Proof. See [Reference LinhLin25, Proposition 2.4].

Proof. See [Reference LinhLin25, Construction 2.6, Lemma 2.7, Proposition 2.11].

Proof. See [Reference LinhLin25, Construction 2.6, Lemma 2.8, Theorem 3.8].

Proof. For (i), see [Reference González-AvilésGA12, Theorem 1.1, Proposition 3.14 (a)]. For (ii) (resp. (iii)), we refer to Theorem 5.4 and Corollary 5.4.1 (resp. Theorems 5.12 and 5.13) in [Reference BorovoiBor98].

Proof.

1. (i) This follows from the fact that t-resolutions are preserved by base change.

2. (ii) It suffices to combine [Reference González-AvilésGA12, Definition 3.1] and [Reference Borovoi and González-AvilésBGA14, Proposition 4.1], noting that the centers of G and $G^{\operatorname {sc}}$ remains unchanged under twisting.

3. (iii) See [Reference González-AvilésGA12, Proposition 3.11].

Proof. The case where $Y = G$ is the trivial torsor was dealt with by González-Avilés in [Reference González-AvilésGA19, Theorem 1.1]. The case where $X = \operatorname {Spec}(k)$ is due to Borovoi–van Hamel [Reference Borovoi and van HamelBvH09, Lemma 5.2]. The proof in the general case is similar; we shall deploy Sansuc’s argument, using the additivity theorem for $\operatorname {UPic}$ . More precisely, the projections $p_Y: Y \times _X G \to Y$ and $p_G: Y \times _X G \to G$ induce a canonical morphism

in $\mathcal {D}^+(X)$ . By [Reference González-AvilésGA18, Proposition 4.4, Remark 5.2 (a)], $\varsigma _Y$ is an isomorphism. Note that the existence of an ‘étale quasi-section’ of $Y \times _X G \to X$ as in (i) of loc. cit. is guaranteed by [Reference GrothendieckGro67, Corollaires 17.16.2 and 17.16.3 (ii)], since this morphism is smooth [Reference MilneMil80, Chapter III, Proposition 4.2] and surjective, and that for every generic point $\eta \in X$ , the ‘étale index’ [Reference González-AvilésGA18, Definition 2.3] of $G_{\eta }$ is $1$ since it has a $\kappa (\eta )$ -rational point. Let $\varphi _Y$ denote the composite

where $\alpha : Y \times _X G \to Y$ is the X-action of G on Y, and $\pi _G$ is the second projection. According to its construction, $\varphi _Y$ is functorial in Y and G. Since the result is known for the trivial torsor $G \to X$ , it remains to show that $\varphi _Y$ is a quasi-isomorphism, that is, $\mathscr {H}^{-1}(\varphi _Y): \operatorname {U}_{Y/X} \to \operatorname {U}_{G/X}$ and $\mathscr {H}^0(\varphi _Y): \operatorname {Pic}_{Y/X} \to \operatorname {Pic}_{G/X}$ are isomorphisms of étale sheaves over X. This follows from the units-Picard-Brauer sequence, see [Reference González-AvilésGA19, Lemmata 2.5 and 3.2, Proposition 3.6, Corollary 3.8] and their proofs.

Proof.

1. (i) When $G^{\operatorname {ss}} = G^{\operatorname {sc}}$ , one has $C \simeq G^{\operatorname {tor}}$ and $\widehat {C} \simeq \widehat {G^{\operatorname {tor}}}[1] = \widehat {G}[1]$ , and the abelianization map $\operatorname {ab}^1: \mathrm {H}^1(X,G) \to \mathrm {H}^1(X,C)$ is simply the pushfoward along the projection ${G \to G^{\operatorname {tor}}}$ . By functoriality, it is enough to prove the claim when $G = T$ is a torus. In this case, $\operatorname {Pic}_{Y/X} = 0$ , and it was shown by Colliot-Thélène and Sansuc that triangle (2.6) is quasi-isomorphic to the short exact sequence

   
   provided by the relative version of Rosenlicht’s lemma, whose class in $\operatorname {Ext}^1_X(\widehat {T},\mathbb {G}_m)$ turns out to be precisely $-[Y]$ (see [Reference Colliot-Thélène and SansucCTS87, Propositions 1.4.2 and 1.4.3]).

