Comparison of Kummer Logarithmic Topologies with Classical Topologies II

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Theorem 0.2 (See also Theorem 2.6). Assume that G is smooth and affine over the underlying scheme of X, and has geometrically connected fibers over the underlying scheme of X. (1) We have R2ε G = lim(R2ε G)[n]= (R2ε G)[l∞], fl∗ −→ fl∗ M fl∗ n l prime 2 2 2 ∞ where (R εfl∗G)[n] denotes the n-torsion subsheaf of R εfl∗G and (R εfl∗G)[l ] 2 denotes the l-primary part of R εfl∗G for a prime number l. 2 ∞ (2) The l-primary part (R εfl∗G)[l ] is supported on the locus where l is invertible. (3) If n is invertible on X, then

2 2 2 (R ε G)[n]= R ε G[n]= G[n](−2) ⊗Z (G /G ) . fl∗ fl∗ ^ m,log m Xfl Theorem 0.3 (See also Theorem 2.7). Assume that G is finite and flat over the underlying scheme of X. Then we have 2 2 ∞ 2 ∞ (1) R εfl∗G = l prime(R εfl∗G)[l ], where (R εfl∗G)[l ] denotes the l-primary 2 L part of R εfl∗G. 2 ∞ (2) The l-primary part (R εfl∗G)[l ] is supported on the locus where l is invertible. (3) Assume X is connected, then the order of G is constant on X. We denote the order by n and assume that n is invertible on X. Then 2 2 R ε G = G(−2) ⊗Z (G /G ) , fl∗ ^ m,log m Xfl where G(−2) denotes the tensor product of G with Z/nZ(−2). Theorem 0.4 (See also Theorem 2.9). Assume that G is an extension of an abelian scheme A by a torus T over X. Then we have 2 2 ∞ 2 ∞ (1) R εfl∗G = l prime(R εfl∗G)[l ], where (R εfl∗G)[l ] denotes the l-primary 2 L part of R εfl∗G. 2 2 (2) We have (R εfl∗G)[n]= R εfl∗G[n]. 2 ∞ (3) The l-primary part (R εfl∗G)[l ] is supported on the locus where l is invertible. (4) If n is invertible on X, then

2 2 2 (R ε G)[n]= R ε G[n]= G[n](−2) ⊗Z (G /G ) , fl∗ fl∗ ^ m,log m Xfl where G[n](−2) denotes the tensor product of G[n] with Z/nZ(−2). As applications of Theorem 0.2, Theorem 0.3, and Theorem 0.4, we have more 2 explicit description of R εfl∗G in certain special cases of the base. The first special case concerns the ranks of the log structure at geometric gp × points of the base. If Y is an fs log scheme over X such that the rank (MY /OY )y is at most one for any geometric point y of Y , then the restriction of the sheaf 2 (Gm,log/Gm)Xfl to (st/Y ) is zero, where (st/Y ) denotes the full subcategory of (fsV /X) consisting of strict fs log schemes over Y . Thus Theorem 0.2 (especially COMPARISON OF KUMMER TOPOLOGIES WITH CLASSICAL TOPOLOGIES II 3 part (3)), Theorem 0.3 (especially part (3)), and Theorem 0.4 (especially part (4)) imply the following theorem. Theorem 0.5 (See also Theorem 2.10). Assume that G satisfies any of the following three conditions: (i) G is smooth and affine over the underlying scheme of X, and has geometrically connected fibers over the underlying scheme of X; (ii) G is finite flat of order n over the underlying scheme of X, and for any prime n number l the kernel of G −→l G is also finite flat over the underlying scheme of X; (iii) G is an extension of an abelian scheme by a torus over the underlying scheme of X. gp × Let Y ∈ (fs/X) be such that the ranks of the stalks of the étale sheaf MY /OY are at most one, and let (st/Y ) be the full subcategory of (fs/X) consisting of strict fs 2 log schemes over Y . Then the restriction of R εfl∗G to (st/Y ) is zero. The rest two special cases involve the characteristic of the base. Assume that the underlying scheme of X is over Q, i.e. any positive integer n 2 is invertible on X. Then we have an explicit description of (R εfl∗G)[n] for any n in the case (i) and the case (iii) from Theorem 0.5 by Theorem 0.2 (3) and Theorem 2 0.4 (4) respectively, as well as an explicit description of R εfl∗G in the case (ii) from Theorem 0.5 by Theorem 0.3 (3). Thus we get the following theorem. Theorem 0.6 (See also Theorem 2.11). Assume that the underlying scheme of X is a Q-scheme, and let G be as in any of the three cases of Theorem 0.5. Then we have 2 2 R ε G = lim G[n](−2) ⊗Z (G /G ) fl∗ −→ m,log m Xfl n ^ in cases (i) and (iii), and 2 2 R ε G = G(−2) ⊗Z (G /G ) fl∗ ^ m,log m Xfl in case (ii).

Assume that the underlying scheme of X is an Fp-scheme for a prime number p. Then any prime l 6= p is invertible on X, hence we have an explicit description of 2 ∞ (R εfl∗G)[l ] in the case (i) and the case (iii) from Theorem 0.5 by Theorem 0.2 (3) 2 ∞ and Theorem 0.4 (4) respectively. And we also have (R εfl∗G)[p ] = 0 in these two cases by Theorem 0.2 (2) and Theorem 0.4 (3) respectively. In a suitable modified version of the case (ii) from Theorem 0.5, we have similar results. Therefore we get the following theorem. Theorem 0.7 (See also Theorem 2.12). Let p be a prime number. Assume that the underlying scheme of X is an Fp-scheme, and G satisfies one of the following three conditions: (i) G is smooth and affine over the underlying scheme of X, and has geometrically connected fibers over the underlying scheme of X; 4 HEER ZHAO

(ii) G is finite flat of order n over the underlying scheme of X, and the kernel pn G[pn] of G −→ G is also finite flat over the underlying scheme of X; (iii) G is an extension of an abelian scheme by a torus over the underlying scheme of X. Then we have 2 2 R ε G = lim G[r](−2) ⊗Z (G /G ) fl∗ −→ ^ m,log m Xfl (r,p)=1 in cases (i) and (iii), and 2 2 n R ε G = (G/G[p ])(−2) ⊗Z (G /G ) fl∗ ^ m,log m Xfl in case (ii). At last we make some computations in Section 3. In particular, over standard Henselian log traits with finite residue field, we compute the first and the second Kummer log flat cohomology with coefficients in group schemes which satisfy any of the conditions of Theorem 0.5.

