Proof. The proof of [Reference Milne10, Chap. III, Prop. 2.9] is purely formal, and it also works here.

Proof. The results follow from Proposition 3.1.

Proof. By Proposition 3.1, there exists a Kummer étale cover

$$ \begin{align*}\left\{Y_i\xrightarrow{g_i} X\right\}_{i\in I}\end{align*} $$

such that $g_i^*(\gamma )=0$ for each $i\in I$ . Let $i_0$ be such that $g_{i_0}\left (Y_{i_0}\right )$ contains the closed point of X. By [Reference Nizioł12, Prop. 2.15], there exists a commutative diagram

such that the image of $g_{i_0}\circ g$ contains the closed point of X, g is Kummer étale, n is a positive integer which is invertible on X, and h is classically étale. Then we have $h^*f_n^*\gamma =g^*g_{i_0}^*\gamma =0$ . Since h is classically étale and the underlying scheme of $X_n$ is strictly Henselian local, h has a section s. It follows that

$$ \begin{align*}f_n^*\gamma=s^*h^*\left(f_n^*\gamma\right)=s^*0=0. \end{align*} $$

Proof. This follows from Corollary 3.3.

Proof. Part (1) follows from [Reference Artin1, Chap. II, Sec. 3, (3.3)].

Part (2) follows from Corollary 3.3. Indeed, we have

$$ \begin{align*}\check{H}_{\mathrm{k\acute{e}t}}^0\left(\mathscr{X}_{{\mathbb N}'},\underline{H}_{\mathrm{k\acute{e}t}}^j(F)\right)\hookrightarrow\varinjlim_{n\in{\mathbb N}'}H_{\mathrm{k\acute{e}t}}^j(X_n,F),\end{align*} $$

and the latter vanishes by Corollary 3.3.

Proof. By the Kummer étale spectral sequence from formula (3.1) and Proposition 3.5(2), it suffices to show that the groups $\varinjlim _{n\in {\mathbb N}'}\check {H}_{\mathrm {k\acute {e}t}}^i\left (X_n/X,\underline {H}_{\mathrm {k\acute {e}t}}^j(F)\right )$ are torsion and p-torsion-free for all $i>0$ and $j\geq 0$ . Define $H_n:=\mathop {\mathrm {Spec}}{\mathbb Z}\left [\left (P^{1/n}\right )^{\mathrm {gp}}/P^{\mathrm {gp}}\right ]$ , which is a group scheme over $\mathop {\mathrm {Spec}}{\mathbb Z}$ such that

$$ \begin{align*}X_n\times_XX_n=X_n\times_{\mathop{\mathrm{Spec}}{\mathbb Z}}H_n\end{align*} $$

(see [Reference Nizioł12, the second paragraph on p522] for more detailed descriptions of $H_n$ ). Regarded as a group scheme over X, $H_n$ is constant; $X_n$ is a Galois cover of X with Galois group $H_n$ . By [Reference Milne10, Example 2.6], we have

$$ \begin{align*}\check{H}_{\mathrm{k\acute{e}t}}^i\left(X_n/X,\underline{H}_{\mathrm{k\acute{e}t}}^j(F)\right)=H^i\left(H_n,H_{\mathrm{k\acute{e}t}}^j(X_n,F)\right),\end{align*} $$

which is torsion and p-torsion-free for $i>0$ . This finishes the proof.

Proof. This follows from Corollary 3.6.

Proof. Part (1) follows from Corollary 3.7.

Part (2) follows from Corollary 3.6.

We are left with part (3). Since n is invertible on X, the sequence

(3.2) $$ \begin{align} 0\rightarrow G[n]\rightarrow G\xrightarrow{n}G\rightarrow0 \end{align} $$

is a short exact sequence of sheaves of abelian groups for the classical flat topology and $G[n]$ is quasi-finite and étale. Indeed, if X is a point, then by the structure theorem of connected algebraic groups [Reference Brion2, Thm. 2.3, Thm. 2.4], we are reduced to checking the cases where G is a torus, a connected unipotent group, or an abelian variety, which are all clearly true. In general, it suffices to show that $G\xrightarrow {n}G$ is an epimorphism for the classical flat topology. This is clear, since it is set-theoretically surjective and flat by the fiberwise criterion of flatness, and therefore it is faithfully flat. Moreover, $G[n]$ being étale over X implies that $G\xrightarrow {n}G$ is even an epimorphism for the classical étale topology, and thus the sequence (3.2) is also exact for the classical étale topology. Since the pullback functor $\varepsilon _{\mathrm {\acute {e}t}}^*$ is exact, the sequence (3.2) remains exact on $(\mathrm {fs}/X)_{\mathrm {k\acute {e}t}}$ and induces a long exact sequence

$$ \begin{align*}\rightarrow R^{i-1}\varepsilon_{\mathrm{\acute{e}t}*}G\xrightarrow{n} R^{i-1}\varepsilon_{\mathrm{\acute{e}t}*}G\rightarrow R^i\varepsilon_{\mathrm{\acute{e}t}*}G[n]\rightarrow R^i\varepsilon_{\mathrm{\acute{e}t}*}G\xrightarrow{n} R^i\varepsilon_{\mathrm{\acute{e}t}*}G.\end{align*} $$