2. (ii) We follow Kottwitz’s argument in [Reference KottwitzKot86, p. 369]. Suppose that $\pi : U \to X$ is a surjective finite étale morphism trivializing Y. Take any t-resolution

   
   of G, and consider the sheaf $T_1:=\pi _\ast T_U$ (which is representable by an X-torus). Let $\partial : \mathrm {H}^1(X,G) \to \mathrm {H}^2(X,T)$ denote the connecting map. Then, the restriction of $\partial ([Y])$ to $\mathrm {H}^2(U,T)$ is $0$ . Now, by the global version of Shapiro’s lemma (which can be obtained by combining [Reference GrothendieckGro73, Corollaire 5.6] and the Leray spectral sequence), one has $\mathrm {H}^2(U,T) \cong \mathrm {H}^2(X,T_1)$ . Hence, the map $i_\ast : \mathrm {H}^2(X,T) \to \mathrm {H}^2(X,T_1)$ (induced by the inclusion $i: T \hookrightarrow T_1$ ) takes $\partial ([Y])$ to $0$ . Let $H_1$ denote the pushout of i and the inclusion $T \hookrightarrow H$ . Then $H_1^{\operatorname {ss}} = H^{\operatorname {ss}} = H^{\operatorname {sc}}$ is simply connected, hence $1 \to T_1 \to H_1 \to G \to 1$ is also a t-resolution of G, and the corresponding connecting map $\delta : \mathrm {H}^1(X,G) \to \mathrm {H}^2(X,T_1)$ takes $[Y]$ to $0$ . It follows that there is a torsor $Z \to X$ under $H_1$ such that the map $\mathrm {H}^1(X,H_1) \to \mathrm {H}^1(X,G)$ takes $[Z]$ to $[Y]$ . We know from (i) that the statement of the proposition holds for Z. By functoriality, it also holds for Y.

Proof. This follows directly from Propositions 2.5 and 2.7. We note that in the case where $f: Y \to X$ is isotrivial, the twisted torsor $f^{\sigma }: Y^{\sigma } \to X$ is also isotrivial for any $\sigma \in \mathrm {Z}^1(k,G)$ . This is because the class $[\sigma ] \in \mathrm {H}^1(k,G)$ is trivialized by a finite extension of k.

Proof.

1. (i) We construct a commutative diagram

   (2.8)

   
   where the bottom row is the distinguished triangle associated with the given t-resolution of G (see [Reference González-AvilésGA19, Proposition 4.12]). The quasi-isomorphisms $\varphi _Y$ and $\varphi _Z$ are given by Proposition 2.6. The unlabeled arrow on the top row of (2.8) is the composite
   
   where $\chi _\pi : \operatorname {U}_{Z/X} \to \widehat {T}$ is the map from the units-Picard-Brauer sequence (cf. [Reference González-AvilésGA19, Corollary 3.8 (i)]), and where the inclusion $\operatorname {U}_{Z/X}[1] \hookrightarrow \operatorname {UPic}_{Z/X}$ is a quasi-isomorphism because $\operatorname {Pic}_{Z/X} = 0$ (note that $H^{\operatorname {sc}} = H^{\operatorname {ss}}$ ). The map $\lambda _\pi : \widehat {T} \to \operatorname {UPic}_{Y/X}[1]$ is the extended type of the torsor $Z \xrightarrow {\pi } Y$ under T, constructed in [Reference Harari and SkorobogatovHS13, Definition 8.2]. Let us verify that diagram (2.8) indeed commutes. To this end, we recall the construction of $\varphi _Y$ and $\varphi _Z$ from the proof of Proposition 2.6. The left square of (2.8) commutes because we have a commutative diagram
   
   where $\alpha _Y: Y \times _X G \to Y$ and $\alpha _Z: Z \times _X H \to Z$ denote the respective action morphisms, and where right horizontal arrows are the second projections. Indeed, the left square commutes because $\alpha _Y \circ (\pi \times _X p) = \pi \circ \alpha _Z$ (the morphism $\pi $ being p-equivariant), and the other two squares obviously commute.

   The middle square of (2.8) commutes because it corresponds to a commutative square

   
   Indeed, $\mathscr {H}^{-1}(\varphi _Z)$ is just the map from the units-Picard-Brauer sequence for Z (cf. [Reference González-AvilésGA19, Corollary 3.8 (i)], and hence the commutativity of the above square follows from the functoriality of the same sequence (see loc. cit., Remarks 3.7 (a)).

   As for the right square of (2.8), since T is central in H, we have a commutative diagram

   
   where $\nabla : T \times _X T \to T$ denotes the multiplication morphism. By functoriality of the extended type construction, we obtain the commutativity of the left square of the following diagram:
   
   The other squares in the above diagram obviously commute. The composite of the top row is $\operatorname {id}_{\widehat {T}}$ (composing the first two arrows yields the diagonal map $\widehat {T} \to \widehat {T} \oplus \widehat {T}$ ), and that of the bottom row is $\varphi _Y$ (when $\operatorname {UPic}_{G/X}$ is identified to $\pi _1^D(G)$ ). Furthermore, by unwinding the definition of the extended type in [Reference Harari and SkorobogatovHS13, Proposition 8.1, Definition 8.2] (see also [Reference LinhLin25, §4.1, pp. 415—416], one checks that the composite $\widehat {T} \xrightarrow {\simeq } \operatorname {UPic}_{T/X} \to \operatorname {UPic}_{G/X}[1]$ is precisely the extended type $\lambda _p$ of the torsor $p: H \to G$ under T. This shows that the right square of (2.8) commutes.

