Y}^{4} (𝒳, ℚ_{p}/ℤ_{p}(2))

→ H4 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))^{G_{𝔽}} → 0

arising from a Hochschild-Serre spectral sequence. We have Ker(ι) ≃ CH_{d−2}(Y) ⊗ ℚ_{p}/ℤ_{p} with d := dim(𝒳 ) by Lemma 4.2.3 (2). Hence to show the inclusion (7.4.1), it suffices to prove

Proposition 7.4.2 (1) Assume that T holds. Then the composite map

CH_{d−2}(Y) ⊗ ℚ_{p}/ℤ_{p} ⟶ H_{Y}^{4} (𝒳, ℚ_{p}/ℤ_{p}(2)) ⟶ H4 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))^{G_{𝔽}}

(7.4.3) is an isomorphism up to finite groups. Consequently, we have

Im(ι) ≃ H^{1}(𝔽, H3 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2)))

up to finite groups.

(2) The image of the composite map

H^{2}(k, H^{1}(X, ℚ_{p}/ℤ_{p}(2))) ⟶ H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶ H_{Y}^{4} (𝒳, ℚ_{p}/ℤ_{p}(2))

contains H^{1}(𝔽, H3 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2)))_{Div}.

We first show the following lemma:

Lemma 7.4.4 (1) Consider the Mayer-Vietoris spectral sequence obtained from the absolute purity (cf. [RZ], [Th], [FG])

E_{1}^{u,v} = H^{2u+v−2}(\overline{Y}^{(−u+1)}, ℚ_{p}/ℤ_{p}(u + 1)) ⟹ Hu+v ; \overline{Y}(𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2)),

(7.4.5)

where \overline{Y}^{(q)} denotes the disjoint union of q-fold intersections of distinct irreducible components of the reduced part of \overline{Y}. Then there are isomorphisms up to finite groups

H^{1}(𝔽, H3 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))) ≃ H^{1}(𝔽, E_{2}^{−1,4}),
H4 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))^{G_{𝔽}} ≃ (E_{2}^{0,4})^{G_{𝔽}}.

(2) As a G_{𝔽}-module, H^{0}(k^{ur}, H^{2}(\overline{X}, ℚ_{p})) has weight ≤ 2.

Proof of Lemma 7.4.4. (1) Since E_{1}^{u,v} = 0 for any (u, v) with u > 0 or 2u + v < 2, there is a short exact sequence

0 ⟶ E_{2}^{0,3} ⟶ H3 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))) ⟶ E_{2}^{−1,4} ⟶ 0,  (7.4.6)

and the edge homomorphism

\xymatrix{E_{2}^{0,4} \ar@{^{(}->}[r] & H4 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))),}  (7.4.7)

where we have E_{2}^{−1,4} = Ker(d_{1}^{−1,4}) and E_{2}^{0,4} = Coker(d_{1}^{−1,4}) and d_{1}^{−1,4} is the Gysin map H^{0}(\overline{Y}^{(2)}, ℚ_{p}/ℤ_{p}) → H^{2}(\overline{Y}^{(1)}, ℚ_{p}/ℤ_{p}(1))). Note that E_{1}^{u,v} is pure of weight v − 4 by Deligne’s proof of the Weil conjecture [De2], so that H^{i}(𝔽, E_{∞}^{u,v}) (i = 0, 1) is finite unless v = 4. The assertions immediately follow from these facts.

(2) By the alteration theorem of de Jong [dJ], we may assume that 𝒳 is projective and has semistable reduction over S. If X is a surface, then the assertion is proved in [RZ]. Otherwise, take a closed immersion 𝒳 ↪ ℙ_{S}^{N} =: ℙ. By [JS] Proposition 4.3 (b), there exists a hyperplane H ⊂ ℙ which is flat over S and for which 𝒵 := 𝒳 ×_{ℙ} H is regular with semistable reduction over S. The restriction map H^{2}(\overline{X}, ℚ_{p}) → H^{2}(\overline{Z}, ℚ_{p}) (\overline{Z} := 𝒵 ×_{𝔬_{k}} \overline{k}) is injective by the weak and hard Lefschetz theorems. Hence the claim is reduced to the case of surfaces. This completes the proof of the lemma. □

Proof of Proposition 7.4.2. (1) Note that the composite map (7.4.3) in question has finite kernel by Lemma 6.4.3 and the arguments in the proof of Lemma 6.4.2. We prove that (7.4.3) has finite cokernel, assuming T. By the Kummer theory, there is a short exact sequence

0 ⟶ Pic(\overline{Y}^{(1)}) ⊗ ℚ_{p}/ℤ_{p} ⟶ H^{2}(\overline{Y}^{(1)}, ℚ_{p}/ℤ_{p}(1)) ⟶ Br(\overline{Y}^{(1)})_{p-tors} ⟶ 0

(7.4.8)

and the differential map d_{1}^{−1,4} of the spectral sequence (7.4.5) factors through the Gysin map

H^{0}(\overline{Y}^{(2)}, ℚ_{p}/ℤ_{p}) ⟶ Pic(\overline{Y}^{(1)}) ⊗ ℚ_{p}/ℤ_{p},

whose cokernel is CH_{d−2}(\overline{Y}) ⊗ ℚ_{p}/ℤ_{p}. Hence in view of the computations in the proof of Lemma 7.4.4 (1), the Gysin map

CH_{d−2}(\overline{Y}) ⊗ ℚ_{p}/ℤ_{p} → H4 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))

(cf. Lemma 4.2.3 (2)) factors through the map (7.4.7) and we obtain a commutative diagram

\xymatrix{CH_{d−2}(Y) ⊗ ℚ_{p}/ℤ_{p} \ar[r]^{(7.4.3)} \ar[d] & H4 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))^{G_{𝔽}} \\ (CH_{d−2}(\overline{Y}) ⊗ ℚ_{p}/ℤ_{p})^{G_{𝔽}} \ar[r] & (E_{2}^{0,4})^{G_{𝔽}}, \ar[u]_{(7.4.7)}}

where the left vertical arrow has finite cokernel (and kernel) by Lemma 6.4.3 and a standard norm argument. The right vertical arrow has finite cokernel (and is injective) by Lemma 7.4.4 (1). Thus it suffices to show that the bottom horizontal arrow has finite cokernel. By the exact sequence (7.4.8), there is a short exact sequence

0 ⟶ CH_{d−2}(\overline{Y}) ⊗ ℚ_{p}/ℤ_{p} ⟶ E_{2}^{0,4} ⟶ Br(\overline{Y}^{(1)})_{p-tors} ⟶ 0.