1. The higher direct images are torsion Let X be an fs log scheme whose underlying scheme is locally noetherian. Let G be a smooth commutative group scheme over the underlying scheme of X. In i this section, we investigate the higher direct images R εﬂ∗G of G. The main result is the following theorem. Theorem 1.1. Let X be an fs log scheme whose underlying scheme is locally noetherian. Let G be a smooth commutative group scheme over the underlying i scheme of X. Then the higher direct image R εﬂ∗G is a torsion sheaf for any i> 0.

i i Apparently the torsionness of R εfl∗G is reduced to the torsionness of Hkfl(X, G) in the case that the underlying scheme of X is Spec R with R a strictly henselian i local ring. We are going to investigate the torsionness of Hkfl(X, G) in this case using Cechˇ cohomology. Now let the underlying scheme of X be Spec R with R a strictly henselian local ring. Let x denote the closed point, let k be the residue field of R, and let P → MX be a chart of the log structure of X with P an fs monoid, such that =∼ × P −→ MX,x/OX,x. Let P 1/n denote the monoid P regarded as a monoid above P via the ho- n 1/n momorphism P −→ P . Let Xn := X ×Spec Z[P ] Spec Z[P ] endowed with the 1/n canonical log structure associated to P and let Hn denote the group scheme 1/n gp gp Spec Z[(P ) /P ] over Spec Z, then Xn is a Kummer log flat cover of X such that Xn ×X Xn =∼ Xn ×Spec Z Hn. We regard x as a log point with respect to the induced log structure. Let xn := Xn ×X x. Then xn is a Kummer log flat cover of x such that xn ×x xn =∼ xn ×Spec Z Hn. COMPARISON OF KUMMER TOPOLOGIES WITH CLASSICAL TOPOLOGIES II 5

The following lemma, which is taken from [Zha21], identifies the Cechˇ coho- ˇ i ˇ i mology group Hkfl(Xn/X,G) with Hkfl(xn/x, G) for any i> 0. Lemma 1.2. Let X be an fs log scheme such that its underlying scheme is Spec R with R a strictly henselian noetherian local ring. Let k be the residue field of R, x the closed point of X, and PX → MX a chart of the log structure MX of X with P an fs monoid (here PX denotes the constant sheaf associated to P ), ∼ = × 1/n such that P −→ MX,x/OX,x. Let P , Xn, and xn be as defined above. Let G be a smooth commutative group scheme over the underlying scheme of X, and we endow G with the induced log structure from X. Then the canonical map ˇ i ˇ i Hkfl(Xn/X,G) → Hkfl(xn/x, G) is an isomorphism for any i> 0. Proof. See [Zha21, Lem. 3.12].

Before going to the next lemma, we recall the standard complex for computing the Hochschild cohomology from [GP11, Expos´eI, §5.1]. Let H be a group scheme over a base scheme S, and let F be a functor of abelian groups on the ﬂat site of S endowed with an H-action. The standard complex

0 1 2 • 0 d 1 d 2 d (1.1) CS (H, F ): CS(H, F ) −→ CS (H, F ) −→ CS (H, F ) −→· · · • for computing the Hochschild cohomology HSfl (H, F ) is given by: i i CS (H, F ) := MorS(H , F ), and the differential i i i+1 d : CS(H, F ) → CS (H, F ) i+1 is given by the alternating sum ∂0 − ∂1 + ··· + (−1) ∂i+1, where i i+1 ∂j : CS (H, F ) → CS (H, F ) i sends c ∈ CS(H, F ) to ∂j (c) defined by h · c(h , ··· ,h ), if j = 0;  1 2 i+1 ∂j (c)(h1, ··· ,hi+1)= c(h1, ··· ,hj · hj+1, ··· ,hi+1), if 1 ≤ j ≤ i; c(h1, ··· ,hi), if j = i + 1.  The following lemma, which is implicitly proved in [Kat19, §4.11, §4.12], can ˇ ˇ i be used to describe the Cech cohomology group Hkfl(xn/x, G) in terms of the i Hochschild cohomology group Hxfl (Hn, G) for i ≥ 1, where the action of Hn on G is the trivial one.

1/n Lemma 1.3. Let X, R, P , P , Xn, and G be as in Lemma 1.2. We further assume that R is artinian. Let F be the sheaf of abelian groups on (fs/X)kﬂ deﬁned by T 7→ Γ(T ×X Xn, G). Then we have the following. 1/n gp gp (1) The group scheme Hn = Spec Z[(P ) /P ] acts on F . 6 HEER ZHAO

ˇ ˇ i (2) For any i ≥ 0, the Cech cohomology group Hkfl(Xn/X,G) is canonically identified with the Hochschild cohomology Hi (H , F ) of H with coeffi- Xfl n n cients in F on the flat site Xfl = (fs/X)fl (see [GP11, ExposéI, §5.1] for the definition of Hochschild cohomology). (3) We endow G with the trivial Hn-action. Then the canonical homomorphism G → F induced by the projection T ×X Xn → T is Hn-equivariant. (4) The homomorphism G → F induces an isomorphism ∼ i = i HXfl (Hn, G) −→ HXfl (Hn, F ) for each i ≥ 1. Proof. See [Kat19, §4.11, §4.12].

Corollary 1.4. Let X, Xn, x, and G be as in Lemma 1.2. Then the Cechˇ ˇ i cohomology group Hkfl(Xn/X,G) is canonically identified with the Hochschild co- i homology group Hxfl (Hn, G) for any i ≥ 1, where G is regarded as an Hn-module with respect to the trivial action. Proof. By Lemma 1.2, we have a canonical isomorphism ∼ ˇ i = ˇ i Hkfl(Xn/X,G) −→ Hkfl(xn/x, G). Since the log point x clearly satisfies the extra condition of Lemma 1.3, we get a canonical isomorphism ∼ ˇ i = i Hkfl(xn/x, G) −→ Hxfl (Hn, G). Then the result follows.

i Now we turn to the study of Hxfl (Hn, G). Lemma 1.5. Let k, G, and x be as in Lemma 1.2. Let p be the characteristic of the field k, and let n = m·pr with (m,p)=1. By the definition of Hochschild cohomology, see [GP11, ExposéI, §5.1], the inclusion Hpr → Hn induces a canonical map i i r Res : Hxfl (Hn, G) → Hxfl (Hp , G), i which we call the restriction map. Then the multiplication by m map on Hxfl (Hn, G) factors through the restriction map Res. Proof. Note that the analogous result for the abstract group cohomology is well-known. We are going to show that the standard proof in the setting of the abstract group cohomology also works in this special case of Hochschild cohomology here. Firstly we construct a corestriction map i i r Cor : Hxfl (Hp , G) → Hxfl (Hn, G) for each i ≥ 0. Since the field k is separable closed and (p,m) = 1, we have Hn =∼ Hm × Hpr with Hm a constant group scheme. Let ι denote the inclusion ∗ r i r Hp ֒→ Hn. We claim that {Hxfl (Hp , ι (−))}i is a universal δ-functor on the category of Hn-modules. It suffices to show that for any Hn-module M, we have COMPARISON OF KUMMER TOPOLOGIES WITH CLASSICAL TOPOLOGIES II 7

∗ i r Hxfl (Hp , ι (EHn (M))) = 0, where EHn (M) is the Hn-module associated to M defined in [GP11, ExposéI, Def. 5.2.0] which contains M as an Hn-submodule. But this follows from [GP11, ExposéI, Lem. 5.2.2] by the reexpression ∗ ι (EHn (M)) = EHpr (Mork(Hm,M)).