   Since the bottom row of (2.8) is a distinguished triangle and since the vertical arrows there are quasi-isomorphisms, the top row is also a distinguished triangle.

2. (ii) The catch here is that the morphism $v: Z' \to Z$ need not correspond to a morphism $H' \to H$ of X-group schemes. Nevertheless, once the maps in (2.7) are explicitly constructed, our claims follow easily from the functoriality of the maps from the units-Picard-Brauer sequence as well as the extended types.

Proof. The main difficulty here is that if G is not commutative, the morphism $\pi $ is not equivariant with respect to any morphism $G \times _X G^P \to G^P$ of X-group schemes. We start with the case where $G = T$ is a torus. In this case, one has $G^P = T$ , $\widehat {C} \simeq \widehat {T}[1]$ and $\pi $ is $\nabla $ -equivariant, where $\nabla : T \times _X T \to T$ is the multiplication morphism. Since $\nabla $ induces the diagonal map $\Delta : \widehat {T} \to \widehat {T} \oplus \widehat {T}$ on character modules, the claim in the toric case follows.

Next, we consider the case where $G^{\operatorname {ss}} = G^{\operatorname {sc}}$ . Then $\widehat {C} \simeq \widehat {G^{\operatorname {tor}}}[1]$ . Let Z (resp. Q) denote the pushforward of Y (resp. P) along the projection $u: G \to G^{\operatorname {tor}}$ , which are torsors under the X-torus $G^{\operatorname {tor}}$ . Then, the projections $u^P: G^P \to (G^P)^{\operatorname {tor}} = G^{\operatorname {tor}}$ and $u \times _X u^P: G \times _X G^P \to (G \times _X G^P)^{\operatorname {tor}} = G^{\operatorname {tor}} \times _X G^{\operatorname {tor}}$ have the respective properties that $\pi ^P_\ast Y^P = Z^Q$ and $(\pi \times _X \pi ^P)_\ast (Y \times _X P) = Z \times _X Q$ . Let $\varpi : Z \times _X Q \to Z^Q$ be the canonical projection. The commutativity of the square

yields the commutativity of the left square in the following diagram:

The right square commutes thanks to the toric case which was dealt with above.

Now, we consider the case where the torsors $Y \to X$ and $P \to X$ are isotrivial. Take a surjective finite étale morphism $U \to X$ trivializing both Y and P. Then, by repeating the argument from the proof of Proposition Proposition 2.7 (ii), we obtain a t-resolution

such that $Y = u_\ast Z$ and $P = u_\ast Q$ for some right torsor $Z \to X$ and some left torsor $Q \to X$ under H. Note that

are also t-resolutions (recall that T is central in H hence remains unchanged under twisting), and that we have $u^Q_\ast Z^Q = Y^P$ and $(u \times _X u^Q)_\ast (Z \times _X Q) = Y \times _X P$ . Let $\varpi : Z \times _X Q \to Z^Q$ (resp. $\nabla : T \times _X T \to T$ ) denote the canonical projection (resp. the multiplication morphism). One easily checks that the diagram

commutes. By Proposition 2.10, we have a morphism of distinguished triangles

Now, since $H^{\operatorname {ss}} = H^{\operatorname {sc}}$ , we deduce from the result in the previous case that the second column in the above diagram can be replaced by the map $\Delta [1]: \widehat {S}[1] \to (\widehat {S} \oplus \widehat {S})[1]$ , where $S:=H^{\operatorname {tor}}$ . Since $\widehat {C}$ is by definition the cone of the morphism $\widehat {S} \to \widehat {T}$ , we obtain the claimed result using the functoriality of the quasi-isomorphisms constructed from Proposition 2.6.

Proof. Take a split maximal torus S of G, and let T be its preimage in $G^{\operatorname {sc}}$ . Then $[\widehat {S} \to \widehat {T}]$ (where $\widehat {S}$ is in degree $-1$ ) is a complex of split tori representing $\widehat {C}$ . Let $K:=k(X)$ and let $V \to \operatorname {Spec}(K)$ be the generic fibre of f. By [Reference Borovoi and KunyavskiiBK00, Corollary B], the group $\operatorname {Br}_{\operatorname {nr}}(V)/ f^\ast (\operatorname {Br}(K))$ injects into the subgroup of $\mathrm {H}^1(K,\widehat {C})$ consisting of elements that become trivial after restriction to every procyclic subgroup of $\Gamma _K$ . Let $r:=\dim (T) = \dim (S)$ , then $\widehat {T} \cong \widehat {S} \cong \mathbb {Z}^r$ as $\Gamma _K$ -modules. In particular, one has $\mathrm {H}^1(K,\widehat {T}) = 0$ . It follows that $\mathrm {H}^1(K,\widehat {C})$ injects into $\mathrm {H}^2(K,\widehat {S})$ . On the other hand, there is an exact sequence

(3.2)

of abelian groups equipped with the trivial actions of $\Gamma _K$ . The term $\mathbb {Q}^r$ is uniquely divisible, hence cohomologically trivial. It follows that $\mathrm {H}^2(K,\widehat {S}) = \mathrm {H}^1(K,(\mathbb {Q}/\mathbb {Z})^r) = \operatorname {Hom}_{\operatorname {cts}}(\Gamma _K, (\mathbb {Q}/\mathbb {Z})^r)$ , and similarly for any procyclic subgroup of $\Gamma _K$ . It follows that , a fortiori , and hence the map $f^\ast : \operatorname {Br}(K) \to \operatorname {Br}_{\operatorname {nr}}(V)$ is surjective.