Our task is to show that (Br(\overline{Y}^{(1)})_{p-tors})^{G_{𝔽}} is finite, which follows from the assumption T and the finiteness of the kernel of the natural map

H^{1}(G_{𝔽}, Pic(\overline{Y}^{(1)}) ⊗ ℚ_{p}/ℤ_{p}) ⟶ H^{1}(G_{𝔽}, H^{2}(\overline{Y}^{(1)}, ℚ_{p}/ℤ_{p}(1)))

(cf. Lemma 7.6.2 in §7.6 below). Thus we obtain the assertion.

(2) Since cd(k^{ur}) = 1, there is a short exact sequence

0 → H^{1}(k^{ur}, H^{1}(\overline{X}, ℚ_{p}/ℤ_{p}(2))) → H^{2}(X^{ur}, ℚ_{p}/ℤ_{p}(2)) → H^{2}(\overline{X}, ℚ_{p}/ℤ_{p}(2))^{G_{k^{ur}}} → 0

arising from a Hochschild-Serre spectral sequence. By Lemma 7.4.4 the last group has weight ≤ −2, and we have isomorphisms up to finite groups

H^{2}(k, H^{1}(\overline{X}, ℚ_{p}/ℤ_{p}(2))) ≃ H^{1}(𝔽, H^{1}(k^{ur}, H^{1}(\overline{X}, ℚ_{p}/ℤ_{p}(2))))
≃ H^{1}(𝔽, H^{2}(X^{ur}, ℚ_{p}/ℤ_{p}(2))).  (7.4.9)

Now we plug the short exact sequence (7.4.6)into the localization exact sequence

H^{2}(X^{ur}, ℚ_{p}/ℤ_{p}(2)) ⟶ H3 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))) ⟶^{α} H^{3}(𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2)).

Note that H^{3}(𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2)) ≃ H^{3}(\overline{Y} , ℚ_{p}/ℤ_{p}(2)), so that it has weight ≤ −1 (cf. [De2]). Let E_{2}^{u,v} be as in (7.4.5). Since E_{2}^{−1,4} is pure of weight 0, the induced map

E_{2}^{−1,4} ⟶ H^{3}(𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2)))/α(E_{2}^{0,3})

has finite image. Hence the composite map

H^{2}(X^{ur}, ℚ_{p}/ℤ_{p}(2)) ⟶ H3 ; \overline{Y} (𝒳 ^{ur}, ℚ_{p}/ℤ_{p}(2))) ⟶ E_{2}^{−1,4}

has finite cokernel, and the following map has finite cokernel as well:

H^{1}(𝔽, H^{2}(X^{ur}, ℚ_{p}/ℤ_{p}(2))) ⟶ H^{1}(𝔽, E_{2}^{−1,4}).

Now Proposition 7.4.2 (2) follows from this fact together with (7.4.9) and the first isomorphism in Lemma 7.4.4 (1). □

Remark 7.4.10 Let J be the set of the irreducible components of Y^{(2)} and put

Δ := Ker(g′ : ℤ^{J} → NS(Y^{(1)})) with NS(Y^{(1)}) := ⨁_{y∈Y^{0}} NS(Y_{y}),

where for y ∈ Y^{0}, Y_{y} denotes the closure \overline{\{y\}} ⊂ Y and NS(Y_{y}) denotes its Néron-Severi group. The arrow g′ arises from the Gysin map ℤ^{J} → Pic(Y^{(1)}). One can easily show, assuming T and using Lemma 7.6.2 in §7.6 below, that the corank of H^{1}(𝔽, E_{2}^{−1,4}) over ℤ_{p} is equal to the rank of Δ over ℤ. Hence Proposition 7.4.2 (2) implies the inequality

dim_{ℚ_{p}}(H^{2}(k, H^{1}(\overline{X}, ℚ_{p}(2))) ≥ dim_{ℚ}(Δ ⊗ ℚ),  (7.4.11)

which will be used in the next subsection.

7.5 Proof of Lemma 7.2.1, Step 2

We prove Lemma 7.2.1, assuming that k is p-adic local (see 1.10 for notation). We first show the following lemma:

Lemma 7.5.1 We have

F^{2}H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⊂ H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}/ℤ_{p}(2)) (= 𝒲 ).

Proof of Lemma 7.5.1. There is a distinguished triangle in D^{b}(𝒳_{ét}, ℤ/p^{r}) by §4.1 (S5)

\xymatrix{i_{∗}ν_{Y,r}^{1}[−3] \ar[r]^{g} & 𝔗_{r}(2) \ar[r]^{t′} & τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2} \ar[r]^{σ} & i_{∗}ν_{Y,r}^{1}[−2].}  (7.5.2)

Applying Ri^{!} to this triangle, we obtain a distinguished triangle in D^{b}(Y_{ét}, ℤ/p^{r})

\xymatrix{ν_{Y,r}^{1}[−3] \ar[r]^{^{Gys{2 \atop i}}} & Ri^{!}𝔗_{r}(2) \ar[r]^{Ri^{!}(t)} & i^{∗}(τ_{≥3}Rj_{∗}μ_{p^{r}}^{⊗2})[−1] \ar[r] & ν_{Y,r}^{1}[−2],} (7.5.3)

where Gys_{i}^{2} := Ri^{!}(g) and we have used the natural isomorphism

i^{∗}(τ_{≥3}Rj_{∗}μ_{p^{r}}^{⊗2})[−1] ≃ Ri^{!}(τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}).