Now for any Hn-module M, we deﬁne the corestriction map in degree 0 to be the norm map Norm : M Hpr → M Hn ,a 7→ g · a. X g∈Hm(k) ∗ i r It extends uniquely to a morphism Cor of the universal δ-functor {Hxfl (Hp , ι (−))}i i into the universal δ-functor {Hxfl (Hn, −)}i. Since the composition Norm M Hn ֒→ M Hpr −−−→ M Hn is the multiplication by m map, so is the composition

i Res i Cor i r Hxfl (Hn,M) −−→ Hxfl (Hp ,M) −−→ Hxfl (Hn,M). Taking M = G ﬁnishes the proof.

i r Next we prove that Hxfl (Hp , G) is torsion for i > 0, which would imply that i Hxfl (Hn, G) is torsion for i> 0 by Lemma 1.5. Lemma 1.6. Let A be a local artinian k-algebra with residue field k such that its m mt maximal ideal A satisfies A =0 for some positive integer t. Let π : G(A) → G(k) (resp. τ : G(k) → G(A)) be the map induced by the projection A → k (resp. the inclusion k → A). Then the multiplication by pt map on G(A) agrees with τ ◦[pt]◦π. Proof. Consider the following commutative diagram G(A) π / G(k) . f τ pt pt G(A) π / G(k) f τ For any a ∈ G(A), we have a − τ(π(a)) ∈ Ker(π). Hence pt · a − τ(pt · π(a)) = pt · (a − τ(π(a))) = 0 by [Kat81, Lem. 1.1.1]. Lemma 1.7. Let k, P , G, and x be as in Lemma 1.2. Let p be the characteristic gp a i ∼ Z r of the field k. Assume that P = . Then the group Hxfl (Hp , G) is killed by r piap for any i> 0. Proof. Now we have no analogous corestriction map available for the inclusion H1 ֒→ Hpr . Instead we are going to show that the map pr • • p : Cx(Hpr , G) → Cx(Hpr , G) 8 HEER ZHAO

• • factors through the canonical restriction map Cx(Hpr , G) → Cx(H1, G), where • • Cx(Hpr , G) (resp. Cx(H1, G)) is the standard complex computing the group cohomology of the trivial Hn-module (resp. H1-module) G over x. • The complex Cx(H1, G) for the trivial group H1 is the complex G(k) −→0 G(k) −→id G(k) −→0 G(k) −→id G(k) −→·0 · · , i so we have Hxfl (H1, G)=0 for i> 0. Consider the following diagram 0 / 1 / 2 / (1.2) Cx(Hpr , G) / Cx(Hpr , G) / Cx(Hpr , G) / ··· ,

Res Res Res G(k) 0 / G(k) G(k) 0 / ···

r r r piap piap piap G(k) 0 / G(k) G(k) 0 / ···

τ τ τ 0 / 1 / 2 / Cx(Hpr , G) / Cx(Hpr , G) / Cx(Hpr , G) / ··· where the upper vertical maps are induced by the inclusion H1 ֒→ Hpr and the maps i τ are as in Lemma 1.6. The maximal ideal of the local artinian ring underlying Hpr is killed by raising to iapr-th power. By Lemma 1.6, the composition of the maps r on the column at degree i is the multiplication by piap map. The upper squares and the middle squares are clearly commutative, we claim that the lower squares are also commutative. We choose the following square

n 1G(k)−1G(k)+···+(−1) 1G(k) G(k) / G(k)

τ τ

n ∂ −∂ +···+(−1) ∂ n−1 0 1 n / n Mork(Hpr , G) / Mork(Hpr , G) to check. It suﬃces to check the commutativity of the square

1G(k) G(k) / G(k)

τ τ

∂ n−1 j / n Mork(Hpr , G) / Mork(Hpr , G) for each j. For a : Spec k → G and 0

Corollary 1.8. Let X, Xn, x, and G be as in Lemma 1.2. Then the Cechˇ ˇ i cohomology group Hkﬂ(Xn/X,G) is torsion for any i ≥ 1. Proof. By Corollary 1.4, it suﬃces to show that the Hochschild cohomology i group Hxfl (Hn, G) is torsion for any i ≥ 1. The latter follows from Lemma 1.5 and Lemma 1.7.

Lemma 1.9. Let X and Xn be as in Lemma 1.2. Let F be a sheaf on (fs/X)kfl. Then the family XN := {Xn → X}n≥1 of Kummer log flat covers of X satisfies the condition (L3) from [Art62, §2], whence a Cech-to-derivedˇ functor spectral sequence ˇ i X j i+j (1.3) Hkfl( N,Hkfl(F )) ⇒ Hkfl (X, F ), where i i Hˇ (XN, −) := lim Hˇ (X /X, −). kfl −→ kfl n n≥1 Proof. This follows from [Art62, Chap. II, Sec. 3, (3.3)].

Now we are ready to prove Theorem 1.1.

Proof of Theorem 1.1. The case i = 1 for a smooth affine commutative 1 group scheme follows from Kato’s computation of R εfl∗G, and the case i =1 fora general smooth commutative group scheme follows from [Zha21, Thm. 3.14]. We i proceed with induction on i. Assume that R εfl∗G is torsion for 0 0 and any d ≥ 0, and the Leray spectral sequence u,v u v v+v E2 = Hfl (Xn ×X ···×X Xn, R εfl∗G) ⇒ Hkfl (Xn ×X ···×X Xn, G) d + 1 folded d + 1 folded | {z } | {z } N+1−i together imply that Hkfl (Xn ×X ···×X Xn, G) is torsion for 0

2. The second higher direct image We have shown that the higher direct images of a smooth commutative group scheme is torsion. In this section, we present further descriptions of the second higher direct image. The following theorem is taken from [Zha21]. Theorem 2.1. Let X be an fs log scheme such that its underlying scheme is Spec R with R a noetherian strictly Henselian local ring. Let PX → MX be a chart of the log structure MX of X with P an fs monoid (here PX denotes the constant =∼ × sheaf associated to P ), such that P −→ MX,x/OX,x. Let G be a smooth commutative group scheme over the underlying scheme of X, and we endow G with the inverse image log structure. Let Xn be as constructed in the beginning of Section 1, x the closed point of X, and xn := Xn ×X x. Then the canonical map ∼ lim Hˇ 2 (X /X,G) −→= H2 (X, G) −→ kﬂ n kﬂ n given by the spectral sequence (1.3) is an isomorphism. Proof. See [Zha21, Thm. 3.19]. COMPARISON OF KUMMER TOPOLOGIES WITH CLASSICAL TOPOLOGIES II 11

Corollary 2.2. Let the notation and the assumptions be as in Theorem 2.1. Then the canonical map ∼ 2 = 2 Hkfl(X, G) −→ Hkfl(x, G) is an isomorphism. Proof. In the canonical commutative diagram ∼ lim Hˇ 2 (X /X,G) = / lim Hˇ 2 (x /x, G) −→n kfl n −→n kfl n

=∼ ∼= 2 / 2 Hkfl(X, G) Hkfl(x, G) the vertical maps are isomorphisms by Theorem 2.1 and the upper horizontal map is an isomorphism by Lemma 1.2. Then the result follows. 2.1. The case of smooth affine commutative group schemes with connected fibers. The following lemma is a generalization of [Zha21, Lem. 3.20] from tori to smooth affine commutative group schemes. Lemma 2.3. Let the notation and the assumptions be as in Theorem 2.1. Let x be the closed point of X, and p the characteristic of the residue field k of R. We further assume that the closed fiber G ×X x of G is affine and connected. Then we have ˇ 2 ˇ 2 (1) Hkfl(Xn/X,G)= Hkét(Xn/X,G) for (n,p)=1; ˇ 2 (2) Hkfl(Xpr /X,G)=0 for r> 0; (3) H2 (X, G) = lim Hˇ 2 (X /X,G) = lim Hˇ 2 (X /X,G)= H2 (X, G), kfl −→ kfl n −→ két n két (n,p)=1 (n,p)=1 2 in particular Hkfl(X, G) is torsion, p-torsion-free and divisible.