Viewing that $k(Y) = K(V)$ , we have $\operatorname {Br}(Y) \subseteq \operatorname {Br}(V) \subseteq \operatorname {Br}(k(Y))$ . Let $\alpha \in \operatorname {Br}_{\operatorname {nr}}(Y) \subseteq \operatorname {Br}_{\operatorname {nr}}(V)$ . From what has been established above, for some . It remains to show that . Indeed, let $\xi \in X^{(1)}$ be any codimension $1$ point. Let $\eta $ be the generic point of the fibre $f^{-1}(\xi ) \to \operatorname {Spec}(k(\xi ))$ , which is a codimension 1 point of Y. Since $\alpha \in \operatorname {Br}(Y)$ , the residue $\partial _\eta (\alpha ) \in \mathrm {H}^1(k(\eta ),\mathbb {Q}/\mathbb {Z})$ vanishes (cf. (1.1)). Since $f^{-1}(\xi )$ is geometrically integral, the map $\mathrm {H}^1(k(\xi ),\mathbb {Q}/\mathbb {Z}) \to \mathrm {H}^1(k(\eta ),\mathbb {Q}/\mathbb {Z})$ is injective. By functoriality of the residue map [Reference Colliot-Thélène and SkorobogatovCTS21, Theorem 3.7.5], we have . It follows that , which concludes the proof.

Proof. Taking hypercohomology of the distinguished triangle

in $\mathcal {D}^+(X)$ yields an exact sequence

(3.3)

On the other hand, by taking cohomology of the distinguished triangle

we have $\mathrm {H}^2(X,\tau _{\le 2} \mathbb {R} f_\ast \mathbb {G}_{m,Y}) = \mathrm {H}^2(X,\mathbb {R} f_\ast \mathbb {G}_{m,Y}) = \mathrm {H}^2(Y,\mathbb {G}_m) = \operatorname {Br}(Y)$ , noting that $\tau _{\ge 3}\mathbb {R} f_\ast \mathbb {G}_{m,Y}$ is acyclic in degrees $0$ , $1$ , $2$ , and that $\mathbb {H}(X,-) \circ \mathbb {R} f_\ast \simeq \mathbb {H}(Y,-)$ [Reference WeibelWei94, Corollary 10.8.3]. It follows that (3.3) yields an identification

Let us check that this group contains $\operatorname {Br}_1(Y/X)$ . First, note that the composite

is $0$ because $f^\ast $ factors through the map

from (3.3). In view of the commutative diagram with exact rows

we see that the inclusion $\operatorname {Br}_1(Y/X) \subseteq \mathrm {H}^2(X,\tau _{\le 1} \mathbb {R} f_\ast \mathbb {G}_{m,Y})$ would follow from the injectivity of the canonical map $\mathrm {H}^0(X,\mathbb {R}^2 f_\ast \mathbb {G}_{m,Y}) \to \mathrm {H}^0(\bar {X},\mathbb {R}^2 f_\ast \mathbb {G}_{m,\bar {Y}})$ . Indeed, since $\mathbb {R}^2 f_\ast \mathbb {G}_{m,Y}$ is the sheaf on associated to the presheaf $(X' \to X) \mapsto \operatorname {Br}(Y')$ (where $Y' := Y \times _X X'$ ), any element $a \in \mathrm {H}^0(X,\mathbb {R}^2 f_\ast \mathbb {G}_{m,Y})$ comes from some element $\alpha ' \in \operatorname {Br}(Y')$ for some étale cover $X' \to X$ . If the image of a in $\mathrm {H}^0(\bar {X},\mathbb {R}^2 f_\ast \mathbb {G}_{m,\bar {Y}})$ is $0$ , then there is a finite field extension $K/k$ and an étale cover $X" \to X^{\prime }_K$ such that the image of $\alpha '$ in $\operatorname {Br}(Y")$ is $0$ , where $Y" := Y_K \times _{X_K} X" = Y \times _X X"$ . This means $\alpha '$ is trivialized by the étale cover $X" \to X$ refining $X'$ , hence $a = 0$ .