Now let us recall the commutative diagram (6.2.3):

H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶  H_{Y}^{4} (𝒳, 𝔗_{∞}(2))  ⟶  ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2))
∥  ↓  ^{λ}↓
H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶^{ϵ_{1}} H^{0}(Y, i^{∗}R^{3}j_{∗}ℚ_{p}/ℤ_{p}(2)) ⟶^{ϵ_{2}} ⨁_{y∈Y^{0}} H^{0}(y, i^{∗}R^{3}j_{∗}ℚ_{p}/ℤ_{p}(2)),

where the middle and the right vertical arrows are induced by Ri^{!}(t) in (7.5.3). By the proof of Lemma 6.2.2, we have H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}/ℤ_{p}(2)) = Ker(ϵ_{2}ϵ_{1}). Hence it suffices to show the image of the composite map

𝔡_{2} : H^{2}(k, H^{1}(\overline{X}, ℚ_{p}/ℤ_{p}(2)) ⟶ H^{3}(X, ℚ_{p}/ℤ_{p}(2)) ⟶^{𝔡} ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2))

is contained in Ker(λ). By the distinguished triangle (7.5.3), Ker(λ) agrees with the image of the Gysin map

Gys := ⨁_{y∈Y^{0}} Gys_{i_{y}}^{2} : ⨁_{y∈Y^{0}} H^{1}(y, W_{∞} Ω_{y,log}^{1}) ↪ ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2)).

On the other hand, 𝔡_{2} factors into the following maps up to a sign, by the commutativity of the central square in [SH] (4.4.2):

H^{2}(k, H^{1}(\overline{X}, ℚ_{p}/ℤ_{p}(2))) ≃ H^{1}(𝔽, H^{1}(k^{ur}, H^{1}(\overline{X}, ℚ_{p}/ℤ_{p}(2))))

⟶ H^{1}(𝔽, H^{2}(X^{ur}, ℚ_{p}/ℤ_{p}(2))) ⟶^{σ} H^{1}(𝔽, H^{0}(\overline{Y} , ν_{\overline{Y} ,∞}^{1}))

⟶ H^{1}(𝔽, ⨁_{η∈\overline{Y}^{0}} H^{0}(η, W_{∞} Ω_{η,log}^{1})) ⟶ ⨁_{y∈Y^{0}} H^{1}(y, W_{∞} Ω_{y,log}^{1})

⟶^{Gys} ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2)),

where ν_{\overline{Y} ,∞}^{1} := \underrightarrow{lim}_{r} ν_{\overline{Y} ,r}^{1}. Thus we obtain the assertion. □

We start the proof of Lemma 7.2.1, i.e., the inclusion (7.4.1), assuming that k is padic local. The triangle (7.5.2) gives rise to the upper exact row of the following commutative diagram with exact rows (cf. [SH] (4.4.2)):

[figure: H^{3}(𝒳, 𝔗_{r}(2)); H^{3}(𝒳, τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2}); −σ; H^{1}(Y, ν_{Y,r}^{1}); _{Gys{2 \atop i}}; H^{3}(𝒳, 𝔗_{r}(2)); H^{3}(X, μ_{p^{r}}^{⊗2}); H_{Y}^{4} (𝒳, 𝔗_{r}(2)),]

where σ (resp. Gys_{i}^{2}) is as in (7.5.2) (resp. (7.5.3)). Hence the map 𝔡 restricted to 𝒲 = H^{3}(𝒳, τ_{≤2}Rj_{∗}ℚ_{p}/ℤ_{p}(2)) factors as

𝒲 ⟶ H^{1}(Y, ν_{Y,∞}^{1}) ⟶ H_{Y}^{4} (𝒳, 𝔗_{∞}(2)) ⟶ ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2)),

where ν_{Y,∞}^{1} := \underrightarrow{lim}_{r} ν_{Y,r}^{1}. By Lemmas 7.5.1 and 6.1.2, it suffices to show that the corank of

Im(H^{1}(Y, ν_{Y,∞}^{1}) → ⨁_{y∈Y^{0}} H_{y}^{4}(𝒳, 𝔗_{∞}(2)))  (7.5.4)

is not greater than dim_{ℚ_{p}} H^{2}(k, H^{1}(\overline{X}, ℚ_{p}(2))). We pursue an analogy to the case p ≠ ch(𝔽) by replacing H_{Y}^{4} (𝒳, ℚ_{p}/ℤ_{p}(2)) with H^{1}(Y, ν_{Y,∞}^{1}). There is an exact sequence

0 ⟶ H^{1}(𝔽, H^{0}(\overline{Y} , ν_{\overline{Y} ,∞}^{1})) ⟶ H^{1}(Y, ν_{Y,∞}^{1}) ⟶ H^{1}(\overline{Y} , ν_{\overline{Y} ,∞}^{1})^{G_{𝔽}} ⟶ 0

arising from a Hochschild-Serre spectral sequence. By [Sat3] Corollary 2.2.7, there is a Mayer-Vietoris spectral sequence

E_{1}^{a,b} = H^{a+b}(\overline{Y}^{(1−a)}, W_{∞} Ω_{\overline{Y}^{(1−a)},log}^{1+a}) ⟹ H^{a+b}(\overline{Y} , ν_{\overline{Y} ,∞}^{1}).

