Proof. (2) $\Rightarrow $ (3) and (2) $\Rightarrow $ (4) are obvious. (3) $\Rightarrow $ (2) holds because $\ker ({\operatorname {CH}}_n(M_K)_\Lambda \to {\operatorname {CH}}_n(M_{\overline {K}})_\Lambda )$ is torsion, where $\overline {K}$ is an algebraic closure of $K \in \operatorname {\mathbf {Fld}}$ . (4) $\Rightarrow $ (2) is seen by taking colimit. We have shown the equivalence (2) $\Leftrightarrow $ (3) $\Leftrightarrow $ (4).

Let us show (1) $\Rightarrow $ (4). By the shown equivalence (3) $\Leftrightarrow $ (4), we are reduced to the case k is algebraically closed (in particular, k is perfect). Take $K \in \operatorname {\mathbf {Fld}}^{\operatorname {fg}}$ . By Nagata’s compactification and de Jong’s alteration (see [Reference Conrad15, Theorem 4.1], [Reference de Jong16, Theorem 4.1]), we can find an integral proper k-scheme $X \in \operatorname {\mathbf {Sch}}$ with $K=k(X)$ and a proper surjective generically finite morphism $f : Y \to X$ with $Y \in \operatorname {\mathbf {SmProj}}$ integral. We then have a sequence of induced maps

$$\begin{align*}{\operatorname{CH}}_{n+d_Y}(M \otimes Y)_\Lambda \twoheadrightarrow {\operatorname{CH}}_{n}(M_{k(Y)})_\Lambda \overset{f_*}{\to} {\operatorname{CH}}_{n}(M_{k(X)})_\Lambda, \end{align*}$$

where $d_Y:=\dim Y$ . The first map is surjective, and the cokernel of the second map is annihilated by $[k(Y): k(X)]$ . Since ${\operatorname {CH}}_{n+d_Y}(M \otimes Y)_\Lambda =\operatorname {\mathbf {Chow}}_\Lambda (\Lambda (n+d_Y), M \otimes Y)$ is torsion by the assumption (1), we conclude that ${\operatorname {CH}}_{n}(M_{k(X)})_\Lambda $ is torsion as well.

It remains to prove (2) $\Rightarrow $ (1), for which we follow [Reference Gorchinskiy and Guletskii22, Lemma 1]. Write $M=(X, \pi , r) \in \operatorname {\mathbf {Chow}}_\Lambda $ with X equidimensional and put $d_X := \dim X$ . We take $N \in \operatorname {\mathbf {Chow}}_\Lambda $ and show that $\operatorname {\mathbf {Chow}}_\Lambda (M, N)$ is torsion. We may assume $N=h^{\operatorname {eff}}(Y)$ for connected $Y \in \operatorname {\mathbf {SmProj}}$ (by replacing r if necessary). Given $Z \in \operatorname {\mathbf {Sch}}$ , we define ${\operatorname {CH}}_n(M \otimes Z)_\Lambda $ as the image of an idempotent operator

$$\begin{align*}{\operatorname{CH}}_n(X \times Z)_\Lambda \to {\operatorname{CH}}_n(X \times Z)_\Lambda, \qquad \alpha \mapsto p_{23 *}(p_{13}^*(\alpha) \cdot_{p_{12}} \pi), \end{align*}$$

where $p_{ij}$ are respective projections on $X \times X \times Z$ , and $\cdot _{p_{12}}$ is the global product along $p_{12}$ defined in [Reference Fulton18, §8.1]; this product exists since $X \times X$ is smooth. We show that ${\operatorname {CH}}_n(M \otimes Z)_\Lambda $ is torsion for any integral $Z \in \operatorname {\mathbf {Sch}}$ and for any n by induction on $d_Z:=\dim Z$ . The case $d_Z=0$ is immediate from the assumption (2). If $d_Z>0$ , from the localization sequence for $X \times Z$ , we deduce an exact sequence

$$\begin{align*}\bigoplus_W {\operatorname{CH}}_n(M \otimes W)_\Lambda \to {\operatorname{CH}}_n(M \otimes Z)_\Lambda \to {\operatorname{CH}}_{n-d_Z}(M_{k(Z)})_\Lambda \to 0, \end{align*}$$

where W runs through integral proper closed subschemes of Z. The claim now follows by induction. Applying this to $Z=Y$ and $n=d_X+r$ , we conclude ${\operatorname {CH}}_{d_X+r}(M \otimes Y)_\Lambda =\operatorname {\mathbf {Chow}}_\Lambda (M, N)$ is torsion.