Proof. Represent C by a complex $[T \to S]$ of k-tori (with T in degree $-1$ ), then $\widehat {C} \simeq [\widehat {S} \to \widehat {T}]$ (with $\widehat {S}$ in degree $-1$ ). We have a commutative diagram

with exact rows. Since $\widehat {S} \cong \mathbb {Z}^{\dim (S)}$ over $\bar {k}$ , we have $\mathrm {H}^1(\bar {X},\widehat {S}) = 0$ . By the Hochschild–Serre spectral sequence $\mathrm {H}^p(k,\mathrm {H}^q(\bar {X},\widehat {S})) \Rightarrow \mathrm {H}^{p+q}(X,\widehat {S})$ , one has $\mathrm {H}^1(k,\widehat {S}) = \mathrm {H}^1(X,\widehat {S})$ and $\mathrm {H}^2(k,\widehat {S}) = \operatorname {Ker}(\mathrm {H}^2(X,\widehat {S}) \to \mathrm {H}^2(\bar {X},\widehat {S}))$ . Similarly, we have $\mathrm {H}^1(\bar {X},\widehat {T}) = 0$ , $\mathrm {H}^1(k,\widehat {T}) = \mathrm {H}^1(X,\widehat {T})$ , and $\mathrm {H}^2(k,\widehat {T}) = \operatorname {Ker}(\mathrm {H}^2(X,\widehat {T}) \to \mathrm {H}^2(\bar {X},\widehat {T}))$ . By a diagram chasing, we check that the map $\mathrm {H}^1(k,\widehat {C}) \to \mathrm {H}^1(X,\widehat {C})$ is injective, and that its image is precisely the kernel of the map $\mathrm {H}^1(X,\widehat {C}) \to \mathrm {H}^1(\bar {X},\widehat {C})$ .

Proof. If G is a torus, this is [Reference Harpaz and WittenbergHW20, Lemme 2.4]. When $G^{\operatorname {ss}} = G^{\operatorname {sc}}$ , we have $C \simeq G^{\operatorname {tor}}$ , and the map $\operatorname {ab}^0$ is just the homomorphism $G(K) \to G^{\operatorname {tor}}(K)$ . By functoriality and the toric case, the result also holds for the case where $G^{\operatorname {ss}} = G^{\operatorname {sc}}$ .

In the general case, it suffices to repeat the argument in [Reference KottwitzKot86, p. 369]. Take any t-resolution

of G and let $\partial : \mathrm {H}^1(k,G) \to \mathrm {H}^2(k,T)$ denote the connecting map. Let $K/k$ be a finite extension trivializing the class $\partial ([Z]) \in \mathrm {H}^2(k,T)$ and let $T_1:=\operatorname {Res}_{K/k}(T_K)$ (where $\operatorname {Res}$ denotes the restriction of scalars à la Weil). Let $i: T \hookrightarrow T_1$ denote the canonical inclusion. By Shapiro’s lemma and [Reference Neukirch, Schmidt and WingbergNSW08, Proposition 1.6.5], the map $i_\ast : \mathrm {H}^2(k,T) \to \mathrm {H}^2(k,T_1) \cong \mathrm {H}^2(K,T)$ takes $\partial ([Z])$ to $0$ . Thus, we obtain a t-resolution

where $H_1$ is the pushforward of H along i, with the property that the connecting map $\mathrm {H}^1(k,G) \to \mathrm {H}^2(k,T_1)$ takes $[Z]$ to $0$ , that is, there exists a k-torsor W under $H_1$ and a $\pi $ -equivariant morphism $W \to Z$ .

Now, we consider the torsors $H_1 \to G$ and $W \to Z$ under $T_1$ , and the fibres $(H_1)_g \to \operatorname {Spec}(K)$ , $W_z \to \operatorname {Spec}(K)$ . Take a finite extension $K'/K$ trivializing the classes $[(H_1)_g], [W_z] \in \mathrm {H}^1(K,T_1)$ . Put $T_2:=\operatorname {Res}_{K'/K}((T_1)_{K'})$ and $T_3:=\operatorname {Res}_{K/k}(T_2) = \operatorname {Res}_{K'/k}(T_{K'})$ , then $(T_3)_K \cong T_2^{[K:k]}$ . Let $j: T_1 \hookrightarrow T_3$ denote the canonical inclusion. By Shapiro’s lemma and [Reference Neukirch, Schmidt and WingbergNSW08, Proposition 1.6.5], the map $j_\ast : \mathrm {H}^1(K,T_1) \to \mathrm {H}^1(K,T_3) \cong \mathrm {H}^1(K,T_2)^{\oplus [K:k]} \cong \mathrm {H}^1(K',T_1)^{\oplus [K:k]}$ takes $[(H_1)_g]$ and $[W_z]$ to $0$ . Thus, we obtain a t-resolution

where $H_3$ denotes the pushforward of $H_1$ along j, with the following properties.

• • There exists a k-torsor $W'$ under $H_3$ such that the map $\mathrm {H}^1(k,H_3) \to \mathrm {H}^1(k,G)$ takes $[W']$ to $[Z]$ .

• • The point $g \in G(K)$ lifts to a point $h \in H_3(K)$ .

• • The point $z \in Z(K)$ lifts to a point $y \in W'(K)$ .