Note that E_{1}^{a,b} is of weight b − 1 so that H^{i}(𝔽, E_{1}^{a,b}) is finite unless b = 1. Thus we obtain isomorphisms up to finite groups

H^{1}(𝔽, H^{0}(\overline{Y} , ν_{\overline{Y} ,∞}^{1})) ≃ H^{1}(𝔽, E_{2}^{−1,1}),  (7.5.5)
H^{1}(\overline{Y} , ν_{\overline{Y} ,∞}^{1})^{G_{𝔽}} ≃ (E_{2}^{0,1})^{G_{𝔽}}  (7.5.6)

with E_{2}^{−1,1} = Ker(d_{1}^{−1,1}) and E_{2}^{0,1} = Coker(d_{1}^{−1,1}), where d_{1}^{−1,1} is the Gysin map

H^{0}(\overline{Y}^{(2)}, ℚ_{p}/ℤ_{p}) ⟶ H^{1}(\overline{Y}^{(1)}, W_{∞} Ω_{\overline{Y}^{(1)},log}^{1}).

There is an exact sequence of G_{𝔽}-modules (cf. (7.6.1)below)

0 ⟶ Pic(\overline{Y}^{(1)}) ⊗ ℚ_{p}/ℤ_{p} ⟶ H^{1}(\overline{Y}^{(1)}, W_{∞} Ω_{\overline{Y}^{(1)},log}^{1}) ⟶ Br(\overline{Y}^{(1)})_{p-tors} ⟶ 0.

Hence we see that the group (7.5.4)coincides with the image of H^{1}(𝔽, H^{0}(\overline{Y} , ν_{Y,∞}^{1})) up to finite groups by the condition T, the same computation as for Proposition 7.4.2 (1) and [CTSS] p. 782 Théorème 3. Now we are reduced to showing

dim_{ℚ_{p}} H^{2}(k, H^{1}(\overline{X}, ℚ_{p}(2))) ≥ corank(H^{1}(𝔽, H^{0}(\overline{Y} , ν_{\overline{Y} ,∞}^{1})))
= corank(H^{1}(𝔽, E_{2}^{−1,1})),

where the last equality follows from (7.5.5). As is seen in Remark 7.4.10, the right hand side is equal to dim_{ℚ}(Δ ⊗ ℚ) under the condition T. On the other hand, by [J2] Corollary 7, the left hand side does not change when one replaces p with another prime p′. Thus the desired inequality follows from (7.4.11). This completes the proof of Lemma 7.2.1 and Theorem 7.1.1. □

7.6 Appendix to Section 7

Let Z be a proper smooth variety over a finite field 𝔽. For a positive integer m, we define the object ℤ/mℤ(1) ∈ D^{b}(Z_{ét}, ℤ/mℤ) as

ℤ/mℤ(1) := μ_{m′} ⊕ (W_{r} Ω_{Z,log}^{1}[−1])

where we factorized m as m′ · p^{r} with (p, m′) = 1. There is a distinguished triangle of Kummer theory for 𝔾_{m} := 𝔾_{m},_{Z} in D^{b}(Z_{ét})

ℤ/mℤ(1) ⟶ 𝔾_{m} ⟶^{×m} 𝔾_{m} ⟶ ℤ/mℤ(1)[1].

So there is a short exact sequence of G_{𝔽}-modules

0 ⟶ Pic(\overline{Z})/m ⟶ H^{2}(\overline{Z}, ℤ/mℤ(1)) ⟶ _{m}Br(\overline{Z}) ⟶ 0,

where \overline{Z} := Z ⊗_{𝔽} \overline{𝔽}. Taking the inductive limit with respect to m ≥ 1, we obtain a short exact sequence of G_{𝔽}-modules

0 ⟶ Pic(\overline{Z}) ⊗ ℚ/ℤ ⟶^{α} H^{2}(\overline{Z}, ℚ/ℤ(1)) ⟶ Br(\overline{Z}) ⟶ 0.  (7.6.1)

Concerning the arrow α, we prove the following lemma, which has been used in this section.

Lemma 7.6.2 The map H^{1}(𝔽, Pic(\overline{Z}) ⊗ ℚ/ℤ) → H^{1}(𝔽, H^{2}(\overline{Z}, ℚ/ℤ(1)) induced by α has finite kernel.

Proof. Note that Pic(\overline{Z}) ⊗ ℚ/ℤ ≃ (NS(\overline{Z})/NS(\overline{Z})_{tors}) ⊗ ℚ/ℤ. By a theorem of Matsusaka [Ma] Theorem 4, the group Div(\overline{Z})/Div(\overline{Z})_{num} is isomorphic to NS(\overline{Z})/NS(\overline{Z})_{tors}, where Div(\overline{Z}) denotes the group of Weil divisors on \overline{Z}, Div(Z)_{num} denotes the subgroup of Weil divisors numerically equivalent to zero. By this fact and the fact that NS(\overline{Z}) is finitely generated, there exists a finite family \{C_{i}\}_{i∈I} of proper smooth curves over 𝔽 which are finite over Z and for which the kernel of the natural map NS(\overline{Z}) → ⨁_{i∈I} NS(\overline{C}_{i}) with \overline{C}_{i} := C_{i} ⊗_{𝔽} \overline{𝔽} is torsion. Now consider a commutative diagram

[figure: H^{1}(𝔽, NS(\overline{Z}) ⊗ ℚ/ℤ); H^{1}(𝔽, H^{2}(\overline{Z}, ℚ/ℤ(1))),; ⨁_{i∈I} H^{1}(𝔽, NS(\overline{C}_{i}) ⊗ ℚ/ℤ); ∼; ⨁_{i∈I} H^{1}(𝔽, H^{2}(\overline{C}_{i}, ℚ/ℤ(1))).]

By a standard norm argument, one can easily show that the left vertical map has finite kernel. The bottom horizontal arrow is bijective, because Br(\overline{C}_{i}) = 0 for any i ∈ I by Tsen’s theorem (cf. [Se] II.3.3). Hence the top horizontal arrow has finite kernel and we obtain the lemma. □

A Relation with Conjectures of Beilinson and Lichtenbaum

Let k, p, S, 𝒳 and K be as in the notation 1.8. We always assume 1.8.1 in what follows. The Zariski site Z_{Zar} on a scheme Z always means (ét/Z)_{Zar}, and Z_{ét} means the usual small étale site. The main result of this appendix is Proposition A.1.3 below.