Proof. This is immediate from what we recalled in §2.3.

Proof. The first statement follows from [Reference Colliot-Thélène, Jacob and Rosenberg12, Theorem 4.1.1] (see also [Reference Rost38, (2.5)]) and [Reference Kahn and Sujatha30, Corollary 6.1.3], and the second from the first. The third statement is obvious from the definition.

Proof. See [Reference Kahn and Sujatha30, Proposition 3.1.1] for (1)–(3). Equivalence of (2) and (4) is obvious from the definition and (2.4) (see also [Reference Kahn29, §2.3]).

Proof. See [Reference Kahn and Sujatha30, Proposition 3.1.4] for the proof of (1) and [Reference Kahn29, Lemma 2.6] for (3). (2) follows from (1), (2.10) and the following Lemma.

Proof. For any $m, n \in \mathbb {Z}$ with $m, n>0$ , we have exact sequences of étale sheaves:

$$ \begin{align*} & 0 \to \mu_{\ell^m} \to \mu_{\ell^{m+n}} \to \mu_{\ell^n} \to 0, & & 0 \to \mu_{\ell^n} \to \mathbb{G}_m \to \mathbb{G}_m \to 0. \end{align*} $$

From the second sequence, we obtain an isomorphism $H^1_{\operatorname {\acute {e}t}}(X, \mu _{\ell ^n}) \cong \operatorname {Pic}(X)[\ell ^n]$ , from which we deduce (2.12) by taking a colimit over n. The upper exact row in the following diagram is obtained in a similar way, while the lower row is obtained by taking a limit over m and a colimit over n of the long exact sequence deduced from the first sequence:

(The limit preserves the exactness of the lower low since $H^i_{\operatorname {\acute {e}t}}(X, \mu _{\ell ^m})$ is finite for each $i, m$ .) The left and the right vertical maps are induced since the composition $\operatorname {Pic}(X) \otimes \mathbb {Q}/\mathbb {Z} \to H^3_{\operatorname {\acute {e}t}}(X, \mathbb {Z}_\ell (1))_{\operatorname {Tor}}$ vanishes (as the source is divisible and the target is finite). The second surjection in (2.13) is obtained as the right vertical map in this diagram, which is bijective if $b_2(X)=\rho (X)$ because so is the left vertical map under this hypothesis.

By a similar argument with different Tate twist, we get an exact sequence

(2.14) $$ \begin{align} 0 \to H^i_{\operatorname{\acute{e}t}}(X, \mathbb{Z}_\ell(r)) \otimes \mathbb{Q}_\ell/\mathbb{Z}_\ell \to H^i_{\operatorname{\acute{e}t}}(X, \mathbb{Q}_\ell/\mathbb{Z}_\ell(r)) \to H^{i+1}_{\operatorname{\acute{e}t}}(X, \mathbb{Z}_\ell(r))_{\operatorname{Tor}} \to 0 \end{align} $$

for any $i, r \in \mathbb {Z}$ . The first surjection in (2.13) is obtained as the second arrow in this sequence for $(i, r)=(1, 0)$ , which is bijective if $b_1(X)=0$ because the first term vanishes under this hypothesis. (We will use (2.14) for other $(i, r)$ later.)

Proof. (1) Proposition 2.14 shows the statement for $i \le 2$ . Then the Poincaré duality $b_{4-i}(X)=b_i(X)$ completes the proof for other i.

(2) All assertions follow from (1), plus a fact $H^1_{\operatorname {\acute {e}t}}(S, \mathbb {Z}_\ell )_{\operatorname {Tor}}=0$ which is seen from (2.14).

(3) Proposition 2.14 shows the first statement. It also shows $\operatorname {NS}(S)_{{\operatorname {Tor}}, \mathbb {Z}_\ell } \cong H^2_{\operatorname {\acute {e}t}}(S, \mathbb {Z}_\ell (1))_{\operatorname {Tor}}$ and $\operatorname {Br}(S)_{\Lambda } \cong H^3_{\operatorname {\acute {e}t}}(S, \mathbb {Z}_\ell (1))_{\operatorname {Tor}}$ ; hence, they are dual to each other by the Poincaré duality.

(4) Proposition 2.14 shows the vanishing of the Picard variety of X, whence the first statement. Since this implies the vanishing of the Albanese variety $\operatorname {Alb}_S$ of S, the last statement of (4) follows from Roitman’s theorem [Reference Rojtman37, p. 565, Consequence III] (which says ${\operatorname {CH}}_0(S_{\overline {K}})[m] \cong \operatorname {Alb}_S(\overline {K})[m]$ for any $m \in \mathbb {Z}$ invertible in k).