Proof. If (n,p) = 1, Xn → X is a Kummer log ´etale cover. Then part (1) is clear by the deﬁnition of Cechˇ cohomology.

ˇ 2 r 2 r r We have Hkﬂ(Xp /X,G)= Hxfl (Hp , G) for r> 0 by Corollary 1.4, where Hp 2 r is as in Corollary 1.4. We are reduced to show that Hxfl (Hp , G)=0. Let

Extalg(Hpr ×X x, G ×X x) denote the group of isomorphism extension classes of Hpr ×X x by G ×X x, and let

Exts(Hpr ×X x, G ×X x) denote the subgroup consisting of the isomorphic extension classes which admit a (not necessarily homomorphic) section, see [DG70, ExposéXVII, §A.2, §A.3]. The 2 r r group Hxfl (Hp , G) can be identified with the group Exts(Hp ×X x, G ×X x) by [DG70, ExposéXVII, Prop. A.3.1]. Hence it suffices to show that

Extalg(Hpr ×X x, G ×X x)=0.

By [BLR90, §9.2, Thm. 2], G ×X x is an extension of a unipotent group U by a torus T over k. Since a torus is smooth and G×X x is smooth over k, the unipotent 12 HEER ZHAO quotient U of G×X x is also smooth over k by descent. By the exact sequence from [DG70, §A.2, Prop. A.2.1 a)], it suﬃces to show that

Extalg(Hpr ×X x, T ) = Extalg(Hpr ×X x, U)=0.

By [DG70, ExposéXVII, Prop. 7.1.1 ((b))], any extension of Hpr ×X x by T must be of multiplicative type, therefore must be commutative. Since the base field k is separable closed, such an extension must be trivial. Therefore the group Extalg(Hpr ×X x, T ) is trivial. Since Hpr ×X x is connected and U is smooth over k, we have Extalg(Hpr ×X x, U) = 0 by [DG70, ExposéXVII, Thm. 5.1.1 (i) (d)]. This finishes the proof of (2). Part (3) can be proven in the same way as [Zha21, Lem. 3.20 (3)]. Before making a corollary to Lemma 2.3, we make a lemma about the first direct image functor.

Lemma 2.4. Let X be a locally noetherian fs log scheme, and let f : G1 → G2 be an epimorphism of smooth commutative group schemes over the underlying scheme of X. Assume that f is as in one of the following two cases.

(1) Both G1 and G2 are affine with geometrically connected fibers over the underlying scheme of X. (2) For i =1, 2, Gi is an extension of an abelian scheme Ai by a torus Ti over the underlying scheme of X, and f is an isogeny. 1 1 1 Then R εfl∗f : R εfl∗G1 → R εfl∗G2 is surjective.

Proof. Let G := (Gm,log/Gm)Xfl . By [Zha21, Thm. 3.14], we have 1 R ε G = lim Hom (Z/nZ(1), G ) ⊗Z G. ﬂ∗ i −→ X i n

By [Zha21, Lem. A.1], the sheaf HomX (Z/nZ(1), Gi) is representable by an étale quasi-finite group scheme over the underlying scheme of X in both cases. Hence it suffices to check the surjectivity for the classical étale topology. Since the stalk at a point x of HomX (Z/nZ(1), Gi) ⊗Z G agrees with Homx(Z/nZ(1), Gi) ⊗Z G, we are further reduced to the case that the underlying scheme of X is Spec k for a separable closed field k. Now we deal with case (1). The group Gi is an extension of a unipotent group Ui by a torus Ti by [BLR90, §9.2, Thm. 2], and thus the map f induces the following commutative diagram

0 / T1 / G1 / U1 / 0

′ ′′ f f f 0 / T2 / G2 / U2 / 0 by [DG70, ExposéXVII, Prop. 2.4 (i)]. Since f is an epimorphism, so are f ′ and f ′′ by [DG70, ExposéXVII, Prop. 2.4 (ii)]. Again by [DG70, ExposéXVII, Prop. 2.4 (i)], the map (2.1) lim Hom (Z/nZ(1), G ) → lim Hom (Z/nZ(1), G ) −→ X 1 −→ X 2 n n COMPARISON OF KUMMER TOPOLOGIES WITH CLASSICAL TOPOLOGIES II 13 induced by f is canonically identified with the map (2.2) lim Hom (Z/nZ(1),T ) → lim Hom (Z/nZ(1),T ) −→ X 1 −→ X 2 n n ′ induced by f . Let X1 (resp. X2) be the character group of T1 (resp. T2), then the map (2.2) can be further identified with

HomZ(X1, Z) ⊗Z Q/Z → HomZ(X2, Z) ⊗Z Q/Z ′ 1 which is surjective by the surjectivity of f . Therefore R ε∗f is surjective. At last we show case (2). If the characteristic of the separable closed ﬁeld k is zero, then the map (2.1) can be identiﬁed with the map

G1(k)tor → G2(k)tor induced by f on the torsion subgroups, which is clearly surjective. We are reduced to consider the case that the characteristic of k is p> 0. In this case the surjectivity of the coprime to p part of the map (2.1) can be proven in the same as in the characteristic zero case. Hence we are reduced to check that the surjectivity of the p-primary part (2.3) lim Hom (Z/pnZ(1), G ) → lim Hom (Z/pnZ(1), G ) −→ X 1 −→ X 2 n n ∞ ∞ of the map (2.1). Let G1[p ]m (resp. G2[p ]m) be the multiplicative part of the p-divisible group of G1 (resp. G2), then the map (2.3) can be identiﬁed with the map (2.4) lim Hom (Z/pnZ(1), G [p∞] ) → lim Hom (Z/pnZ(1), G [p∞] ) −→ X 1 m −→ X 2 m n n induced by f. ∞ ∞ Now we digress to understand the p-divisible groups G1[p ]m and G2[p ]m. Since HomX (T1, A2) = 0 by [Bri17, Thm. 4.3.2 (1)], the map f induces the following commutative diagram

0 / T1 / G1 / A1 / 0 .