Since the claim in the lemma holds for $W'$ (because $H_3^{\operatorname {ss}} = H_3^{\operatorname {sc}}$ ), it also holds for Z by functoriality.

Proof.

1. (i) Since $\mathrm {H}^3(k,\mathbb {G}_m) = 0$ , the map $\varphi $ from (3.6) is surjective; hence $\varphi ^{\sigma }(\alpha ^{\sigma }) = \varphi (\alpha )$ for some $\alpha \in \operatorname {Br}_1(Z)$ . Take a nonempty open subset $U \subseteq X$ such that the restriction $V:=f^{-1}(U) \xrightarrow {f|_V} U$ is an isotrivial torsor under G. When we replace $f^{\sigma }$ by $f^{\sigma }|_{V^{\sigma }}$ in the sequence (3.5), the map $\delta : \mathrm {H}^1(k,\widehat {C}) \to \mathrm {H}^3(U,\mathbb {G}_m)$ is given by

   
   thanks to Corollary 2.9 and the fact that $[\sigma ] \cup \widehat {c} \in \mathrm {H}^3(k,\mathbb {G}_m) = 0$ (the cup-product being induced by the pairing (2.3)). Thus, this map does not depend on $\sigma $ . It follows that the subgroup $\varphi ^{\sigma }(\operatorname {Br}_1(V^{\sigma }/U)) \subseteq \mathrm {H}^1(k,\widehat {C})$ does not depend on $\sigma $ . Hence, we may write $\varphi ^{\sigma }(\alpha ^{\sigma }) = \varphi ^1(\alpha ^1)$ for some $\alpha ^1 \in \operatorname {Br}_1(V/U)$ . Let us now make the necessary modifications to $\alpha ^1$ and $\alpha $ so that the conditions in the statement are fulfilled. From now on, the argument is the same as in [Reference Harpaz and WittenbergHW20, Proposition 2.3 (i) and (ii)]. Let $q: Z \to \operatorname {Spec}(k)$ denote the structure morphism and let $h:= f \times _k q = f \circ p_1 = f^{\sigma } \circ \pi $ . Then $h: Y \times _k Z \to X$ is a torsor under $G \times _k G^{\sigma }$ (a connected reductive k-group with dual algebraic fundamental complex $\widehat {C} \oplus \widehat {C}$ ). Let us now consider the distinguished triangles (2.6) (deduced from (2.5) and Proposition 2.6) associated with the torsors $f^{\sigma }|_{V^{\sigma }}: V^{\sigma } \to U$ , $h|_{V \times _k Z}: V \times _k Z \to U$ , $f|_{V}: V \to U$ , and $q: Z \to \operatorname {Spec}(k)$ . By functoriality and by Proposition 2.11, we have a commutative diagram
   
   where $\Delta : \widehat {C}_U \to \widehat {C}_U \oplus \widehat {C}_U$ denotes the diagonal map and where $\nabla : \mathbb {G}_{m,U} \oplus \mathbb {G}_{m,U} \to \mathbb {G}_{m,U}$ is the addition map. Now, the exact sequences of the shape (3.4) associated with the rows of the above diagram are given by
   
   Since $\Delta (\varphi ^{\sigma }(\alpha ^{\sigma })) = (\varphi ^{\sigma }(\alpha ^{\sigma }),\varphi ^{\sigma }(\alpha ^{\sigma })) = (\varphi ^1(\alpha ^1),\varphi (\alpha ))$ , the elements $p_1^\ast \alpha ^1 + p_2^\ast \alpha $ and $\pi ^\ast \alpha ^{\sigma }$ have the same image in $\mathrm {H}^1(k,\widehat {C}) \oplus \mathrm {H}^1(k,\widehat {C})$ . By exactness of the middle row, there exists such that , or
   
   On the other hand, since $\varphi ^1 \circ f^\ast = 0$ . Replacing $\alpha ^1$ by , we may assume that $\varphi ^{\sigma }(\alpha ^{\sigma }) + \varphi ^1(\alpha ^1) = \varphi (\alpha )$ and $p_1^\ast \alpha ^1 + p_2^\ast \alpha = \pi ^\ast \alpha ^\sigma $ . It remains to check that this implies $\alpha ^1 \in \operatorname {Br}_{\operatorname {nr}}(Y)$ . Indeed, since $\alpha ^{\sigma } \in \operatorname {Br}_{\operatorname {nr}}(V^\sigma ) = \operatorname {Br}_{\operatorname {nr}}(Y^{\sigma })$ , we have $p_1^\ast \alpha ^1 + p_2^\ast \alpha \in \operatorname {Br}_{\operatorname {nr}}(Y \times _k Z)$ . Let $\xi $ be any codimension $1$ point on a smooth compactification of Y, and let $\eta $ denote the generic point $\xi \times _k Z$ (which is a $k(\xi )$ -torsor under $G_{k(\xi )}$ ). Then $\eta $ is a codimension $1$ point on a smooth compactification of $Y \times _k Z$ , hence $\partial _{\eta }(p_1^\ast \alpha ^1 + p_2^\ast \alpha ) = 0$ . It follows that $\partial _{\eta }(p_1^\ast \alpha ^1) = 0 \in \mathrm {H}^1(k(\eta ),\mathbb {Q}/\mathbb {Z})$ because $p_2^\ast \alpha \in \operatorname {Br}(Y \times _k Z)$ . Since $\xi \times _k Z$ is geometrically integral, the restriction map $\mathrm {H}^1(k(\xi ),\mathbb {Q}/\mathbb {Z}) \to \mathrm {H}^1(k(\eta ),\mathbb {Q}/\mathbb {Z})$ is injective. By functoriality of the residue map [Reference Colliot-Thélène and SkorobogatovCTS21, Theorem 3.7.5], we have $\partial _\xi (\alpha ^1) = 0 \in \mathrm {H}^1(k(\xi ),\mathbb {Q}/\mathbb {Z})$ . Since $\xi $ is arbitrary, we conclude that $\alpha ^1 \in \operatorname {Br}_{\operatorname {nr}}(Y)$ .