Proof. It suffices to show that $\delta \in \mathbb {Z}_\ell ^\times $ for any prime number $\ell \not = p$ . By Proposition 2.14 we have an isomorphism $\operatorname {NS}(S)_{\mathbb {Z}_\ell } \cong H^2_{\operatorname {\acute {e}t}}(S, \mathbb {Z}_\ell (1))$ which is compatible with the intersection pairing and the cup product. Therefore, it suffices to show that the cup product induces an isomorphism

$$\begin{align*}H^2_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(1))_{\operatorname{fr}} \overset{\cong}{\longrightarrow} \operatorname{Hom}_{\mathbb{Z}_\ell}(H^2_{\operatorname{\acute{e}t}}(S, \mathbb{Z}_\ell(1))_{\operatorname{fr}}, \mathbb{Z}_\ell), \end{align*}$$

where we put $M_{\operatorname {fr}} := M/M_{\operatorname {Tor}}$ for a $\mathbb {Z}_\ell $ -module M. This follows [Reference Zarhin49, Corollary 1.3].

Proof. The statement without the condition (3) is shown by Gorchinskiy-Orlov in (the proof of) [Reference Gorchinskiy and Orlov23, Proposition 2.3, Remark 2.5] when $k=\mathbb {C}$ , and the full statement by Vishik in [Reference Vishik44, Proposition 4.1] when S is the classical Godeaux surface. The same proof works without any essential change, but for the sake of completeness, we give a brief account.

Let $\rho :=\rho (S)$ and take $e_1, \dots , e_\rho \in \operatorname {NS}(S)$ such that their classes form a $\mathbb {Z}$ -basis of $\operatorname {NS}(S)/\operatorname {NS}(S)_{\operatorname {Tor}}$ . Let $a_{ij}$ be as in Lemma 3.3, and set $A:=(a_{ij}) \in {\operatorname {GL}}_\rho (\Lambda )$ . Write $A^{-1}=(b_{ij}) \in {\operatorname {GL}}_\rho (\Lambda )$ . Take also a closed point $x_0 \in S_{(0)}$ . We then define orthogonal projectors

$$ \begin{align*} & \pi_L := [S \times x_0] + [x_0 \times S], \quad \pi_N := \sum_{i, j} b_{ij} [e_i \times e_j] \quad \in \operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}(S, S)={\operatorname{CH}}_2(S \times S)_\Lambda. \end{align*} $$

Set $L:=(S, \pi _L, 0), N:=(S, \pi _N, 0), M:=(S, 1-\pi _L-\pi _N) \in \operatorname {\mathbf {Chow}}^{\operatorname {eff}}_\Lambda $ . Then we have (1) and (3). Observe that (1) and Lemma 3.2 imply that for any $\overline {K} \in \operatorname {\mathbf {Fld}}^{\operatorname {ac}}$ ,

(3.1) $$ \begin{align} {\operatorname{CH}}_1(M_{\overline{K}})_\Lambda=\operatorname{NS}(S)_{\Lambda, {\operatorname{Tor}}} \quad \text{and}\quad {\operatorname{CH}}_i(M_{\overline{K}})_\Lambda=0 ~\text{for}~ i \not= 1. \end{align} $$

It then follows by Lemma 2.2 that M satisfies (2) too. We are done.

Setting 3.5. In what follows, we denote by $M \in \operatorname {\mathbf {Chow}}^{\operatorname {eff}}_\Lambda $ a Chow motive constructed in Proposition 3.4. Observe that we have $S=M$ in $\operatorname {\mathbf {Chow}}^{\operatorname {nor}}_\Lambda $ , because $\Lambda (r)$ vanishes in $\operatorname {\mathbf {Chow}}^{\operatorname {nor}}_\Lambda $ for any $r \ge 0$ by Remark 2.3.

Proof. (1) (Compare [Reference Gorchinskiy and Orlov23, Proposition 2.3].) We consider a commutative diagram

The maps a and c are surjective by definition. Therefore, it suffices to prove the injectivity of e. Take $f \in \operatorname {\mathbf {Chow}}_\Lambda ^{\operatorname {eff}}(T, M)$ such that $e(f)=0$ . By Proposition 2.14 (2) and Lemma 2.15, this implies that, for any prime number $\ell \not = p$ , we have

(3.3) $$ \begin{align} f^*=0 : H^i_{\operatorname{\acute{e}t}}(M, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}} \to H^i_{\operatorname{\acute{e}t}}(T, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}} \quad \text{for } i=2, 3. \end{align} $$