This further induces a short exact sequence

$$ \begin{align*}0\rightarrow R^{i-1}\varepsilon_{\mathrm{\acute{e}t}*}G\otimes_{{\mathbb Z}}{\mathbb Z}/n{\mathbb Z}\rightarrow R^{i}\varepsilon_{\mathrm{\acute{e}t}*}G[n]\rightarrow \left(R^{i}\varepsilon_{\mathrm{\acute{e}t}*}G\right)[n]\rightarrow 0\end{align*} $$

for each $i>0$ . We have $R^i\varepsilon _{\mathrm {\acute {e}t}*}G[n]=G[n](-i)\otimes _{{\mathbb Z}}\bigwedge ^i\left (\mathbb {G}_{\mathrm {m,log}}/\mathbb {G}_{\mathrm {m}}\right )_{X_{\mathrm {\acute {e}t}}}$ by Theorem 2.6. To finish the proof, it suffices to prove that the sheaf $R^{i-1}\varepsilon _{\mathrm {\acute {e}t}*}G$ is n-divisible. We proceed by induction. For $i=1$ , this is clear, since $G\xrightarrow {n}G$ is an epimorphism of sheaves of abelian groups for the classical étale topology. Assume $R^{j}\varepsilon _{\mathrm {\acute {e}t}*}G$ is n-divisible; then we have $R^{j+1}\varepsilon _{\mathrm {\acute {e}t}*}G[n]\xrightarrow {\cong } \left (R^{j+1}\varepsilon _{\mathrm {\acute {e}t}*}G\right )[n]$ . Therefore

$$ \begin{align*} \varinjlim_r\left(R^{j+1}\varepsilon_{\mathrm{\acute{e}t}*}G\right)[n^r]&=\varinjlim_rR^{j+1}\varepsilon_{\mathrm{\acute{e}t}*}G[n^r] \\ &=\varinjlim_rG[n^r](-j-1)\otimes_{{\mathbb Z}}\bigwedge^{j+1}\left(\mathbb{G}_{\mathrm{m,log}}/\mathbb{G}_{\mathrm{m}}\right)_{X_{\mathrm{\acute{e}t}}}, \end{align*} $$

where the second equality follows from Theorem 2.6. Hence $\varinjlim _r\left (R^{j+1}\varepsilon _{\mathrm {\acute {e}t}*}G\right )[n^r]$ is n-divisible. It follows by part (1) that $R^{j+1}\varepsilon _{\mathrm {\acute {e}t}*}G$ is n-divisible. This finishes the induction.

Proof. This follows from Theorem 3.8.

Proof. This follows from Theorem 3.8.

Proof. By definition, $\underline {C}^{i}(G)$ is the functor

$$ \begin{align*}(\mathrm{st}/X)\rightarrow\mathrm{Ab}, Y\mapsto G(Y_m\times_Y\cdots\times_YY_m)=G\left(Y\times_X X_m\times_{\mathop{\mathrm{Spec}}{\mathbb Z}}H_m^i\right);\end{align*} $$

that is, it is $\mathring {\pi }_*G$ , where $\pi $ denotes the map $X_m\times _{\mathop {\mathrm {Spec}}{\mathbb Z}}H_m^i\rightarrow X$ and $\mathring {\pi }$ denotes the underlying map of schemes of $\pi $ . Since $\mathring {\pi }$ is clearly finite and faithful flat, $\underline {C}^{i}(G)$ is therefore represented by the Weil restriction of scalars of $G\times _X\left (X_m\times _{\mathop {\mathrm {Spec}}{\mathbb Z}}H_m^i\right )$ , which is representable by a group algebraic space by [Reference Milne10, Chap. V, 1.4 (a)]. Hence the functor $\underline {Z}^i(G)$ , as the kernel of a map $d^i:\underline {C}^i(G)\rightarrow \underline {C}^{i+1}(G)$ of group algebraic spaces over X, is representable by a group algebraic space over X.

Now we prove the smoothness of $d^{i-1}:\underline {C}^{i-1}(G)\rightarrow \underline {Z}^i(G)$ . It suffices to show that for any affine X-scheme T, closed subscheme $T_0$ of T defined by an ideal I of square zero, and $z\in \underline {Z}^i(G)(T)$ whose image $z_0$ in $\underline {Z}^i(G)(T_0)$ arises from an element $c_0\in \underline {C}^{i-1}(G)(T_0)$ , there exists $c\in \underline {C}^{i-1}(G)(T)$ such that c maps to $c_0$ . Let N be the functor

$$ \begin{align*}(\mathrm{Sch}/T)\rightarrow\mathrm{Ab},Y\mapsto\mathrm{ker}(G(Y)\rightarrow G(Y\times_TT_0)).\end{align*} $$

Let $C^{\cdot }(T_m/T,N)$ be the Čech complex for the Kummer flat cover $T_m:=T\times _XX_m\rightarrow T$ with coefficients in N. Then we have the following commutative diagram with exact rows:

where the exactness property at the right-hand side follows from the smoothness of G. Let $Z^i(T_m/T,N)$ be the kernel of

$$ \begin{align*}C^{i}(T_m/T,N)\rightarrow C^{i+1}(T_m/T,N);\end{align*} $$

then the diagram induces the following commutative diagram with exact rows:

By an easy diagram chasing, for the existence of c it suffices to show that the map $C^{i-1}(T_m/T,N)\rightarrow Z^{i}(T_m/T,N)$ is surjective – that is, that $\check {H}^i(T_m/T,N)=0$ , $i\geq 1$ .