   We have shown that $\varphi ^{\sigma }(\operatorname {Br}_{\operatorname {nr}}(Y^{\sigma })) \subseteq \varphi ^{1}(\operatorname {Br}_{\operatorname {nr}}(Y))$ for any torsor $Y \to X$ under any reductive group G and any $\sigma \in \mathrm {Z}^1(k,G)$ . If $\sigma ,\tau \in \mathrm {Z}^1(k,G)$ , then $\varphi ^{\sigma }(\operatorname {Br}_{\operatorname {nr}}(Y^{\sigma })) \subseteq \varphi ^{\tau }(\operatorname {Br}_{\operatorname {nr}}(Y^\tau ))$ (resp. $\varphi ^{\tau }(\operatorname {Br}_{\operatorname {nr}}(Y^\tau )) \subseteq \varphi ^{\sigma }(\operatorname {Br}_{\operatorname {nr}}(Y^{\sigma }))$ ) thanks to a twisting argument by $\tau $ (resp. $\sigma $ ). It follows that $\varphi ^{\tau }(\operatorname {Br}_{\operatorname {nr}}(Y^\tau )) = \varphi ^{\sigma }(\operatorname {Br}_{\operatorname {nr}}(Y^{\sigma }))$ .

2. (ii) It suffices to repeat the proof of [Reference Harpaz and WittenbergHW20, Proposition 2.3 (iii)].

Proof of Theorem 3.4.

For any Galois cocycle $\sigma \in \mathrm {Z}^1(k,G)$ , the subgroup $A \subseteq \operatorname {Br}(X)$ is the preimage of $\operatorname {Br}_{\operatorname {nr}}(Y^{\sigma })$ by $(f^\sigma )^\ast : \operatorname {Br}(X) \to \operatorname {Br}(Y^{\sigma })$ , thanks to Proposition 3.6 (ii). The inclusion

is then obvious, by functoriality of the Brauer–Manin pairing (BM_𝔸 ). Let us take a family $x_{\mathbb {A}} = (x_v)_{v \in \Omega _k} \in X(\mathbb {A}_k)^A$ and show that it can be lifted to $Y^{\sigma }(\mathbb {A}_k)^{\operatorname {Br}_{\operatorname {nr}} (Y^{\sigma })}$ for some $\sigma \in \mathrm {Z}^1(k,G)$ . First, we try to lift $x_{\mathbb {A}}$ to $Y^{\sigma }(\mathbb {A}_k)$ by exploiting the orthogonality to $\operatorname {Ker}(f^\ast )$ . This has already been done in [Reference Cao, Demarche and XuCDX19, Proof of Theorem 3.5]. We present here an alternative proof in the spirit of [Reference Harpaz and WittenbergHW20, Proposition 2.5] for the sake of uniformity.

Twisting by the cocycle provided by Lemma 3.7, we may assume that $x_{\mathbb {A}} \in f(Y(\mathbb {A}_k))$ . Then $x_{\mathbb {A}} \in f^{\sigma }(Y^\sigma (\mathbb {A}_k))$ for any cocycle $\sigma $ such that . For such a cocycle, we define a map

(3.7)

where $y_{\mathbb {A}} = (y_v)_{v \in \Omega _k} \in Y^{\sigma }(\mathbb {A}_k) = Y(\mathbb {A}_k)$ is any lifting of $x_{\mathbb {A}}$ and where $\alpha ^{\sigma } \in \operatorname {Br}_{\operatorname {nr}} (Y^\sigma )$ is such that $\varphi ^\sigma (\alpha ^{\sigma }) = \alpha '$ (which exists by Proposition 3.6 (i)). The map $\varepsilon _\sigma $ is well-defined:

• • It does not depend on the choice of $\alpha ^{\sigma }$ . Indeed, upon replacing $\alpha ^{\sigma }$ by (in view of the exact sequence (3.5)) for some , one has

  
  where the last equality is due to the fact that $f^{\sigma }(y_{\mathbb {A}}) = x_{\mathbb {A}} \in X(\mathbb {A}_k)^A$ .