However, we have a commutative diagram

Here, ${\operatorname {cyc}}$ is the cycle map. The upper horizontal injective map is the one constructed by Bloch (see [Reference Colliot-Thélène and Ballico11, Théorème 4.3]). The upper right triangle is commutative by [Reference Colliot-Thélène, Sansuc and Soulé13, Corollaire 4]. The right vertical map is bijective since we have $H^*_{\operatorname {\acute {e}t}}(M \otimes T, \mathbb {Q}_\ell (2))=0$ (as M is torsion). We have shown the injectivity of ${\operatorname {cyc}}$ . We consider isomorphisms

$$ \begin{align*} H^4_{\operatorname{\acute{e}t}}(M \otimes T, \mathbb{Z}_\ell(2))_{{\operatorname{Tor}}} &\cong \underset{i=2, 3}{\bigoplus} {\operatorname{Tor}}(H^{5-i}_{\operatorname{\acute{e}t}}(M, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}}, H^i_{\operatorname{\acute{e}t}}(T, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}}) \\ &\cong \underset{i=2, 3}{\bigoplus} \operatorname{Hom}(H^i_{\operatorname{\acute{e}t}}(M, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}}, H^i_{\operatorname{\acute{e}t}}(T, \mathbb{Z}_\ell(1))_{{\operatorname{Tor}}}) \end{align*} $$

induced by the Künneth formula, Poincaré duality (together with Proposition 3.4 (3)), and Lemma 8.3 below. Their composition sends $\alpha $ to the correspondence action (that is, $\beta \mapsto \operatorname {pr}_{2*}(\operatorname {pr}_1^*(\beta ) \cup \alpha )$ , where $\operatorname {pr}_i$ are projections on $M \otimes T$ ). Hence, it fits in the right vertical arrow of a commutative diagram

where the lower horizontal map is induced by the functors $H^i_{\operatorname {\acute {e}t}}(-, \mathbb {Z}_\ell (1))_{\operatorname {Tor}}$ for $i=2, 3$ . Now (3.3) shows that $f=0$ in $\operatorname {\mathbf {Chow}}^{\operatorname {eff}}_\Lambda (T, M)_{\mathbb {Z}_\ell }$ . We are done.

(2) The relations

$$\begin{align*}\exp(\operatorname{NS}(S)_{{\operatorname{Tor}}, \Lambda})= \exp(\operatorname{Br}(S)_\Lambda) ~|~ {\operatorname{Tor}}_\Lambda^{\operatorname{nor}}(S) = {\operatorname{Tor}}_\Lambda^{\operatorname{nor}}(M) ~|~ {\operatorname{Tor}}_\Lambda^{\operatorname{eff}}(M) \end{align*}$$

are seen by Lemma 3.2 (3), Propositions 2.9 and 2.14 (3) applied to $F=\operatorname {Br}(-)_\Lambda $ , the equality $S=M$ in $\operatorname {\mathbf {Chow}}^{\operatorname {nor}}_\Lambda $ , and Remark 2.16 (2), respectively. To conclude, it suffices to apply (1) to $T=S$ and $f=m \cdot {\operatorname {id}}_S$ with $m \in \mathbb {Z}_{>0}$ to get ${\operatorname {Tor}}_\Lambda ^{\operatorname {eff}}(M) ~|~ \exp (\operatorname {NS}(S)_{{\operatorname {Tor}}, \Lambda })$ .

Proof. (1) and (2) follows from Propositions 3.6 and 3.4, respectively.

Proof. Vishik proved (5.6) in [Reference Vishik44, Proposition 4.2] assuming $k=\mathbb {C}$ and m is a prime, although his proof did not use those assumptions. For the sake of completeness, we include a short proof. We may replace N by $X \in \operatorname {\mathbf {Sm}}$ . Consider the spectral sequence

(5.7) $$ \begin{align} E_2^{a, b}= H^a_{{\operatorname{Gal}}}(K, H^b_{{\operatorname{\acute{e}t}}}(X_{\overline{K}})) \Rightarrow H^{a+b}_{{\operatorname{\acute{e}t}}}(X_K), \end{align} $$

where $\overline {K}$ is a separable closure of K. By the smooth base change theorem, we have $H^b_{{\operatorname {\acute {e}t}}}(X_{\overline {K}}) \cong H^b_{{\operatorname {\acute {e}t}}}(X)$ on which the absolute Galois group of K acts trivially, and hence,

$$\begin{align*}E_2^{a, b}= H^a_{{\operatorname{Gal}}}(K, H^b_{{\operatorname{\acute{e}t}}}(X_{\overline{K}})) \cong H^a_{{\operatorname{Gal}}}(K) \otimes H^b_{{\operatorname{\acute{e}t}}}(X). \end{align*}$$