To finish the proof, we compute $\check {H}^i(T_m/T,N)$ for $i\geq 1$ . Let $\tilde {N}$ be the functor

$$ \begin{align*}(\mathrm{Sch}/T)\rightarrow\mathrm{Ab},U\mapsto N(U\times_TT_m).\end{align*} $$

Since $T_m\times _TT_m\cong T_m\times H_m$ , the group scheme $H_m$ acts on the functor $\tilde {N}$ , and the Čech complex $C^{\cdot }(T_m/T,N)$ can be identified with the standard complex $C^{\cdot }\left (H_m,\tilde {N}\right )$ computing the cohomology of the $H_m$ -module $\tilde {N}$ . We claim that $\tilde {N}$ is coherent; then the vanishing of $\check {H}^i(T_m/T,N)$ follows from $H_m$ being diagonalizable by [Reference Gille and Polo4, Exposé I, Thm. 5.3.3]. For any $U\in (\mathrm {Sch}/T)$ , the smoothness of G implies that

$$ \begin{align*} \tilde{N}(U)=N(U\times_TT_m)&=\mathrm{ker}(G(U\times_TT_m)\rightarrow G(U\times_TT_m\times_TT_0)) \\ &=\mathrm{Lie}(G)\otimes_{\Gamma\left(T,\mathcal{O}_T\right)}\Gamma\left(T_m,I\mathcal{O}_{T_m}\right)\otimes_{\Gamma\left(T,\mathcal{O}_T\right)}\Gamma(U,\mathcal{O}_U). \end{align*} $$

Therefore $\tilde {N}$ is coherent.

Proof. Since G is smooth and $X_m\times _X\cdots \times _XX_m\cong X_m\times _{\mathop {\mathrm {Spec}}{\mathbb Z}} H_m^r$ is a disjoint union of spectra of Henselian local rings, the maps

$$ \begin{align*}C^{i}(X_m/X,G)\rightarrow C^{i}(x_m/x,G)\end{align*} $$

are surjective by [Reference Milne10, Chap. I, 4.13]. Thus we are reduced to showing that the complex $\mathrm {ker}(C^{\cdot }(X_m/X,G)\rightarrow C^{\cdot }(x_m/x,G))$ is exact. Let

$$ \begin{align*}z\in \mathrm{ker}\left(C^{i}(X_m/X,G)\xrightarrow{d^i} C^{i+1}(X_m/X,G)\right)\end{align*} $$

have image $z_0=0$ in $C^{i}(x_m/x,G)$ . We seek a $c\in C^{i-1}(X_m/X,G)$ with $c_0=0$ such that $d^{i-1}(c)=z$ . By Lemma 3.11 we know that $\left (d^{i-1}\right )^{-1}(z)$ is representable by a smooth scheme over X. But $\left (d^{i-1}\right )^{-1}(z)$ has a section over x, namely the zero section, and as X is Henselian, this lifts to a section of $\left (d^{i-1}\right )^{-1}(z)$ over X. This finishes the proof.

Proof. In the proof of [Reference Nizioł12, Prop. 3.13], the affineness of G is only used in the paragraph before [Reference Nizioł12, Cor. 3.17]. Its use is to extend from the complete local case [Reference Nizioł12, Lem. 3.15]) to the Henselian local case. Note that the proof of [Reference Nizioł12, Lem. 3.15] deals with the Artinian local case first, then passes to the complete local case. With the help of Lemma 3.12, we can pass from the Artinian local case directly to the Henselian local case, and hence no affineness of G is needed.

Proof. The statement for the finite flat case is the same as in [Reference Kato8, Thm. 4.1]. We only need to deal with the other case, which follows from Theorem 3.13 in the same way as [Reference Nizioł12, Thm. 3.12] follows from [Reference Nizioł12, Prop. 3.13].

Proof. We prove only $\varinjlim _nH_{\mathrm {kfl}}^1(X_{n},G)=0$ ; the other statement can be proven in the same way. By Theorem 3.13, we have

$$ \begin{align*} H_{\text{kfl}}^{1} (X_n,G)\cong \varinjlim_{r} \mathrm{Hom}_{X_{n}}({\mathbb Z}/r{\mathbb Z}(1),G)\otimes_{{\mathbb Z}} \left(P^{{1}/{n}}\right)^{\mathrm{gp}} \end{align*} $$

for each $n>0$ . As n varies, these isomorphisms fit into commutative diagrams of the form

with the second row induced by the canonical inclusion $P^{1/n}\hookrightarrow P^{1/mn}$ . The group $\mathrm {Hom}_{X_{n}}({\mathbb Z}/r{\mathbb Z}(1),G)$ is clearly torsion, and thus it follows that

$$ \begin{align*} \varinjlim_{n}H_{\mathrm{kfl}}^1(X_n,G)=\varinjlim_{n}\varinjlim_{r}\mathrm{Hom}_{X_{n}}({\mathbb Z}/r{\mathbb Z}(1),G)\otimes_{{\mathbb Z}}\left(P^{{1}/{n}}\right)^{\mathrm{gp}} =0. \end{align*} $$