A.1 Motivic Complex and Conjectures

Let ℤ(2)_{Zar} = ℤ(2)_{Zar}^{𝒳} be the motivic complex on 𝒳_{Zar} defined by using Bloch’s cycle complex, and let ℤ(2)_{ét} be its étale sheafification, which are, by works of Levine ([Le1], [Le2]), considered as strong candidates for motivic complexes of Beilinson-Lichtenbaum ([Be], [Li1]) in Zariski and étale topology, respectively (see also [Li2], [Li3]). We put

H_{Zar}^{∗}(𝒳, ℤ(2)) := H_{Zar}^{∗}(𝒳, ℤ(2)_{Zar}), H_{ét}^{∗}(𝒳, ℤ(2)) := H_{ét}^{∗}(𝒳, ℤ(2)_{ét}).

In this appendix, we observe that the finiteness of H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)) is deduced from the following conjectures on motivic complexes:

Conjecture A.1.1 Let ϵ : 𝒳_{ét} → 𝒳_{Zar} be the natural continuous map of sites. Then:

(1) (Beilinson-Lichtenbaum conjecture). We have

ℤ(2)_{Zar} ^{∼} ⟶ τ_{≤2}Rϵ_{∗}ℤ(2)_{ét} in D(𝒳_{Zar}).

(2) (Hilbert’s theorem 90). We have R^{3}ϵ_{∗}ℤ(2)_{ét} = 0.

(3) (Kummer theory on 𝒳 [p^{−1}]_{ét}). We have (ℤ(2)_{ét})|_{𝒳 [p^{−1}]} ⊗^{𝕃} ℤ/p^{r} ≃ μ_{p^{r}}^{⊗2}.

This conjecture holds if 𝒳 is smooth over S by a result of Geisser [Ge1] Theorem 1.2 and the Merkur’ev-Suslin theorem [MS] (see also [GL2] Remark 5.9).

Conjecture A.1.2 Let γ^{2} be the canonical map

γ^{2} : CH^{2}(𝒳 ) = H_{Zar}^{4}(𝒳, ℤ(2)) ⟶ H_{ét}^{4}(𝒳, ℤ(2)).

Then the p-primary torsion part of Coker(γ^{2}) is finite.

This conjecture is based on Lichtenbaum’s conjecture [Li1] that H_{ét}^{4}(𝒳, ℤ(2)) is a finitely generated abelian group (by the properness of 𝒳 /S). The aim of this appendix is to prove the following:

Proposition A.1.3 If Conjectures A.1.1 and A.1.2 hold, then H_{ur}^{3}(K, ℚ_{p}/ℤ_{p}(2)) is finite.

This proposition is reduced to the following lemma:

Lemma A.1.4 (1) If Conjecture A.1.1 holds, then for r ≥ 1 there is an exact sequence

0 ⟶ Coker(α_{r}) ⟶ H_{ur}^{3}(K, ℤ/p^{r}(2)) ⟶ Ker(ϱ_{r}^{2}) ⟶ 0,

where α_{r} denotes the map

α_{r} : _{p^{r}}CH^{2}(𝒳 ) ⟶ _{p^{r}}H_{ét}^{4}(𝒳, ℤ(2))

induced by γ^{2}, and ϱ_{r}^{2} denotes the cycle class map

ϱ_{r}^{2} : CH^{2}(𝒳 )/p^{r} ⟶ H_{ét}^{4}(𝒳, 𝔗_{r}(2)).

(2) If Conjectures A.1.1 and A.1.2 hold, then Coker(α_{ℚ_{p}/ℤ_{p}}) and Ker(ϱ_{ℚ_{p}/ℤ_{p}}^{2}) are finite, where α_{ℚ_{p}/ℤ_{p}} := \underrightarrow{lim}_{r≥1} α_{r} and ϱ_{ℚ_{p}/ℤ_{p}}^{2} := \underrightarrow{lim}_{r≥1} ϱ_{r}^{2}.

To prove this lemma, we need the following sublemma, which is a variant of Geisser’s arguments in [Ge1] §6 and provides a Kummer theory on the whole 𝒳 extending Conjecture A.1.1 (3):

Sublemma A.1.5 Put ℤ/p^{r}(2)_{ét} := ℤ(2)_{ét} ⊗^{𝕃} ℤ/p^{r}. If Conjecture A.1.1 holds, then there is a unique isomorphism

ℤ/p^{r}(2)_{ét} ^{∼} ⟶ 𝔗_{r}(2) in D(𝒳_{ét}, ℤ/p^{r})

that extends the isomorphism in Conjecture A.1.1 (3).

We prove Sublemma A.1.5 in §A.2 below and Lemma A.1.4 in §A.3 below.

A.2 Proof of Sublemma A.1.5

By Conjecture A.1.1 (3), we have only to consider the case where p is not invertible on S. Let us note that

(∗) ℤ/p^{r}(2)_{ét} is concentrated in degrees ≤ 2

by Conjecture A.1.1 (1) and (2). Let V, Y, i and j be as follows:

[figure: V := 𝒳 [p^{−1}]; j; 𝒳; Y,; i]

where Y denotes the union of the fibers of 𝒳 /S of characteristic p. In étale topology, we define Ri^{!} and Rj_{∗} for unbounded complexes by the method of Spaltenstein [Spa]. We will prove

τ_{≤3}Ri^{!}ℤ/p^{r}(2)_{ét} ≃ ν_{Y,r}^{1}[−3] in D(Y_{ét}, ℤ/p^{r}),  (A.2.1)

using (∗) (see (S5) in §4.1 for ν_{Y,r}^{1}). We first prove Sublemma A.1.5 admitting this isomorphism. Since (ℤ(2)_{ét})|_{V} ⊗^{𝕃} ℤ/p^{r} ≃ μ_{p^{r}}^{⊗2} by Conjecture A.1.1 (3), we obtain a distinguished triangle from (A.2.1)and (∗)

i_{∗}ν_{Y,r}^{1}[−3] ⟶ ℤ/p^{r}(2)_{ét} ⟶ τ_{≤2}Rj_{∗}μ_{p^{r}}^{⊗2} ⟶ i_{∗}ν_{Y,r}^{1}[−2].