Observe that $E_2^{*, *}$ is generated by $H^*_{\operatorname {\acute {e}t}}(X)$ as a $H^*_{\operatorname {Gal}}(K)$ -module, and the differential maps $d_r^{*, *} : E_r^{*, *} \to E_r^{*+r, *-r+1}$ are $H^*_{\operatorname {Gal}}(K)$ -linear. It follows from the commutative diagram

that the edge maps $H^j_{\operatorname {\acute {e}t}}(X_K) \to E_2^{0, j}$ are surjective for all j, whence $E_2^{0, j}=E_\infty ^{0, j}$ . We conclude that (5.7) degenerates at $E_2$ -terms and induces the desired isomorphism.

Proof. Apply Proposition 5.3 to $N=M$ and use (4.6), (5.1) and (5.5).

Proof. This is taken from [Reference Tezuka and Yagita42, Lemma 2.1, Theorem 2.4]. Here, we only recall that (1) is a consequence of (5.5), (2) is due to Deligne and Paranjape (see [Reference Bloch and Ogus7, p.195, footnote], [Reference Paranjape36, Corollary 4.4]), and (3) is a consequence of (2).

Proof. The m-th power map $H^0_{\operatorname {Gal}}(k, \mu _{m^2}) \to H^0_{\operatorname {Gal}}(k, \mu _m)$ is surjective since k is algebraically closed, and hence, $Q(\zeta )=0$ for any $\zeta \in \mu _m$ . Thus, (1) follows from a formal property of the Bockstein operator $Q(x \cup y)=Q(x) \cup y \pm x \cup Q(y)$ by taking $y=\zeta $ (since $\tau = - \cup \zeta $ by definition). The same formal property reduces (2) to the surjectivity of $H_{\operatorname {Gal}}^a(K, \mu _{m^2}^{\otimes a}) \to H_{\operatorname {Gal}}^a(K, \mu _{m}^{\otimes a})$ , which is a consequence of the norm residue isomorphism theorem (see [Reference Voevodsky48, Theorem 6.16]).

Proof of Proposition 5.6.

The injectivity of $\tau $ is a part of the Beilinson-Lichtenbaum conjecture (proved by Voevodsky in [Reference Voevodsky48, Theorem 6.17]). Since $H^{-1}_{\operatorname {Zar}}(S_K, \mathscr {H}^a_m)=0$ and $H^0_{\operatorname {Zar}}(S_K, \mathscr {H}^a_m)=H^a_{{\operatorname {ur}}, m}(S_K)$ by the definition (2.9), we obtain from Theorem 5.8 (1) with $a=b$ an exact sequence sitting in the upper row of a diagram:

(5.10)

(This reproves the desired injectivity.) The square in (5.10) is commutative by Lemma 5.9 (1). The lower horizontal arrow in the diagram is an isomorphism by (5.5). By (5.8), we find that $H^{a-1, a-1}_{\mathscr {M}}(M_K)$ and $H^{a, a}_{\mathscr {M}}(M_K)$ are respectively decomposed as

$$ \begin{align*} &(H^{a-2}_{{\operatorname{Gal}}}(K) \otimes N_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes \widetilde{B}_S) \oplus (H^{a-4}_{{\operatorname{Gal}}}(K) \otimes B_S), \\ &(H^{a-1}_{{\operatorname{Gal}}}(K) \otimes N_S) \oplus (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes \widetilde{B}_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes B_S). \end{align*} $$

By Lemma 5.9 (2) and (5.10), we get

$$\begin{align*}\rho(H^{a-2}_{{\operatorname{Gal}}}(K) \otimes QN_S) = \rho(Q\tau(H^{a-2}_{{\operatorname{Gal}}}(K) \otimes N_S)) = \rho(\tau Q(H^{a-2}_{{\operatorname{Gal}}}(K) \otimes N_S))=0. \end{align*}$$

Similarly, we obtain $\rho (H^{a-3}_{{\operatorname {Gal}}}(K) \otimes B_S)=0$ since $B_S=Q\widetilde {B}_S$ . To conclude (5.9), it suffices now to note that $H^a_{{\operatorname {ur}}, m}(M_K)=H^a_{{\operatorname {ur}}}(M_K)$ by (2.8) and use Corollary 3.7 (1).