′ ′′ f f f 0 / T2 / G2 / A2 / 0 Since f is an isogeny and there is no non-trivial homomorphism from abelian variety ′ ′′ to torus, both f and f are isogeny. Let r be the dimension of T1 and let s be the p-rank of the dual of A1, then the dimension of T2 is also r and the p-rank of the ∞ r+s dual of A2 is also s. Since k is separable closed, we have Gi[p ]m =∼ (Qp/Zp(1)) for i =1, 2. Now we can ﬁnish the proof. We have lim Hom (Z/pnZ(1), G [p∞] ) ∼ (Q /Z )r+s −→ X 1 m = p p n ∞ ∞ for i = 1, 2. Since the map G1[p ]m → G2[p ]m induced by f is an isogeny, the 1 map (2.4) is surjective. It follows that R εﬂ∗f is surjective as well in case (2). 14 HEER ZHAO

Corollary 2.5. Let the notation and the assumptions be as in Theorem 2.1 and Lemma 2.3. We further assume that G is affine. Let n be a positive integer n which is invertible on X, and let G[n] denote the kernel of the multiplication by n map on G. Then G[n] is a quasi-finite étale group scheme over the underlying scheme of X and 2 ∼ gp Hkfl(X, G[n]) = Γ(X, G[n](−2)) ⊗Z P Proof. By the proof of [Zha21, Thm. 3.8 (3)], G[n] is a quasi-finite étale group scheme over the underlying scheme of X, and we have a short exact sequence n 0 → G[n] → G −→ G → 0 of sheaves on both (fs/X)két and (fs/X)kfl. This exact sequence induces two exact sequences 1 2 2 0 → Hkfl(X, G) ⊗Z Z/nZ → Hkfl(X, G[n]) → Hkfl(X, G)[n] → 0 and 1 2 2 0 → Hkét(X, G) ⊗Z Z/nZ → Hkét(X, G[n]) → Hkét(X, G)[n] → 0. i 1 ∼ i 1 Since Hfl(X, R ε∗G) = Hét(X, R ε∗G)=0 for any i> 0, we get 1 ∼ 0 1 Hkfl(X, G) = Hfl(X, R ε∗G). 1 n 1 Thus the map Hkfl(X, G) −→ Hkfl(X, G) can be identified with 0 1 n 0 1 Hfl(X, R ε∗G) −→ Hfl(X, R ε∗G). 1 n 1 By Lemma 2.4 (1), the map R ε∗G −→ R ε∗G is surjective. Since i 1 ∼ i 1 Hfl(X, (R ε∗G)[n]) = Hét(X, (R ε∗G)[n]) = 0, 0 1 n 0 1 the map Hfl(X, R ε∗G) −→ Hfl(X, R ε∗G) is surjective. It follows that 1 Hkfl(X, G) ⊗Z Z/nZ =0, 2 2 and thus Hkfl(X, G[n]) = Hkfl(X, G)[n]. By the proof of [Zha21, Thm. 3.8 (3)], 1 n 1 the map R εét∗G −→ R εét∗G is also surjective. By a similar argument as above, 1 2 2 one can show that Hkét(X, G) ⊗Z Z/nZ = 0. Thus Hkét(X, G[n]) = Hkét(X, G)[n]. Therefore 2 2 2 2 Hkfl(X, G[n]) = Hkfl(X, G)[n]= Hkét(X, G)[n]=Hkét(X, G[n]) 2 gp =Γ(X, G[n](−2)) ⊗Z P , ^ where the second equality follows from Lemma 2.3 (3) and the last equality follows from [KN99, Thm. 2.4].

Theorem 2.6. Let X be a locally noetherian fs log scheme, and let G be a smooth commutative aﬃne group scheme with geometrically connected ﬁbers over the underlying scheme of X. COMPARISON OF KUMMER TOPOLOGIES WITH CLASSICAL TOPOLOGIES II 15

(1) We have R2ε G = lim(R2ε G)[n]= (R2ε G)[l∞], fl∗ −→ fl∗ M fl∗ n l prime 2 2 2 ∞ where (R εfl∗G)[n] denotes the n-torsion subsheaf of R εfl∗G and (R εfl∗G)[l ] 2 denotes the l-primary part of R εfl∗G for a prime number l. 2 ∞ (2) The l-primary part (R εfl∗G)[l ] is supported on the locus where l is invertible. (3) If n is invertible on X, then 2 2 2 (R ε G)[n]= R ε G[n]= G[n](−2) ⊗Z (G /G ) . fl∗ fl∗ ^ m,log m Xfl 2 Proof. By Theorem 1.1, R εfl∗G is torsion. Hence part (1) follows. Part (2) follows from Lemma 2.3 (3). We are left with part (3). By [Swa99, §3], in particular the part between [Swa99, Cor. 3.7] and [Swa99, Thm. 3.8], we have cup-product for the higher direct image functors for the map of sites εfl : (fs/X)kfl → (fs/X)fl. The cup- product induces homomorphisms 2 1 2 2 G[n] ⊗Z Z R ε Z/nZ → G[n] ⊗Z Z R ε Z/nZ → R ε G[n]. /n ^ fl∗ /n fl∗ fl∗ Since 2 2 1 G[n] ⊗Z Z R ε Z/nZ = G[n](−2) ⊗Z (G /G ) , /n ^ fl∗ ^ m,log m Xfl we get a canonical homomorphism 2 2 G[n](−2) ⊗Z (G /G ) → R ε G[n]. ^ m,log m Xfl fl∗ By Corollary 2.5, this homomorphism is an isomorphism. This finishes the proof of part (3).

2 2 Remark 2.1. In the case that G is a torus, we have (R εfl∗G)[n]= R εfl∗G[n] for any positive integer n by [Zha21, Thm. 3.23 (2)]. In Theorem 2.6 we only have this identification for positive integer n which is invertible on the base. One reason for this is that we always have a short exact sequence 0 → G[n] → G −→n G → 0 for n ≥ 1 in the torus case. 2.2. The case of certain finite flat group schemes. Theorem 2.7. Let X be a locally noetherian fs log scheme, and F a finite flat group scheme over the underlying scheme of X. Then we have 2 2 ∞ 2 ∞ (1) R εfl∗F = l prime(R εfl∗F )[l ], where (R εfl∗F )[l ] denotes the l-primary 2 L part of R εfl∗F . 2 ∞ (2) The l-primary part (R εfl∗F )[l ] is supported on the locus where l is invertible. (3) Assume X is connected, then the order of F is constant on X. We denote the order by n and assume that n is invertible on X. Then 2 2 R ε F = F (−2) ⊗Z (G /G ) , fl∗ ^ m,log m Xfl 16 HEER ZHAO

where F (−2) denotes the tensor product of F with Z/nZ(−2).

Proof. Part (1) is clear. Let θ 0 → F → G1 −→ G2 → 0 be the canonical smooth resolution of F by smooth affine commutative group schemes, see [Bég81, Prop. 2.2.1] or [Mil06, Thm. A.5]. By the construction of loc. cit., the group scheme G1 is actually the Weil restriction of Gm from the underlying scheme of X to that of F . We claim that G1 has geometrically connected fiber over X. It suffices to check this for the case that X is a log point. Then the result follows from [CGP15, Prop. A.5.9]. It follows that G2 also has connected fiber over X. Applying the functor εfl∗, we get an exact sequence

1 2 1 R εfl∗θ 1 2 2 R εfl∗θ 2 (2.5) R εfl∗G1 −−−−−→ R εfl∗G2 → R εfl∗F → R εfl∗G1 −−−−−→ R εfl∗G2.