Hence comparing this distinguished triangle with that of (S5) in §4.1, we obtain the desired isomorphism in the sublemma, whose uniqueness follows from [SH] Lemmas 1.1 and 1.2 (1).

In what follows, we prove (A.2.1). Put 𝒦 := ℤ(2)_{Zar} ⊗^{𝕃} ℤ/p^{r} and ℒ := ℤ/p^{r}(2)_{ét} for simplicity. Let ϵ : 𝒳_{ét} → 𝒳_{Zar} be as in Conjecture A.1.1. In Zariski topology, we define Ri_{Zar}^{!} and Rj_{Zar∗} for unbounded complexes in the usual way by the finiteness of cohomological dimension. Because ℒ = ϵ^{∗}𝒦 is concentrated in degrees ≤ 2 by (∗), there is a commutative diagram with distinguished rows in D^{b}(𝒳_{ét}, ℤ/p^{r})

[figure: ϵ^{∗}𝒦; τ_{≤2}ϵ^{∗}Rj_{Zar∗}j_{Zar}^{∗}𝒦; α; (τ_{≤3}ϵ^{∗}i_{Zar∗}Ri_{Zar}^{!}𝒦 )[1]; β; ϵ^{∗}𝒦 [1]; ℒ; τ_{≤2}Rj_{ét∗}j_{ét}^{∗}ℒ; (τ_{≤3}i_{ét∗}Ri_{ét}^{!}ℒ)[1]; ℒ [1],]

where the upper (resp. lower) row is obtained from the localization triangle in the Zariski (resp. étale) topology and the arrows α and β are canonical base-change morphisms. Since α is an isomorphism ([MS], [SV], [GL2]), β is an isomorphism as well. Hence (A.2.1)is reduced to showing

τ_{≤3}Ri_{Zar}^{!}𝒦 ≃ ϵ_{Y ∗}ν_{Y,r}^{1}[−3] in ∈ D(Y_{Zar}, ℤ/p^{r}),  (A.2.2)

where ϵ_{Y} : Y_{ét} → Y_{Zar} denotes the natural continuous map of sites and we have used the base-change isomorphism ϵ^{∗}i_{Zar∗} = i_{ét∗}ϵ_{Y}^{∗} ([Ge1] Proposition 2.2 (a)). Finally we show (A.2.2). Consider the local-global spectral sequence in the Zariski topology

E_{1}^{u,v} = ⨁_{x∈𝒳^{u}∩Y} R^{u+v}i_{x∗}(Ri_{x}^{!}Ri_{Zar}^{!}𝒦 ) ⟹ R^{u+v}i_{Zar}^{!}𝒦 ,

where for x ∈ Y, i_{x} denotes the natural map x → Y. We have

E_{1}^{u,v} ≃ \begin{cases}⨁_{x∈𝒳^{u}∩Y} i_{x∗}ϵ_{x∗}W_{r} Ω_{x,log}^{2−u} & (if v = 2) \\ 0 & (otherwise) \end{cases}

by the localization sequence of higher Chow groups [Le1] and results of Geisser-Levine ([GL1] Proposition 3.1, Theorem 7.1), where for x ∈ Y, ϵ_{x} denotes the natural continuous map x_{ét} → x_{Zar} of sites. By this description of E_{1}-terms and the compatibility of boundary maps ([GL2] Lemma 3.2, see also [Sz] Appendix), we obtain (A.2.2). This completes the proof of Sublemma A.1.5.

A.3 Proof of Lemma A.1.4

(1) By Sublemma A.1.5, there is an exact sequence

0 ⟶ H_{ét}^{3}(𝒳, ℤ(2))/p^{r} ⟶ H_{ét}^{3}(𝒳, 𝔗_{r}(2)) ⟶ _{p^{r}}H_{ét}^{4}(𝒳, ℤ(2)) ⟶ 0.

By Conjecture A.1.1 (1) and (2), we have

H_{ét}^{3}(𝒳, ℤ(2)) ≃ H_{Zar}^{3}(𝒳, ℤ(2)) ≃ CH^{2}(𝒳, 1).

Thus we get an exact sequence

0 ⟶ CH^{2}(𝒳, 1)/p^{r} ⟶ H_{ét}^{3}(𝒳, 𝔗_{r}(2)) ⟶ _{p^{r}}H_{ét}^{4}(𝒳, ℤ(2)) ⟶ 0.

On the other hand, there is an exact sequence

0 → N^{1}H_{ét}^{3}(𝒳, 𝔗_{r}(2)) → H_{ét}^{3}(𝒳, 𝔗_{r}(2)) → H_{ur}^{3}(K, ℤ/p^{r}(2)) → Ker(ϱ_{r}^{2}) → 0

by Proposition 4.3.2 (see Lemma 4.2.3 for N^{1}). In view of these facts and the short exact sequence in Lemma 4.2.3 (1), we get the desired exact sequence.

(2) By Conjecture A.1.1 (1) and (2), the map γ^{2} in Conjecture A.1.2 is injective. Hence we get an exact sequence

0 ⟶ Coker(α_{r}) ⟶ _{p^{r}}Coker(γ^{2}) ⟶ CH^{2}(𝒳 )/p^{r} ⟶^{γ^{2}/p^{r}} H_{ét}^{4}(𝒳, ℤ(2))/p^{r}.