Proof. Since $H^{a, b}_{\mathscr {M}}(M_K, \Lambda )$ is annihilated by m for any $a, b \in \mathbb {Z}$ , a commutative diagram with an exact row

shows that the complex $(H^{\bullet , b}_{\mathscr {M}}(M_K), Q)$ is exact. Consider a diagram

which is commutative by Lemma 5.9 (1). Since $H^{a+1, a-2}_{\mathscr {M}}(M_K)=0$ by (5.3), the previous remark shows that the left vertical map in the diagram is surjective. The rest of the proof goes along the same lines as Proposition 5.6. We apply (5.9) to obtain direct sum decompositions of $H^{a-1, a-2}_{\mathscr {M}}(M_K)$ and $H^{a, a-1}_{\mathscr {M}}(M_K)$ respectively as

$$ \begin{align*} & (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-4}_{{\operatorname{Gal}}}(K) \otimes B_S) \oplus \ker(\alpha_{a-1}), \\ & (H^{a-2}_{{\operatorname{Gal}}}(K) \otimes QN_S) \oplus (H^{a-3}_{{\operatorname{Gal}}}(K) \otimes B_S) \oplus \ker(\alpha_a). \end{align*} $$

By Lemma 5.9 (2), the summand $(H^{a-3}_{{\operatorname {Gal}}}(K) \otimes QN_S) \oplus (H^{a-4}_{{\operatorname {Gal}}}(K) \otimes B_S)$ of $H^{a-1, a-2}_{\mathscr {M}}(M_K)$ is killed by the left vertical map because $Q^2=0$ and $B_S=Q\widetilde {B}_S$ . However, $\tau \circ Q$ maps $\ker (\alpha _{a-1})$ injectively into the summand $(H^{a-2}_{{\operatorname {Gal}}}(K) \otimes QN_S) \oplus (H^{a-3}_{{\operatorname {Gal}}}(K) \otimes B_S)$ of $H^{a, a-1}_{\mathscr {M}}(M_K)$ , showing the first statement.

In particular, we have shown the injectivity of $\tau : H^{a+1, a-1}_{\mathscr {M}}(X_K) \to H^{a+1, a}_{\mathscr {M}}(X_K)$ . Thus, the exact sequence from Theorem 5.8 (1) applied with $a=b$ shows that $\rho : H^{a, a}_{\mathscr {M}}(X_K) \to H_{\operatorname {ur}}^a(M_K)$ is surjective. The same exact sequence together with Proposition 5.6 shows that $\rho ((H^{a-2}_{{\operatorname {Gal}}}(K) \otimes QN_S) \oplus (H^{a-3}_{{\operatorname {Gal}}}(K) \otimes B_S))=0$ . This completes the proof of the last statement.

Proof of Theorem 5.2.

As the unramified cohomology is normalized, birational and motivic (Proposition 2.9), we have $H^i_{{\operatorname {ur}}}(S)=H^i_{{\operatorname {ur}}}(M)$ and $H^i_{{\operatorname {ur}}}(K(S)/K)=H^i_{{\operatorname {ur}}}(M_K)$ . Now Propositions 5.6 and 5.10 complete the proof.

Proof. We may assume $m=\ell ^e$ for a prime number $\ell \not = p$ and $e \in \mathbb {Z}_{>0}$ . Recall that a module over an Artin local ring is free if and only if it is flat (see, for example, [Reference Anderson and Pablos Romo2, Proposition 2.1.4]). By the norm residue isomorphism theorem (see [Reference Voevodsky48, Theorem 6.16]), $K_{j-1}^M(E) \otimes \mu _{\ell ^\infty }$ surjects onto $K_j^M(E)_{{\operatorname {Tor}}} \otimes \mathbb {Z}_{(\ell )}$ , and hence, $K_j^M(E)_{{\operatorname {Tor}}}$ is divisible by $\ell $ . It follows that $K_j^M(E)$ is the direct sum of an $\ell $ -divisible group and a flat $\mathbb {Z}_{(\ell )}$ -module. Thus, $K_j^M(E) \otimes \mathbb {Z}/m\mathbb {Z} \cong H^j_{\operatorname {Gal}}(E)$ is a flat $\mathbb {Z}/m\mathbb {Z}$ -module.