Proof. By Proposition 3.1, we can find a Kummer flat cover $T\rightarrow X$ such that $\gamma $ dies in $H_{\mathrm {kfl}}^2(T,G)$ . By [Reference Nizioł12, Cor. 2.16] we may assume that for some n, we have a factorization $T\rightarrow X_n\rightarrow X$ with $T\rightarrow X_n$ a classical flat cover. It follows that the class $\gamma $ on $X_n$ is trivialized by a classical flat cover – that is, $\gamma \in \mathrm {ker}\left (H_{\mathrm {kfl}}^2(X_n,G)\rightarrow H_{\mathrm {fl}}^0\left (X_n,R^2\varepsilon _{\mathrm {fl}*} G\right )\right )$ . The seven-term exact sequence of the spectral sequence $H_{\mathrm {fl}}^i\left (X_n,R^j\varepsilon _{\mathrm {fl}*}G\right )\Rightarrow H_{\mathrm {kfl}}^{i+j}(X_n,G)$ gives an exact sequence

$$ \begin{align*}\cdots\rightarrow H_{\mathrm{fl}}^2(X_n,G)\rightarrow \mathrm{ker}\left(H_{\mathrm{kfl}}^2(X_n,G)\rightarrow H_{\mathrm{fl}}^0\left(X_n,R^2\varepsilon_{\mathrm{fl}*} G\right)\right)\rightarrow H_{\mathrm{fl}}^1\left(X_n,R^1\varepsilon_{\mathrm{fl}*} G\right).\end{align*} $$

Hence, to show that $\gamma =0$ in $H_{\mathrm {kfl}}^2(X_n,G)$ , it suffices to show that $H_{\mathrm {fl}}^2(X_n,G)=H_{\mathrm {fl}}^1\left (X_n,R^1\varepsilon _{\mathrm {fl}*} G\right )=0$ .

Since G is smooth and A is strictly Henselian, we have

$$ \begin{align*}H_{\mathrm{fl}}^2(X_n,G)=H_{\mathrm{\acute{e}t}}^2(X_n,G)=0.\end{align*} $$

We have $R^1\varepsilon _{\mathrm {fl}*}G=\varinjlim _m\mathcal {H}om_{X_n}({\mathbb Z}/m{\mathbb Z}(1),G)\otimes \left (\mathbb {G}_{\mathrm {m,log}}/\mathbb {G}_{\mathrm {m}}\right )_{X_{n,\mathrm {fl}}}$ by Theorem 3.14. By Lemma A.1, the sheaf $\mathcal {H}om_{X_n}({\mathbb Z}/m{\mathbb Z}(1),G)$ is representable by a quasi-finite étale separated group scheme over $X_n$ . It follows that

$$ \begin{align*} H_{\mathrm{fl}}^1\left(X_n,R^1\varepsilon_{\mathrm{fl}*} G\right)&=H_{\mathrm{fl}}^1\left(X_n,\varinjlim_m\mathcal{H}om_{X_n}({\mathbb Z}/m{\mathbb Z}(1),G)\otimes \left(\mathbb{G}_{\mathrm{m,log}}/\mathbb{G}_{\mathrm{m}}\right)_{X_{n,\mathrm{fl}}}\right) \\ &=\varinjlim_m H_{\mathrm{fl}}^1\left(X_n,\mathcal{H}om_{X_n}({\mathbb Z}/m{\mathbb Z}(1),G)\otimes \left(\mathbb{G}_{\mathrm{m,log}}/\mathbb{G}_{\mathrm{m}}\right)_{X_{n,\mathrm{fl}}}\right) \\ &=\varinjlim_m H_{\mathrm{fl}}^1\left(X_n,\theta^*\left(\mathcal{H}om_{X_n}({\mathbb Z}/m{\mathbb Z}(1),G)\otimes_{{\mathbb Z}}\left(\mathbb{G}_{\mathrm{m,log}}/\mathbb{G}_{\mathrm{m}}\right)_{X_{n,\mathrm{\acute{e}t}}}\right)\right) \\ &=\varinjlim_m H_{\mathrm{\acute{e}t}}^1\left(X_n,\mathcal{H}om_{X_n}({\mathbb Z}/m{\mathbb Z}(1),G)\otimes_{{\mathbb Z}}\left(\mathbb{G}_{\mathrm{m,log}}/\mathbb{G}_{\mathrm{m}}\right)_{X_{n,\mathrm{\acute{e}t}}}\right) \\ &=0, \end{align*} $$

where $\theta :(\mathrm {fs}/X_n)_{\mathrm {fl}}\rightarrow (\mathrm {fs}/X_n)_{\mathrm {\acute {e}t}}$ denotes the forgetful map between these two sites. This finishes the proof.

Proof. This follows from Proposition 3.16.

Proof. We denote by $X_{n,i}$ the fiber product of $i+1$ copies of $X_n$ over X. We have $X_{n,i}=H_n^i\times _{\mathop {\mathrm {Spec}}{\mathbb Z}}X_n=(H_n)_{X}^i\times _XX_n$ , where $H_n:=\mathop {\mathrm {Spec}}{\mathbb Z}\left [\left (P^{1/n}\right )^{\mathrm {gp}}/P^{\mathrm {gp}}\right ]$ and $(H_n)_X:=H_n\times _{\mathop {\mathrm {Spec}}{\mathbb Z}}X$ . Note that $(H_n)_X$ is a constant group scheme associated to the abstract group $H_n(X)$ over X if $(n,p)=1$ .