Noting that the composite of γ^{2}/p^{r} and the injective map

[figure: H_{ét}^{4}(𝒳, ℤ(2))/p^{r}; H_{ét}^{4}(𝒳, 𝔗_{r}(2))]

obtained from Sublemma A.1.5 coincides with ϱ_{r}^{2}, we get a short exact sequence

0 ⟶ Coker(α_{r}) ⟶ _{p^{r}}Coker(γ^{2}) ⟶ Ker(ϱ_{r}^{2}) ⟶ 0,

which implies the finiteness of Coker(α_{ℚ_{p}/ℤ_{p}}) and Ker(ϱ_{ℚ_{p}/ℤ_{p}}^{2}) under Conjecture A.1.2. This completes the proof of Lemma A.1.4 and Proposition A.1.3. □

B Zeta Value of Threefolds over Finite Fields

In this appendix B, all cohomology groups of schemes are taken over the étale topology. Let X be a projective smooth geometrically integral threefold over a finite field 𝔽_{q}, and let K be the function field of X (the case of fourfolds is treated in a recent paper of Kohmoto [Ko]). We define the unramified cohomology H_{ur}^{n+1}(K, ℚ/ℤ(n)) in the same way as in 1.8. We show that the groups

H_{ur}^{2}(K, ℚ/ℤ(1)) = Br(X) and H_{ur}^{3}(K, ℚ/ℤ(2))

are related with the value of the Hasse-Weil zeta function ζ(X, s) at s = 2:

ζ^{∗}(X, 2) := lim_{s→2} ζ(X, s)(1 − q^{2−s})^{−ϱ_{2}}, where ϱ_{2} := ord_{s=2} ζ(X, s).

Let

θ : CH^{2}(X) ⟶ Hom(CH^{1}(X), ℤ)

be the map induced by the intersection pairing and the degree map

CH^{2}(X) × CH^{1}(X) ⟶ CH^{3}(X) = CH_{0}(X) ⟶^{deg} ℤ.

The map θ has finite cokernel by a theorem of Matsusaka [Ma] Theorem 4. We define

ℛ := |Coker(θ)|.

We prove the following formula (compare with the formula in [Ge2]):

Theorem B.1 Assume that Br(X) and H_{ur}^{3}(K, ℚ/ℤ(2)) are finite. Then ζ^{∗}(X, 2) equals the following rational number up to a sign:

q^{χ(X,𝒪_{X},2)} · \frac{|H_{ur}^{3}(K, ℚ/ℤ(2))|}{|Br(X)| · ℛ} · ∏_{i=0}^{3} |CH^{2}(X, i)_{tors}|^{(−1)^{i}} · ∏_{i=0}^{1} |CH^{1}(X, i)_{tors}|^{(−1)^{i}},

where CH^{2}(X, i) and CH^{1}(X, i) denote Bloch’s higher Chow groups [Bl3] and χ(X, 𝒪_{X}, 2) denotes the following integer:

χ(X, 𝒪_{X}, 2) := ∑_{i,j}(−1)^{i+j} (2 − i) dim_{𝔽_{q}} H^{j}(X, Ω_{X}^{i}) (0 ≤ i ≤ 2, 0 ≤ j ≤ 3).

This theorem follows from a theorem of Milne ([Mi2] Theorem 0.1) and Proposition B.2 below. For integers i, n ≥ 0, we define

H^{i}(X, ℤ̂(n)) := ∏_{all ℓ} H^{i}(X, ℤ_{ℓ}(n)),

where ℓ runs through all prime numbers, and H^{i}(X, ℤ_{p}(n)) (p := ch(𝔽_{q})) is defined as

H^{i}(X, ℤ_{p}(n)) := \underleftarrow{lim}_{r≥1} H^{i−n}(X, W_{r} Ω_{X,log}^{n}).

Proposition B.2 (1) We have

H^{i}(X, ℤ̂(2)) ≃ \begin{cases}CH^{2}(X, 4 − i)_{tors} & (i = 0, 1, 2, 3) \\ (CH^{1}(X, i − 6)_{tors})^{∗} & (i = 6, 7), \end{cases}  (B.3)

where for an abelian group M, we put

M^{∗} := Hom(M, ℚ/ℤ).

Furthermore, CH^{1}(X, j)_{tors} and CH^{2}(X, j)_{tors} are finite for any j ≥ 0, and we have

CH^{1}(X, j)_{tors} = 0 for j ≥ 2 and CH^{2}(X, j)_{tors} = 0 for j ≥ 4.

(2) Assume that Br(X) is finite. Then we have

H^{5}(X, ℤ̂(2))_{tors} ≃ Br(X)^{∗},

and the cycle class map

CH^{2}(X) ⊗ ℤ_{ℓ} ⟶ H^{4}(X, ℤ_{ℓ}(2))

has finite cokernel for any prime number ℓ.

(3) Assume that Br(X) and H_{ur}^{3}(K, ℚ/ℤ(2)) are finite. Then the following map given by the cup product with the canonical element 1 ∈ ℤ̂ ≃ H^{1}(𝔽_{q}, ℤ̂) has finite kernel and cokernel:

ϵ^{4} : H^{4}(X, ℤ̂(2)) ⟶ H^{5}(X, ℤ̂(2)),

and we have the following equality of rational numbers:

\frac{|Ker(ϵ^{4})|}{|Coker(ϵ^{4})|} = \frac{|H_{ur}^{3}(K, ℚ/ℤ(2))| · |CH^{2}(X)_{tors}|}{|Br(X)| · ℛ}.

Proof of Proposition B.2. (1) By standard arguments on limits, there is a long exact sequence

⋯ ⟶ H^{i}(X, ℤ̂(2)) ⟶ H^{i}(X, ℤ̂(2)) ⊗_{ℤ} ℚ ⟶ H^{i}(X, ℚ/ℤ(2))
⟶ H^{i+1}(X, ℤ̂(2)) ⟶ · · · .

By [CTSS] p. 780 Théorème 2, p. 782 Théorème 3, we see that

H^{i}(X, ℤ̂(2)) and H^{i}(X, ℚ/ℤ(2)) are finite for i ≠ 4, 5.

Hence we have

H^{i}(X, ℤ̂(2)) ≃ H^{i−1}(X, ℚ/ℤ(2)) for i ≠ 4, 5, 6.