Proof. For $i=1$ , (6.4) is nothing other than the paring in Lemma 3.2 (3), whence the result. Assume now $i=2$ . We claim that $Q_2'(H_{\operatorname {ur}}^1(S))$ is the exact annihilator of itself with respect to $\langle -, - \rangle $ . For this, we first note that $\langle Q_2'(H_{\operatorname {ur}}^1(S)), Q_2'(H_{\operatorname {ur}}^1(S)) \rangle =0$ because

$$\begin{align*}Q(a) \cup Q(b) =Q(a) \cup Q(b) - a \cup Q^2(b) = Q(a \cup Q(b)) = 0 \end{align*}$$

for $a, b \in H_{\operatorname {\acute {e}t}}^1(S)$ . Here, the first (resp. third) equality holds because $Q^2=0$ (resp. $Q : H^3_{\operatorname {\acute {e}t}}(S) \to H^4_{\operatorname {\acute {e}t}}(S)$ is the zero map, as $H^4_{\operatorname {\acute {e}t}}(S, \mathbb {Z}/m\mathbb {Z}) \to H^4_{\operatorname {\acute {e}t}}(S, \mathbb {Z}/m^2\mathbb {Z})$ is injective). We then use the fact $|Q_2'(H_{\operatorname {ur}}^1(S))|=|H^1_{\operatorname {ur}}(S)|=|H^2_{\operatorname {ur}}(S)|=|H^2_{\operatorname {\acute {e}t}}(S)/Q_2'(H_{\operatorname {ur}}^1(S))|$ to conclude the claim. It follows that $\langle -, - \rangle $ induces the perfect paring in the statement characterized by (6.4).

Proof. This follows from Lemmas 6.3, 6.4 and 8.1 (2).

Proof. We consider a commutative diagram with exact row:

where U is an open dense subscheme of S. Since the map in question is obtained as the colimit of $\iota _U \circ \gamma _U$ as U ranges over such schemes, it suffices to show the vanishing of the lower left group for sufficiently small $U \subset S$ . For this, we take a (possibly reducible) curve $C \subset S$ whose components generate $\operatorname {NS}(S)$ . Then we find $\operatorname {Pic}(U_E)=0$ as soon as $U \subset S \setminus C$ because we have $\operatorname {Pic}(S_E)=\operatorname {NS}(S)$ by Lemma 3.2 (4). We are done.

Proof. We decompose (6.5) as follows:

$$ \begin{align*} \bigoplus_{i=1, 2} H^{i-1}_{\operatorname{Gal}}(E) \otimes H^{3-i}_{\operatorname{ur}}(S) &\hookrightarrow \bigoplus_{i=1, 2} H^{i-1}_{\operatorname{Gal}}(E) \otimes H^{3-i}_{\operatorname{Gal}}(k(S)) \\ &\hookrightarrow H^2_{\operatorname{\acute{e}t}}(\operatorname{Spec}(E \otimes_k k(S))) \hookrightarrow H^2_{\operatorname{Gal}}(E(S)). \end{align*} $$

The injectivity of the first map follows from Lemma 6.3 since $H^i_{\operatorname {ur}}(S)=H^i_{\operatorname {ur}}(k(S)/k)$ is a subgroup of $H^i_{\operatorname {Gal}}(k(S))$ by definition (see (2.6), (2.11)). The second (resp. third) map is also injective by Remark 5.4 (resp. Lemma 6.6).

Proof. Apply Theorem 6.1 to $K=k(T)$ and use (2.11). Note that the injectivity of $\Phi $ follows also from Proposition 3.6 together with (6.2).

Proof. The implications (1) $\Rightarrow $ (2) $\Rightarrow $ (3) are obvious, and (3) $\Rightarrow $ (1) follows from Theorem 7.1 and Lemma 7.4 below.

Proof. Put $K:=k(T)$ . We use a cartesian diagram

where $\operatorname {pr}_i$ are the projections and $s_i$ are the structure maps. We first show, by a standard argument, the commutativity of the diagram

(7.3)

where ${\operatorname {cr}}$ is the correspondence action (that is, ${\operatorname {cr}}(\xi )(a)=\operatorname {pr}_{2*}(\operatorname {pr}_1^*(a) \cup \xi )$ ), ${\operatorname {k\ddot {u}}}$ is the Künneth isomorphism, and ${\operatorname {pd}}$ is the isomorphism from Lemma 6.5. We take $a \in H^i_{\operatorname {\acute {e}t}}(S)$ , $b \in H^{4-i}_{\operatorname {\acute {e}t}}(S)$ and $x \in H^i_{\operatorname {Gal}}(K)$ and compute

$$ \begin{align*} ({\operatorname{cr}} \circ {\operatorname{k\ddot{u}}})(b \otimes x)(a) &= \operatorname{pr}_{2*}(\operatorname{pr}_1^*(a) \cup \operatorname{pr}_1^*(b) \cup \operatorname{pr}_2^*(x)) \\ &= \operatorname{pr}_{2*}(\operatorname{pr}_1^*(a \cup b) \cup \operatorname{pr}_2^*(x)) \overset{(1)}{=} \operatorname{pr}_{2*}(\operatorname{pr}_1^*(a \cup b)) \cup x \\ & \overset{(2)}{=} s_2^* s_{1*}(a \cup b) \cup x ={\operatorname{pd}}(b \otimes x)(a). \end{align*} $$