We first compute the Čech cohomology group $\check {H}_{\mathrm {kfl}}^i\left (X_n/X,\underline {H}_{\mathrm {kfl}}^1(G)\right )$ for $n=p^r$ . In this case, the underlying scheme of the group scheme $\left (H_{p^r}\right )_{X}$ is strictly Henselian local. Let $A_{p^r}$ (resp., $x_{p^r}$ ) be the underlying ring (resp., the closed point) of $X_{p^r}$ , and let $\mathfrak {m}_{A_{p^r}}$ be the maximal ideal of $A_{p^r}$ . Since $A_{p^r}$ is a finite local A-algebra, it is also Henselian local by [15, Tag 04GH], and thus $\left (A_{p^r},\mathfrak {m}_{A_{p^r}}\right )$ is a Henselian pair [15, Tag 09XE]. Let B be the underlying ring of $\left (H_{p^r}\right )_{X}^i\times _XX_{p^r}$ . Clearly B is a finite local $A_{p^r}$ -algebra. Therefore any finite B-algebra C is also a finite $A_{p^r}$ -algebra, and we have $\left (\mathfrak {m}_{A_{p^r}}B\right )C=\mathfrak {m}_{A_{p^r}}C$ . Then by an easy exercise, one can see that the equivalence [15, Tag 09XI] implies that $\left (B,\mathfrak {m}_{A_{p^r}}B\right )$ is also a Henselian pair. Since the Hom-sheaf $\mathcal {H}om_X({\mathbb Z}/m{\mathbb Z}(1),G)$ is torsion, Gabber’s theorem [15, Tag 09ZI]) implies that the vertical maps in the canonical commutative diagram

are isomorphisms. Since $\left (H_{p^r}\right )_{X}^i\times _Xx_{p^r}$ is an infinitesimal thickening of $x_{p^r}$ , the lower horizontal map in the diagram is also an isomorphism by [15, Tag 03SI]. Thus the canonical map

$$ \begin{align*}\mathrm{Hom}_{X_{p^r}}({\mathbb Z}/m{\mathbb Z}(1),G)\xrightarrow{\cong}\mathrm{Hom}_{\left(H_{p^r}\right)_{X}^i\times_XX_{p^r}}({\mathbb Z}/m{\mathbb Z}(1),G)\end{align*} $$

is an isomorphism. Combining with Theorem 3.13, we get

$$ \begin{align*} H_{\mathrm{kfl}}^1\left(X_{p^r,i},G\right)&=H_{\mathrm{kfl}}^1\left(\left(H_{p^r}\right)_{X}^i\times_XX_{p^r},G\right) \\ &=\varinjlim_m\mathrm{Hom}_{\left(H_{p^r}\right)_{X}^i\times_XX_{p^r}}({\mathbb Z}/m{\mathbb Z}(1),G)\otimes_{{\mathbb Z}}\left(P^{{1}/{p^r}}\right)^{\mathrm{gp}} \\ &=\varinjlim_m\mathrm{Hom}_{X_{p^r}}({\mathbb Z}/m{\mathbb Z}(1),G)\otimes_{{\mathbb Z}}\left(P^{{1}/{p^r}}\right)^{\mathrm{gp}} \\ &=H_{\mathrm{kfl}}^1\left(X_{p^r},G\right). \end{align*} $$

The Čech complex for $\underline {H}_{\mathrm {kfl}}^1(G)$ with respect to the cover $X_{p^r}/X$ can be identified with

$$ \begin{align*}H_{\mathrm{kfl}}^1\left(X_{p^r},G\right)\xrightarrow{0}H_{\mathrm{kfl}}^1\left(X_{p^r},G\right)\xrightarrow{\mathrm{Id}}H_{\mathrm{kfl}}^1\left(X_{p^r},G\right)\xrightarrow{0} H_{\mathrm{kfl}}^1\left(X_{p^r},G\right)\xrightarrow{\mathrm{Id}}\cdots ,\end{align*} $$

hence

(3.4) $$ \begin{align} \check{H}_{\mathrm{kfl}}^i\left(X_{p^r}/X,\underline{H}_{\mathrm{kfl}}^1(G)\right)=\begin{cases}H_{\mathrm{kfl}}^1\left(X_{p^r},G\right)&\text{if } i=0, \\ 0 &\text{if } i>0.\end{cases} \end{align} $$

In general, we write $n=p^r\cdot n'$ with $(p,n')=1$ ; then we have

$$ \begin{align*}X_{n,i}=\left(H_{n'}\right)_{X}^i\times_X\left(\left(H_{p^r}\right)_{X}^i\times_XX_n\right)\end{align*} $$

and

$$ \begin{align*}H_{\mathrm{kfl}}^1\left(X_{n,i},G\right)=\prod_{x\in H_{n'}\left(X\right)^i}H_{\mathrm{kfl}}^1(X_{n},G)=\mathrm{Map}\left(H_{n'}(X)^i,H_{\mathrm{kfl}}^1(X_{n},G)\right).\end{align*} $$

The Čech complex for $\underline {H}_{\mathrm {kfl}}^1(G)$ with respect to the cover $X_{n}/X$ can be identified with the standard complex that computes the cohomology of $H_{\mathrm {kfl}}^1(X_{n},G)$ regarded as a trivial $H_{n'}(X)$ -module. Hence we get

$$ \begin{align*}\check{H}_{\mathrm{kfl}}^i\left(X_{n}/X,\underline{H}_{\mathrm{kfl}}^1(G)\right)=H^i\left(H_{n'}(X),H_{\mathrm{kfl}}^1(X_{n},G)\right).\end{align*} $$

Finally, we get

$$ \begin{align*} \varinjlim_{n}\check{H}_{\mathrm{kfl}}^i\left(X_n/X,\underline{H}_{\mathrm{kfl}}^1(G)\right)&=\varinjlim_{n=p^r\cdot n'} H^i\left(H_{n'}(X),H_{\mathrm{kfl}}^1(X_{n},G)\right) \\ &=H^i\left(\varprojlim_{n'}H_{n'}(X),\varinjlim_{n=p^r\cdot n'}H_{\mathrm{kfl}}^1(X_{n},G)\right), \end{align*} $$

where the second identification follows from [Reference Serre13, §2, Prop. 8]. By Proposition 3.15 we have $\varinjlim _{n}H_{\mathrm {kfl}}^1(X_{n},G)=0$ , and thus $\varinjlim _{n}\check {H}_{\mathrm {kfl}}^i\left (X_n/X,\underline {H}_{\mathrm {kfl}}^1(G)\right )=0$ .