On the other hand, there is an exact sequence

0 ⟶ CH^{2}(X, 5 − i) ⊗ ℚ/ℤ ⟶ H^{i−1}(X, ℚ/ℤ(2)) ⟶ CH^{2}(X, 4 − i)_{tors} ⟶ 0

for i ≤ 3 ([MS], [SV], [GL1], [GL2]), where CH^{2}(X, 5 − i) ⊗ ℚ/ℤ must be zero because it is divisible and finite. Thus we get the isomorphism (B.3) for i ≤ 3, the finiteness of CH^{2}(X, j)_{tors} for j ≥ 1 and the vanishing of CH^{2}(X, j)_{tors} for j ≥ 4. The finiteness of CH^{2}(X, 0)_{tors} = CH^{2}(X)_{tors} (cf. [CTSS] p. 780 Théorème 1) follows from the exact sequence

0 ⟶ CH^{2}(X, 1) ⊗ ℚ/ℤ ⟶ N^{1}H^{3}(X, ℚ/ℤ(2)) ⟶ CH^{2}(X)_{tors} ⟶ 0

(cf. Lemma 3.2.2), where we put

N^{1}H^{3}(X, ℚ/ℤ(2)) := Ker(H^{3}(X, ℚ/ℤ(2)) → H^{3}(K, ℚ/ℤ(2))).

As for the case i = 6, 7 of (B.3), we have

H^{i}(X, ℤ̂(2))^{∗} ≃ H^{7−i}(X, ℚ/ℤ(1))

by a theorem of Milne [Mi2] Theorem 1.14 (a). It remains to show

CH^{1}(X, j)_{tors} ≃ H^{1−j}(X, ℚ/ℤ(1)) for j ≥ 0,

which can be checked by similar arguments as before.

(2) We have H^{5}(X, ℤ̂(2))^{∗} ≃ H^{2}(X, ℚ/ℤ(1)) and an exact sequence

0 ⟶ CH^{1}(X) ⊗ ℚ/ℤ ⟶ H^{2}(X, ℚ/ℤ(1)) ⟶ Br(X) ⟶ 0.  (B.4)

Hence we have (H^{5}(X, ℤ̂(2))_{tors})^{∗} ≃ Br(X), assuming Br(X) is finite. To show the second assertion for ℓ ≠ ch(𝔽_{q}), it is enough to show that the cycle class map

CH^{2}(X) ⊗ ℚ_{ℓ} ⟶ H^{4}(\overline{X}, ℚ_{ℓ}(2))^{Γ}

is surjective, where Γ := Gal(\overline{𝔽_{q}}/𝔽_{q}). The assumption on Br(X) implies the bijectivity of the cycle class map

CH^{1}(X) ⊗ ℚ_{ℓ} ^{∼} ⟶ H^{2}(\overline{X}, ℚ_{ℓ}(1))^{Γ}

by [Ta2] Proposition (4.3) (see also [Mi1] Theorem 4.1), and the assertion follow from [Ta2] Proposition (5.1). As for the case ℓ = ch(𝔽_{q}), one can easily pursue an analogy using crystalline cohomology, whose details are left to the reader.

(3) The finiteness assumption on Br(X) implies the condition SS(X, 1, ℓ) in [Mi2] for all prime numbers ℓ by loc. cit. Proposition 0.3. Hence SS(X, 2, ℓ) holds by the Poincaré duality, and ϵ^{4} has finite kernel and cokernel by loc. cit. Theorem 0.1.

To show the equality assertion, we put

ĈH^{2}(X) := \underleftarrow{lim}_{n≥1} CH^{2}(X)/n,

and consider the following commutative square (cf. [Mi3] Lemma 5.4):

\xymatrix{ĈH^{2}(X) \ar[r]^{Θ} \ar[d]_{α} & Hom(CH^{1}(X), ℤ̂) \\ H^{4}(X, ℤ̂(2)) \ar[r]^{ϵ^{4}} & H^{5}(X, ℤ̂(2)), \ar[u]_{β}}

where the top arrow Θ denotes the map induced by θ. The arrow α denotes the cycle class map of codimension 2, and β denotes the Pontryagin dual of the cycle class map with ℚ/ℤ-coefficients in (B.4). The arrow α is injective (cf. (4.3.3)) and we have

|Coker(α)| = |H_{ur}^{3}(K, ℚ/ℤ(2))|

by the finiteness assumption on H_{ur}^{3}(K, ℚ/ℤ(2)) and (2) (cf. Proposition 4.3.5). The arrow β is surjective and we have

Ker(β) = H^{5}(X, ℤ̂(2))_{tors} ≃^{(2)} Br(X)^{∗},

by Milne’s lemma ([Mi3] Lemma 5.3) and the isomorphism CH^{1}(X) ⊗ ℤ̂ ≃ H^{2}(X, ℤ̂(1)) (cf. [Ta2] Proposition (4.3)), where we have used again the finiteness assumption on Br(X). Therefore in view of the finiteness of Ker(ϵ^{4}), the map Θ has finite kernel and we obtain

Ker(Θ) = ĈH^{2}(X)_{tors} = CH^{2}(X)_{tors},

where we have used the finiteness of CH^{2}(X)_{tors} in (1). Finally the assertion follows from the following equality concerning the above diagram:

\frac{|Ker(Θ)|}{|Coker(Θ)|} = \frac{|Ker(α)|}{|Coker(α)|} · \frac{|Ker(ϵ^{4})|}{|Coker(ϵ^{4})|} · \frac{|Ker(β)|}{|Coker(β)|}

This completes the proof of Proposition B.2 and Theorem B.1. □

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Shuji Saito Department of Mathematics University of Tokyo Komaba, Meguro-ku Tokyo 153-8914, JAPAN sshuji@ms.u-tokyo.ac.jp Kanetomo Sato Graduate School of Mathematics, Nagoya University Furo-cho, Chikusa-ku Nagoya 464-8602 JAPAN kanetomo@math.nagoyau.ac.jp