Here, we have used the projection formula for étale cohomology and the base change property in [Reference Artin, Grothendieck, Verdier, Deligne and Saint-Donat3, Exposé XVIII, Théorème 2.9] at $(1)$ and $(2)$ , respectively. We have shown the commutativity of (7.3).

We now consider the following diagram:

Here, ${\operatorname {cyc}}$ is the cycle map, and $\Pi $ is the direct sum of the compositions

$$\begin{align*}\operatorname{Hom}(H^i_{\operatorname{ur}}(S), H^i_{\operatorname{ur}}(T)) \hookrightarrow \operatorname{Hom}(H^i_{\operatorname{ur}}(S), H^i_{\operatorname{Gal}}(K)) \overset{\pi_i^*}{\hookrightarrow} \operatorname{Hom}(H^i_{\operatorname{\acute{e}t}}(S), H^i_{\operatorname{Gal}}(K)), \end{align*}$$

where $\pi _i^*$ is induced by $\pi _i$ in (6.3) (which is split surjective). If we set ${\operatorname {pd}} \circ {\operatorname {k\ddot {u}}}^{-1}$ at $(*)$ and $\Phi $ at $(**)$ , then the diagram commutes by Theorem 5.8 (3) and Lemma 6.4. However, if we set ${\operatorname {cr}}$ at $(*)$ and the induced map by $H^*_{\operatorname {ur}}$ at $(**)$ , then the diagram commutes by definition. Hence, the assertion follows from the commutativity of (7.3).

Proof. Let $M \in \operatorname {\mathbf {Chow}}^{\operatorname {eff}}_\Lambda $ be the Chow motive constructed in Proposition 3.4. Since ${\operatorname {CH}}_0(S_{k(S)})_{{\operatorname {Tor}}, \Lambda } = \operatorname {\mathbf {Chow}}^{\operatorname {nor}}_\Lambda (S, S) = \operatorname {\mathbf {Chow}}^{\operatorname {nor}}_\Lambda (M, M)$ , Proposition 3.6 (1) and (7.1) yields an exact sequence

(7.4) $$ \begin{align} 0 \to \operatorname{\mathbf{Chow}}_\Lambda^{\operatorname{eff}}(M, M) \overset{\Phi}{\to} \bigoplus_{i=1, 2} \operatorname{Hom}(H_{\operatorname{ur}}^i(S), H_{\operatorname{ur}}^i(S)) \to H^3_{\operatorname{ur}}(S \times S) \to 0. \end{align} $$

We know ${\operatorname {id}}_M \in \operatorname {\mathbf {Chow}}^{\operatorname {eff}}_\Lambda (S, S)$ has order $\ell $ by Proposition 3.6 (2). Thus, it suffices to show $\Phi $ is not surjective. If it were surjective, then by (7.4), there should be a projector $\pi : M \to M$ in $\operatorname {\mathbf {Chow}}^{\operatorname {eff}}_\Lambda $ such that $N:=\operatorname {Im}(\pi ) \subset M$ satisfies $\operatorname {Pic}(N)=0$ and $\operatorname {Br}(N) \cong \mathbb {Z}/\ell \mathbb {Z}$ , but this would contradict the following result of Vishik.

Proof. See [Reference Vishik44, Corollary 4.22].

Proof. Set $X:=S \times S$ . We claim that ${\operatorname {CH}}_0(X) \cong \mathbb {Z}$ . For this, it suffices to show that $\ker ({\operatorname {CH}}_0(X) \to \mathbb {Z})$ is torsion by Roitman’s theorem, but Proposition 3.4 implies that

$$\begin{align*}\ker({\operatorname{CH}}_0(X) \to \mathbb{Z}) \cong \operatorname{\mathbf{Chow}}^{\operatorname{eff}}_\Lambda(\Lambda(0), M \otimes M), \end{align*}$$

which is obviously killed by $\ell $ . Now the corollary is a consequence of Theorem 7.8 and the following result of Colliot-Thélène and Voisin [Reference Colliot-Thélène and Voisin14].

Proof. See [Reference Colliot-Thélène and Voisin14, Théorème 3.9].