1 2 The map R εfl∗θ is surjective by Lemma 2.4 (1). It follows that R εfl∗F is a sub- 2 2 ∞ 2 sheaf of R εfl∗G1. Since the l-primary part (R εfl∗G1)[l ] of R εfl∗G1 is supported 2 ∞ on the locus where l is invertible by Theorem 2.6 (2), so is (R εfl∗F )[l ]. This finishes the proof of part (2). 2 Now we prove part (3). Since F is killed by its order n, so is the sheaf R εfl∗F . 2 ∞ 2 Let (R εfl∗Gi)[n ] denote the n-power torsion subsheaf of R εfl∗Gi for i = 1, 2. Then the exact sequence (2.5) induces an exact sequence

2 2 2 ∞ R εfl∗θ 2 ∞ (2.6) 0 → R εfl∗F → (R εfl∗G1)[n ] −−−−−→ (R εfl∗G2)[n ].

2 r We abbreviate (Gm,log/Gm)Xfl as Λ, and let Gi[n ](−2) denote the tensor prod- r V r uct of Gi[n ] with Z/n Z(−2) for i = 1, 2. The exact sequence (2.6) extends to a diagram (2.7) 0 / F (−2) ⊗Z Λ / lim G [nr](−2) ⊗Z Λ / lim G [nr](−2) ⊗Z Λ , −→r 1 −→r 2

α1 ∼= α2 ∼= α3

2 2 2 ∞ R εfl∗θ 2 ∞ 0 / R εfl∗F / (R εfl∗G1)[n ] / (R εfl∗G2)[n ] where the vertical maps are given by the cup-product construction as in the proof of Theorem 2.6 (3), the diagram is commutative by the functoriality of cup-product, and the maps α2 and α3 are isomorphisms by Theorem 2.6 (3). In order to prove part (3), i.e. the map α1 is an isomorphism, it suffices to prove the lemma below by the five lemma.

Lemma 2.8. The ﬁrst row of the diagram (2.7) is exact. COMPARISON OF KUMMER TOPOLOGIES WITH CLASSICAL TOPOLOGIES II 17

Proof. The multiplication by nr maps give rise to the following commutative diagram θ 0 / F / G1 / G2 / 0

0 nr nr θ 0 / F / G1 / G2 / 0 with exact rows, which induces an exact sequence r r 0 → F → G1[n ] → G2[n ]. We regard this sequence as an exact sequence of Z/nrZ-modules and tensor it with Z/nrZ(−2), and thus get another exact sequence

r θr r 0 → F (−2) → G1[n ](−2) −→ G2[n ](−2). 2 Since the sheaf Λ = (Gm,log/Gm)Xfl takes values in free abelian groups, tensoring 2 V with Λ = (Gm,log/Gm)X gives rise to an exact sequence V fl r r 0 → F (−2) ⊗Z Λ → G1[n ](−2) ⊗Z Λ → G2[n ](−2) ⊗Z Λ. Passing to direct limit, we get an exact sequence which is nothing but the first row of the diagram (2.7). 2.3. The case of extensions of abelian schemes by tori. Theorem 2.9. Let X be a locally noetherian fs log scheme, and G a group scheme over the underlying scheme of X which is an extension of an abelian scheme A by a torus T over X. Then we have 2 2 ∞ 2 ∞ (1) R εfl∗G = l prime(R εfl∗G)[l ], where (R εfl∗G)[l ] denotes the l-primary 2 L part of R εfl∗G. 2 2 (2) We have (R εfl∗G)[n]= R εfl∗G[n]. 2 ∞ (3) The l-primary part (R εfl∗G)[l ] is supported on the locus where l is invertible. (4) If n is invertible on X, then 2 2 2 (R ε G)[n]= R ε G[n]= G[n](−2) ⊗Z (G /G ) , fl∗ fl∗ ^ m,log m Xfl where G[n](−2) denotes the tensor product of G[n] with Z/nZ(−2).

2 Proof. By Theorem 1.1, R εfl∗G is torsion. Hence part (1) follows. Since G is an extension of an abelian scheme by a torus, we have an exact n sequence 0 → G[n] → G −→ G → 0 of sheaves of abelian groups on (fs/X)kfl and G[n] is a finite flat group scheme over the underlying scheme of X. Applying the functor εfl∗, we get an exact sequence 1 2 2 0 → R εfl∗G ⊗Z Z/nZ → R εfl∗G[n] → (R εfl∗G)[n] → 0. 1 By Lemma 2.4 (2), we have R εfl∗G ⊗Z Z/nZ = 0. Thus part (2) follows. Since G[n] is a finite flat group scheme, part (3) and part (4) follow from part (2) and Theorem 2.7. 18 HEER ZHAO

2.4. Special cases of the base. We can give more explicit description of the second higher direct image in some special cases of the base. The first case concerns the ranks of the log structure at geometric points of the base. For an fs log scheme X and a geometric point x of X, by the rank of the gp × log structure of X at x we mean the rank of the free abelian group (MX /OX )x. Theorem 2.10. Let X be a locally noetherian fs log scheme, and let G be a commutative group scheme over the underlying scheme of X which satisfies any of the following three conditions: (i) G is smooth, affine, and of geometrically connected fibers over the underlying scheme of X; (ii) G is finite, flat, and of order n over the underlying scheme of X, and for any n prime number l the kernel of G −→l G is also finite flat over the underlying scheme of X; (iii) G is an extension of an abelian scheme by a torus over the underlying scheme of X. gp × Let Y ∈ (fs/X) be such that the ranks of the stalks of the étale sheaf MY /OY are at most one, and let (st/Y ) be the full subcategory of (fs/X) consisting of strict fs 2 log schemes over Y . Then the restriction of R εfl∗G to (st/Y ) is zero. Proof. By Theorem 2.6 (1), Theorem 2.7 (1), and Theorem 2.9 (1), it suffices 2 ∞ to show that the restriction of the l-primary part (R εfl∗G)[l ] to (st/Y ) is zero 2 ∞ for any prime number l. Since (R εfl∗G)[l ] is supported on the locus where l is invertible by Theorem 2.6 (2), Theorem 2.7 (2), and Theorem 2.9 (3), we are reduced to the case that l is invertible on Y . By Theorem 2.6 (3) and Theorem 2.9 (4), we have

2 2 ∞ r (R ε G)[l ] = lim G[l ](−2) ⊗Z (G /G ) fl∗ −→ m,log m Xfl r ^ in cases (i) and (iii). In case (ii), we have a short exact sequence 0 → G[ln] → G → G/G[ln] → 0 of finite flat group schemes over the underlying scheme of X. Since the order of n i n ∞ G/G[l ] is coprime to l, we get (R εfl∗(G/G[l ]))[l ] =0 for any i> 0. Hence

2 2 ∞ 2 n ∞ n (R ε G)[l ] = (R ε G[l ])[l ]= G[l ](−2) ⊗Z (G /G ) fl∗ fl∗ ^ m,log m Xfl gp × by Theorem 2.7 (3). Since the ranks of the stalks of the étale sheaf MY /OY are at 2 most one, the restriction of the sheaf (Gm,log/Gm)Xfl to (st/Y ) is zero. Therefore 2 V ∞ the restriction of the sheaf (R εfl∗G)[l ] to (st/Y ) is zero in all the three cases. This finishes the proof.