Proof. The result follows from Lemma 3.18 and the exact sequence (3.3).

Proof. (1) Since $X_n$ is a Kummer étale cover of X whenever $(p,n)=1$ , this is clear.

(2) By Lemma 3.12, we may assume that X is a log point $\mathop {\mathrm {Spec}} k$ with k separably closed. By [Reference Nizioł12, page 521-523, in particular Lem. 3.16], we have

$$ \begin{align*}\check{H}_{\mathrm{kfl}}^i\left(X_{p^r}/X,G\right)\cong H^i_{X_{\mathrm{fl}}}\left(H_{p^r},G\right)\end{align*} $$

for $i\geq 1$ , where $H^i_{X_{\mathrm {fl}}}\left (H_{p^r},G\right )$ denotes the ith cohomology group of the group scheme $H_{p^r}=\mathop {\mathrm {Spec}}{\mathbb Z}\left [\left (P^{1/p^r}\right )^{\mathrm {gp}}/P^{\mathrm {gp}}\right ]$ acting trivially on G over the flat site $X_{\mathrm {fl}}$ . The group $H^2_{X_{\mathrm {fl}}}\left (H_{p^r},G\right )$ can be identified with the group of extension classes of $H_{p^r}$ by G which admit a (not necessarily homomorphic) section [Reference Demazure and Grothendieck14, Exposé XVII, Prop. A.3.1]. By [Reference Demazure and Grothendieck14, Exposé XVII, Prop. 7.1.1], such extensions must be of multiplicative type, and therefore must be commutative. Since the base field k is separably closed, such extensions must be trivial. It follows that $\check {H}_{\mathrm {kfl}}^2\left (X_{p^r}/X,G\right )\cong H^2_{X_{\mathrm {fl}}}\left (H_{p^r},G\right )=0$ .

(3) First we show that any class $\gamma \in H_{\mathrm {kfl}}^2(X,G)$ vanishes in $H_{\mathrm {kfl}}^2(X_m,G)$ for some positive integer m with $(m,p)=1$ . By Corollary 3.17, $\gamma $ is annihilated by some cover $X_{m\cdot p^r}$ with $(m,p)=1$ . Let $\gamma '$ be the image of $\gamma $ in $H_{\mathrm {kfl}}^2(X_m,G)$ ; we want to show that it is zero. The Čech-to-derived functor spectral sequence for the cover $X_{m\cdot p^r}/X_{m}$ gives rise to an exact sequence

$$ \begin{align*} \cdots&\rightarrow \check{H}_{\mathrm{kfl}}^2\left(X_{m\cdot p^r}/X_m,G\right)\rightarrow \mathrm{ker}\left(H_{\mathrm{kfl}}^2(X_m,G)\rightarrow H_{\mathrm{kfl}}^2\left(X_{m\cdot p^r},G\right)\right) \\ &\rightarrow \check{H}_{\mathrm{kfl}}^1\left(X_{m\cdot p^r}/X_m,\underline{H}_{\mathrm{kfl}}^1(G)\right)\rightarrow\cdots. \end{align*} $$

We have $\gamma '\in \mathrm {ker}\left (H_{\mathrm {kfl}}^2(X_m,G)\rightarrow H_{\mathrm {kfl}}^2\left (X_{m\cdot p^r},G\right )\right )$ . By part (2), we have $\check {H}_{\mathrm {kfl}}^2(X_{m\cdot p^r}/$ $X_m,G)=0$ . By equation (3.4) we have $\check {H}_{\mathrm {kfl}}^1\left (X_{m\cdot p^r}/X_m,\underline {H}_{\mathrm {kfl}}^1(G)\right )=0$ . Hence we get $\gamma '=0$ . It follows that

(3.5) $$ \begin{align} \varinjlim_{\left(m,p\right)=1}\mathrm{ker}\left(H_{\mathrm{kfl}}^2(X,G)\rightarrow H_{\mathrm{kfl}}^2(X_m,G)\right)\xrightarrow{\cong}H_{\mathrm{kfl}}^2(X,G). \end{align} $$

Now consider the exact sequences

$$ \begin{align*} &\rightarrow \check{H}_{\mathrm{kfl}}^0\left(X_m/X,\underline{H}_{\mathrm{kfl}}^1(G)\right)\rightarrow \check{H}_{\mathrm{kfl}}^2(X_{m}/X,G)\rightarrow \mathrm{ker}\left(H_{\mathrm{kfl}}^2(X,G)\rightarrow H_{\mathrm{kfl}}^2(X_m,G)\right) \\ &\rightarrow \check{H}_{\mathrm{kfl}}^1\left(X_m/X,\underline{H}_{\mathrm{kfl}}^1(G)\right)\rightarrow \end{align*} $$

arising from the spectral sequence $\check {H}_{\mathrm {kfl}}^i\left (X_m/X,\underline {H}_{\mathrm {kfl}}^j(G)\right )\Rightarrow H_{\mathrm {kfl}}^{i+j}(X_m,G)$ for the Kummer flat covers $X_m/X$ with $(m,p)=1$ . Taking the direct limit, we get an exact sequence