Next we investigate the cases involving the characteristic of the base. COMPARISON OF KUMMER TOPOLOGIES WITH CLASSICAL TOPOLOGIES II 19

Theorem 2.11. Let X be a locally noetherian fs log scheme such that the underlying scheme of X is a Q-scheme, and let G be as in any of the three cases of Theorem 2.10. Then we have 2 2 R ε G = lim G[n](−2) ⊗Z (G /G ) ﬂ∗ −→ m,log m Xfl n ^ in cases (i) and (iii), and

2 2 R ε G = G(−2) ⊗Z (G /G ) ﬂ∗ ^ m,log m Xfl in case (ii). Proof. Since the underlying scheme of X is a Q-scheme, the results follow from Theorem 2.6, Theorem 2.7, and Theorem 2.9 easily.

Theorem 2.12. Let p be a prime number. Let X be a locally noetherian fs log scheme such that the underlying scheme of X is an Fp-scheme, and G a commutative group scheme over the underlying scheme of X which satisfies any one of the following three conditions: (i) G is smooth, affine, and of geometrically connected fibers over the underlying scheme of X; (ii) G is finite, flat, and of order n over the underlying scheme of X, and the pn kernel G[pn] of G −→ G is also finite flat over the underlying scheme of X; (iii) G is an extension of an abelian scheme by a torus over the underlying scheme of X. Then we have 2 2 R ε G = lim G[r](−2) ⊗Z (G /G ) fl∗ −→ ^ m,log m Xfl (r,p)=1 in cases (i) and (iii), and

2 2 n R ε G = (G/G[p ])(−2) ⊗Z (G /G ) ﬂ∗ ^ m,log m Xfl in case (ii).

2 Proof. Since the underlying scheme of X is an Fp-scheme, R εfl∗G is p-power torsion free by Theorem 2.6 (2), Theorem 2.7 (2), and Theorem 2.9 (3). By Theorem 2 1.1, R εfl∗G is torsion for case (i) and case (iii). Hence we have R2ε G = lim (R2ε G)[r] = lim R2ε G[r] fl∗ −→ fl∗ −→ fl∗ (r,p)=1 (r,p)=1 2 = lim G[r](−2) ⊗Z (G /G ) −→ ^ m,log m Xfl (r,p)=1 by Theorem 2.6 (3) (resp. Theorem 2.9 (4)) for case (i) (resp. (iii)). 20 HEER ZHAO

For case (ii), we have a short exact sequence

0 → G[pn] → G → G/G[pn] → 0 of ﬁnite ﬂat group schemes over the underlying scheme of X. This short exact sequence induces an exact sequence

2 n 2 2 n 3 n R εfl∗G[p ] → R εfl∗G → R εfl∗(G/G[p ]) → R εfl∗G[p ].

Since the order of G/G[pn] is coprime to p, hence invertible on X, we get

2 i n n R ε (G/G[p ]) = (G/G[p ])(−2) ⊗Z (G /G ) ﬂ∗ ^ m,log m Xfl by Theorem 2.7 (3), which is p-power torsion free. Since G[pn] is killed by pn, i n n R εﬂ∗G[p ] is p -torsion for any i> 0. It follows that

2 2 2 n n R ε ∗G =∼ R ε ∗(G/G[p ]) = (G/G[p ])(−2) ⊗Z (G /G ) . fl fl ^ m,log m Xfl This finishes the proof for case (ii).

3. Examples 3.1. Discrete valuation rings. Let R be a Henselian discrete valuation ring with fraction field K and residue field k. Let π be a uniformizer of R, and we endow X = Spec R with the log structure associated to the homomorphism N → R, 1 7→ π. Let x be the closed point of X and i the closed immersion x ֒→ X, and we endow x with the induced log structure from X. Let η be the generic point of X and j .the open immersion η ֒→ X Let G be a commutative group scheme over Spec R which satisfies one of the following three conditions: (i) G is smooth and affine over Spec R and has geometrically connected fibers over Spec R; (ii) G is finite, flat, and of order n over Spec R, and for any prime number l the n kernel of G −→l G is also finite flat over Spec R; (iii) G is an extension of an abelian scheme by a torus over Spec R. We consider the Leray spectral sequence

s t s+t (3.1) Hfl(X, R εfl∗G) ⇒ Hkfl (X, G).

We have

1 R ε G = lim Hom (Z/nZ(1), G) ⊗Z (G /G ) ﬂ∗ −→ X m,log m Xfl n COMPARISON OF KUMMER TOPOLOGIES WITH CLASSICAL TOPOLOGIES II 21

∼ by [Zha21, Thm. 3.14]. Then on (st/X), we have (Gm,log/Gm)Xfl = i∗Z. Therefore s 1 s H (X, R ε G)=H (X, lim Hom (Z/nZ(1), G) ⊗Z i Z) fl fl∗ fl −→ X ∗ n =Hs(x, lim Hom (Z/nZ(1), G × x)) fl −→ x X (3.2) n = lim Hs(x, Hom (Z/nZ(1), G × x)) −→ fl x X n = lim Hs (x, Hom (Z/nZ(1), G × x)) −→ ét x X n for s ≥ 0, where the last equality follows from [Sta21, Tag 0DDU] and the fact that Homx(Z/nZ(1), G ×X x) is representable by an étale group scheme (see [Zha21, Lem. A.1]). We also have s 2 s (3.3) Hfl(X, R εfl∗G)= Hfl(X, 0)=0 for s ≥ 0 by Theorem 2.10. Then the spectral sequence (3.1) gives rise to an exact sequence 1 1 α 0 1 2 0 → H (X, G) → H (X, G) −→ H (X, R εfl∗G) → H (X, G) (3.4) fl kfl fl fl 2 1 1 3 → Hkfl(X, G) → Hfl(X, R εfl∗G) → Hfl(X, G). Proposition 3.1. Assume that k is a finite field, and G satisfies condition (i). Let Tx be the torus part of Gx := G ×X x, and N := Homx(Gm,Tx). Then we have 1 0 0 H (X, G) ∼ lim H (x, N ⊗Z Z/nZ)= H (x, N ⊗Z Q/Z) kfl = −→ ét ét n and 2 1 1 H (X, G) ∼ lim H (x, N ⊗Z Z/nZ)= H (x, N ⊗Z Q/Z). kfl = −→ ét ét n s s s Proof. We have Hfl(X, G)= Hét(X, G)= Hét(x, Gx) by [Sta21, Tag 0DDU] and [Mil80, Chap. III, Rmk. 3.11 (a)]. Since k as a finite field has cohomological s dimension 1, we have Hét(x, Gx)=0for s> 1. Since Gx is connected and smooth, 1 we also have Hét(x, Gx) = 0 by Lang’s theorem. Then the exact sequence (3.4) s ∼ s−1 1 gives rise to Hkfl(X, G) = Hfl (X, R εfl∗G) for s =1, 2. By (3.2), we get Hs (X, G) ∼ lim Hs−1(x, Hom (Z/nZ(1), G )) kfl = −→ ét x x n for s =1, 2. Since Gx is smooth and connected, it fits into a short exact sequence 0 → Tx → Gx → Ux by [BLR90, §9.2, Thm. 2], where Tx is the torus part of Gx and Ux is unipotent. We have

Homx(Z/nZ(1), Gx)= Homx(Z/nZ(1),Tx) by [DG70, Expos´eXVII, Prop. 2.4 (i)]. Applying the functor Homx(−,Tx) to the short exact sequence n 0 → Z/nZ(1) → Gm −→ Gm → 0, we get an exact sequence n 1 0 → N −→ N → Homx(Z/nZ(1),Tx) →Extx(Gm,Tx). 22 HEER ZHAO

1 We have Extx(Gm,Tx) = 0 by [Gro72, ExposéVIII, Prop. 3.3.1]. Therefore Homx(Z/nZ(1),Tx)= N ⊗Z Z/nZ. It follows that s s−1 s−1 H (X, G) ∼ lim H (x, N ⊗Z Z/nZ)= H (x, N ⊗Z Q/Z) kfl = −→ ét ét n for s =1, 2.