(3.6) $$ \begin{align} \begin{array}{ll} &\rightarrow\displaystyle \varinjlim_{\left(m,p\right)=1}\check{H}_{\mathrm{kfl}}^0\left(X_m/X,\underline{H}_{\mathrm{kfl}}^1(G)\right)\xrightarrow{\alpha} \varinjlim_{\left(m,p\right)=1}\check{H}_{\mathrm{kfl}}^2(X_{m}/X,G)\rightarrow H_{\mathrm{kfl}}^2(X,G) \\ &\rightarrow \displaystyle\varinjlim_{\left(m,p\right)=1}\check{H}_{\mathrm{kfl}}^1\left(X_m/X,\underline{H}_{\mathrm{kfl}}^1(G)\right) \end{array} \end{align} $$

by the identification (3.5). Similar to the general case of the proof of Lemma 3.18, we can show that $\varinjlim _{\left (m,p\right )=1}\check {H}_{\mathrm {kfl}}^i\left (X_m/X,\underline {H}_{\mathrm {kfl}}^1(G)\right )=0$ for any $i\geq 0$ . Then the exact sequence (3.6) tells us that

$$ \begin{align*}H_{\mathrm{kfl}}^2(X,G)=\varinjlim_{\left(m,p\right)=1}\check{H}_{\mathrm{kfl}}^2(X_m/X,G)=\varinjlim_{\left(m,p\right)=1}\check{H}_{\mathrm{k\acute{e}t}}^2(X_m/X,G)=H_{\mathrm{k\acute{e}t}}^2(X,G).\end{align*} $$

The group $H_{\mathrm {k\acute {e}t}}^2(X,G)$ is torsion and p-torsion-free by Corollary 3.6, and n-divisible for $(n,p)=1$ by Theorem 3.8(3). Therefore $H_{\mathrm {kfl}}^2(X,G)$ is torsion, p-torsion-free, and divisible.

Proof. We have a short exact sequence

$$ \begin{align*}0\rightarrow H_{\mathrm{kfl}}^1(X,G)\otimes_{{\mathbb Z}}{\mathbb Z}/n{\mathbb Z}\rightarrow H_{\mathrm{kfl}}^2(X,G[n])\rightarrow H_{\mathrm{kfl}}^2(X,G)[n]\rightarrow 0.\end{align*} $$

Since G is a torus, the group $H_{\mathrm {kfl}}^1(X,G)=\varinjlim _{n}\mathrm {Hom}_{X_{\mathrm {kfl}}}({\mathbb Z}/n{\mathbb Z}(1),G)\otimes _{{\mathbb Z}}P^{\mathrm {gp}}$ is divisible, whence $H_{\mathrm {kfl}}^1(X,G)\otimes _{{\mathbb Z}}{\mathbb Z}/n{\mathbb Z}=0$ . Therefore for $n=m\cdot p^r$ with $(m,p)=1$ , we have

$$ \begin{align*} H_{\mathrm{kfl}}^2(X,G[n])&=H_{\mathrm{kfl}}^2(X,G)[n]=H_{\mathrm{k\acute{e}t}}^2(X,G)[n]=H_{\mathrm{k\acute{e}t}}^2(X,G)[m] \\ &=H_{\mathrm{k\acute{e}t}}^2(X,G[m])=G[m](-2)(X)\otimes_{{\mathbb Z}}\bigwedge^2P^{\mathrm{gp}}. \end{align*} $$

Proof. This follows from Corollary 3.21.

Proof. By Lemma 3.20, $R^2\varepsilon _{\mathrm {fl}*}G$ is torsion. Hence part (1) follows.

We prove part (2). Since G is a torus, we have a short exact sequence $0\to G[n]\to G\xrightarrow {n}G\to 0$ for any $n\geq 1$ . This short exact sequence induces a short exact sequence

$$ \begin{align*}0\rightarrow R^1\varepsilon_{\mathrm{fl}*}G\otimes_{{\mathbb Z}}{\mathbb Z}/n{\mathbb Z}\rightarrow R^2\varepsilon_{\mathrm{fl}*}G[n]\rightarrow \left(R^2\varepsilon_{\mathrm{fl}*}G\right)[n]\rightarrow 0.\end{align*} $$

Since the sheaf $R^1\varepsilon _{\mathrm {fl}*}G$ is divisible, we get $\left (R^2\varepsilon _{\mathrm {fl}*}G\right )[n]=R^2\varepsilon _{\mathrm {fl}*}G[n]$ .

Part (3) follows from part (2) and Corollary 3.21.

We are left with part (4). By [Reference Swan16, §3], in particular the part between [Reference Swan16, Cor. 3.7] and [Reference Swan16, Thm. 3.8], we have a cup product for the higher direct image functors for the map of sites $\varepsilon _{\mathrm {fl}}:(\mathrm {fs}/X)_{\mathrm {kfl}}\rightarrow (\mathrm {fs}/X)_{\mathrm {fl}}$ . The cup product induces homomorphisms

$$ \begin{align*}G[n]\otimes_{{\mathbb Z}/n{\mathbb Z}}\bigwedge^2R^1\varepsilon_{\mathrm{fl}*}{\mathbb Z}/n{\mathbb Z} \rightarrow G[n]\otimes_{{\mathbb Z}/n{\mathbb Z}}R^2\varepsilon_{\mathrm{fl}*}{\mathbb Z}/n{\mathbb Z}\rightarrow R^2\varepsilon_{\mathrm{fl}*}G[n].\end{align*} $$

Since $G[n]\otimes _{{\mathbb Z}/n{\mathbb Z}}\bigwedge ^2R^1\varepsilon _{\mathrm {fl}*}{\mathbb Z}/n{\mathbb Z}=G[n](-2)\otimes _{{\mathbb Z}}\bigwedge ^2\left (\mathbb {G}_{\mathrm {m,log}}/\mathbb {G}_{\mathrm {m}}\right )_{X_{\mathrm {fl}}}$ , we get a canonical homomorphism

$$ \begin{align*}G[n](-2)\otimes_{{\mathbb Z}}\bigwedge^2\left(\mathbb{G}_{\mathrm{m,log}}/\mathbb{G}_{\mathrm{m}}\right)_{X_{\mathrm{fl}}}\rightarrow R^2\varepsilon_{\mathrm{fl}*}G[n].\end{align*} $$

By Corollary 3.21, this homomorphism is an isomorphism. This finishes the proof of part (4).

Proof. In part (2), since G is smooth, U has to be smooth by fppf descent. Hence part (2) follows from part (1) and we are left to prove part (1).

By Theorem 3.14, we have

$$ \begin{align*}R^1\varepsilon_{\mathrm{fl},*}G=\varinjlim_{n}\mathcal{H}om_{X}({\mathbb Z}/n{\mathbb Z}(1),G)\otimes_{{\mathbb Z}}\left(\mathbb{G}_{\mathrm{m,log}}/\mathbb{G}_{\mathrm{m}}\right)_{X_{\mathrm{fl}}},\end{align*} $$

which is zero by [Reference Demazure and Grothendieck14, Exposé XVII, Lem. 2.5]. To prove $R^2\varepsilon _{\mathrm {fl},*}G=0$ , it suffices to prove $H^2_{\mathrm {kfl}}(X,G)=0$ for the case where the underlying scheme of X is $\mathop {\mathrm {Spec}} A$ with A a strictly Henselian ring and X admitting a chart $P\rightarrow M_X$ with $P\xrightarrow {\cong }M_{X,x}/\mathcal {O}_{X,x}^{\times }$ . By Theorem 3.19, we have $\varinjlim _n\check {H}_{\mathrm {kfl}}^2(X_n/X,G)\xrightarrow {\cong } H_{\mathrm {kfl}}^2(X,G)$ . By Lemma 3.12, we are further reduced to the case where X is a log point with A a separably closed field. We have $\check {H}_{\mathrm {kfl}}^2(X_n/X,G)=H^2_{X_{\mathrm {fl}}}(H_n,G)$ by [Reference Nizioł12, page 521-523, in particular Lem. 3.16]. But $H^2_{X_{\mathrm {fl}}}(H_n,G)=0$ for any positive integer n by [Reference Demazure and Grothendieck14, Exposé XVII, Thm. 5.1.1 (1) (c)] and [Reference Demazure and Grothendieck14, Exposé XVII, Appendice I, Prop. 3.1]. This finishes the proof.

Proof. We have

$$ \begin{align*}H^s_{\mathrm{fl}}(X,\mathbb{G}_{\mathrm{m}})=H^s_{\mathrm{\acute{e}t}}(X,\mathbb{G}_{\mathrm{m}})=0\end{align*} $$

for $s>0$ by [Reference Milne11, Chap. II, Prop. 1.5 (a)]. Since k is a finite field, it has absolute Galois group $\hat {{\mathbb Z}}$ . Therefore we can identify the group $H^1_{\mathrm {\acute {e}t}}(x,{\mathbb Q}/{\mathbb Z})$ with the Galois cohomology

$$ \begin{align*}H^1\left(\hat{{\mathbb Z}},{\mathbb Q}/{\mathbb Z}\right)=\mathrm{Hom}\left(\hat{{\mathbb Z}},{\mathbb Q}/{\mathbb Z}\right)\cong{\mathbb Q}/{\mathbb Z}.\end{align*} $$

Then the results follow from the spectral sequence (5.1)) with the help of equations (5.2) and (5.3).

Proof. We are left with checking $H^2_{\mathrm {fl}}(X,\mathbb {G}_{\mathrm {m}})=0$ for X a smooth projective curve over an algebraically closed field, and $H^1_{\mathrm {fl}}(x,{\mathbb Q}/{\mathbb Z})\cong {\mathbb Q}/{\mathbb Z}$ for x a point with finite residue field. The first follows from [15, Tag 03RM], and the second follows from

$$ \begin{align*}H^1_{\mathrm{fl}}(x,{\mathbb Q}/{\mathbb Z})\cong H^1_{\mathrm{\acute{e}t}}(x,{\mathbb Q}/{\mathbb Z})\cong\mathrm{Hom}\left(\hat{{\mathbb Z}},{\mathbb Q}/{\mathbb Z}\right)\cong{\mathbb Q}/{\mathbb Z}. \\[-38pt] \end{align*} $$