Proposition 3.2. Assume that k is a finite field, and G satisfies condition (ii). mul Let Gx be the maximal multiplicative subgroup of Gx := G ×X x whose existence is guaranteed by [DG70, ExposéXVII, Thm. 7.2.1]. Let n be an integer which kills mul G, and let N := Homx(Z/nZ(1), Gx ). Then we have 1 ∼ 1 0 Hkfl(X, G) = Hfl(X, G) ⊕ Hét(x, N) and 2 ∼ 1 Hkfl(X, G) = Hét(x, N). s Proof. By [Mil06, Chap. III, Lem. 1.1], we have Hfl(X, G) = 0 for s ≥ 2. Then the exact sequence (3.4) and the identification (3.2) together give rise to a short exact sequence 0 → H1(X, G) → H1 (X, G) → lim H0 (x, Hom (Z/rZ(1), G )) → 0 fl kfl −→ ét x x r and an isomorphism H2 (X, G) ∼ lim H1 (x, Hom (Z/rZ(1), G )). kfl = −→ ét x x r Since n kills G, we get lim Hs (x, Hom (Z/rZ(1), G )) = Hs (x, Hom (Z/nZ(1), G )) −→ ét x x ét x x r mul for s =0, 1. Since Gx/Gx is unipotent, we get mul Homx(Z/nZ(1), Gx)= Homx(Z/nZ(1), Gx )= N by [DG70, ExposéXVII, Prop. 2.4]. Thus we get a short exact sequence 1 1 0 (3.5) 0 → Hfl(X, G) → Hkfl(X, G) → Hét(x, N) → 0 and an isomorphism 2 ∼ 1 Hkfl(X, G) = Hét(x, N). The short exact sequence (3.5) is actually split by [WZ20, App. D, Lem. D.1]. Thus we get 1 ∼ 1 0 Hkfl(X, G) = Hfl(X, G) ⊕ Hét(x, N).

Remark 3.1. Let the situation be as in Proposition 3.2. The restriction map 1 1 Hfl(X, G) → Hfl(Spec K, G×X Spec K) is injective by [Mil06, Chap. III, Lem. 1.1]. The analogue for the Kummer log flat topology is also true, i.e. the restriction map 1 1 Hkfl(X, G) → Hfl(Spec K, G ×X Spec K) is injective, see [Gil12, Prop. 3.6]. COMPARISON OF KUMMER TOPOLOGIES WITH CLASSICAL TOPOLOGIES II 23

Proposition 3.3. Assume that k is a finite field, and G satisfies condition (iii). Then we have Hs (X, G) ∼ lim Hs−1(x, Hom (Z/nZ(1), G )) kfl = −→ ét x x n for s =1, 2. Proof. The same argument in the beginning of the proof of Proposition 3.1 shows also s s s Hfl(X, G)= Hét(X, G)= Hét(x, Gx)=0 for s ≥ 1. Then the result follows from the exact sequence (3.4) and the identification (3.2). Remark 3.2. By Proposition 3.1, Proposition 3.2, and Proposition 3.3, we s reduce the computation of Hkfl(X, G) for s = 1, 2 to the computation of certain étale cohomology groups over the finite field k, which can be easily computed via Galois cohomology over k. 3.2. Global Dedekind domains. Through this subsection, let K be a global field. When K is a number field, X denotes the spectrum of the ring of integers in K, and when K is a function field, k denotes the field of constants of K and X denotes the unique connected smooth projective curve over k having K as its ,function field. Let S be a finite set of closed points of X, U := X − S, j : U ֒→ X and ix : x ֒→ X for each closed point x ∈ X. We endow X with the log structure × j∗OU ∩OX → OX . In the case of number field, let S∞ := S∪{infinite places of K}, and in the case of function field, we just let S∞ := S. Proposition 3.4. Let X be as above. Let G be a commutative group scheme over the underlying scheme of X which satisfies any of the following three conditions: (i) G is smooth and affine over the underlying scheme of X, and has geometrically connected fibers; (ii) G is finite flat of order n over the underlying scheme of X, and for any prime n number l the kernel of G −→l G is also finite flat over the underlying scheme of X; (iii) G is an extension of an abelian scheme by a torus over the underlying scheme of X.

For x ∈ S, let Gx := G ×X x. Then we have the following exact sequence 1 1 α 0 1 0 →Hfl(X, G) → Hkfl(X, G) −→ Hfl(X, R εfl∗G) 2 2 1 1 (3.6) →Hfl(X, G) → Hkfl(X, G) → Hfl(X, R εfl∗G) 3 3 →Hfl(X, G) → Hkfl(X, G), and i 1 i H (X, R ε ∗G) =∼ H (x, lim Hom (Z/nZ(1), G )) fl fl M ét −→ x x x∈S n for i =0, 1. 24 HEER ZHAO

2 Proof. On (st/X), we have R εfl∗G = 0 by Theorem 2.10. Thus the Leray spectral sequence s t s+t Hfl(X, R εfl∗G) ⇒ Hkfl (X, G) gives rise to the long exact sequence (3.6). We have R1ε G = i lim Hom (Z/nZ(1), G ) fl∗ M x,∗ −→ x x x∈S n by [Zha21, Thm. 3.14]. It follows that Hi (X, R1ε G)=Hi (X, i lim Hom (Z/nZ(1), G )) fl fl∗ fl M x,∗ −→ x x x∈S n = Hi (x, lim Hom (Z/nZ(1), G )) M fl −→ x x x∈S n =∼ Hi (x, lim Hom (Z/nZ(1), G )), M ét −→ x x x∈S n where the last isomorphism follows from [Zha21, Lem. A.1].

In the case that G = Gm, the map α of the exact sequence (3.6) has been shown to be surjective in [Zha21, §5.2]. The map α of the exact sequence (3.4) is also surjective if the residue ﬁeld k of the base ring R is ﬁnite by Proposition 3.1, Proposition 3.2, and Proposition 3.3. It is not clear to the author at this moment if the map α of (3.6) is surjective beyond the case G = Gm. The method of [Zha21, §5.2] does not work even for general tori.

Acknowledgement The author thanks Professor Ulrich G¨ortz for very helpful discussions.

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Fakultat¨ fur¨ Mathematik, Universitat¨ Duisburg-Essen, Essen 45117, Germany Email address: [email protected]