• • It does not depend on the choice of $y_{\mathbb {A}}$ . Indeed, since $Y \to X$ is a torsor under G, upon replacing $y_{\mathbb {A}}$ by $y_{\mathbb {A}} \cdot g_{\mathbb {A}}:=(y_v \cdot g_v)_{v \in \Omega _k}$ for some $g_{\mathbb {A}} = (g_v)_{v \in \Omega _k} \in \mathbb {G}(\mathbb {A}_k)$ , one has

  
  by Lemma 3.5 applied to each fibre $(f^\sigma )^{-1}(x_v) \to \operatorname {Spec}(k_v)$ . But $\operatorname {loc}_v(\alpha ') = 0$ for all $v \in \Omega _k$ since , hence $\left <y_{\mathbb {A}} \cdot g_{\mathbb {A}}, \alpha ^{\sigma }\right>_{\operatorname {BM}} = \left <y_{\mathbb {A}}, \alpha ^{\sigma }\right>_{\operatorname {BM}}$ .

In what follows, we denote by the global Poitou–Tate pairing induced by (2.3), which is a perfect duality of finite groups (see [Reference DemarcheDem11b, Théorème 5.7]).

Lemma 3.8. Let . Let Z be the k-torsor under $G^{\sigma }$ in Proposition 3.6 (hence $Z(\mathbb {A}_k) \neq \varnothing $ ) and $\varphi $ the map from (3.6). For any (cf. (1.2)) and $z_\Omega \in Z(k_\Omega )$ , one has

Lemma 3.8 allows us to relate the maps $\varepsilon _1$ and $\varepsilon _\sigma $ from (3.7) as follows. Let Z be as above, then $Y^{\sigma } = Y \times _k^G Z$ . Let $\pi : Y \times _k Z \to Y^{\sigma }$ , $p_1: Y \times _k Z \to Y$ and $p_2: Y \times _k Z \to Z$ be the canonical projections. For any , by Proposition 3.6 (i), there are $\alpha \in \operatorname {Br}_1(Z)$ , $\alpha ^1 \in \operatorname {Br}_{\operatorname {nr}}(Y)$ and $\alpha ^{\sigma } \in \operatorname {Br}_{\operatorname {nr}}(Y^{\sigma })$ such that $\varphi (\alpha ) = \varphi ^1(\alpha ^1) = \varphi ^{\sigma }(\alpha ^{\sigma }) = \alpha '$ and $p_1^\ast \alpha ^1 + p_2^\ast \alpha = \pi ^\ast \alpha ^\sigma $ . Take any $z_{\mathbb {A}} \in Z(\mathbb {A}_k)$ , then

Since the group $\mathbb {Q}/\mathbb {Z}$ is divisible, the homomorphism $\varepsilon _1$ can be extended to the whole group . Since the Poitou–Tate pairing is a perfect duality and since the map $\operatorname {ab}^1: \mathrm {H}^1(k,G) \to \mathrm {H}^1(k,C)$ induces a bijection by Proposition 2.4 (iii), there exists such that $\varepsilon _1(\alpha ') = \left <\operatorname {ab}^1([\sigma ]), \alpha '\right>_{\operatorname {BM}}$ for all . By the above calculation, we have $\varepsilon _\sigma = 0$ . Twisting by the cocycle $\sigma $ as above, we may assume that $\varepsilon _1 = 0$ . The rest of the proof is to show that this implies $x_{\mathbb {A}} \in f(Y(\mathbb {A}_k)^{\operatorname {Br}_{\operatorname {nr}}(Y)})$ . As always, we fix an adelic point $y_{\mathbb {A}} \in Y(\mathbb {A}_k)$ lifting $x_{\mathbb {A}}$ , and consider the homomorphism $\varepsilon : \operatorname {Br}_{\operatorname {nr}}(Y) \to \mathbb {Q}/\mathbb {Z}$ given by $\varepsilon (\alpha ^1):=\left <y_{\mathbb {A}},\alpha ^1\right>_{\operatorname {BM}}$ . Then, one has

Hence, $\varepsilon $ factors through a homomorphism . On the other hand, the map $\varphi ^1$ induces an injection . Since $\mathbb {Q}/\mathbb {Z}$ is a divisible group, there exists a map such that $\bar {\varepsilon }' \circ \bar {\varphi }^1 = \bar {\varepsilon }$ , or equivalently, a map $\varepsilon ': \mathrm {H}^1(k,\widehat {C}) \to \mathbb {Q}/\mathbb {Z}$ such that and $\varepsilon ' \circ \varphi ^1 = \varepsilon $ . By Proposition 2.4 (i), one has a commutative diagram with exact rows

Actually, the first column from the left is a complex, which is exact at the term $\mathrm {H}^1(k,\widehat {C})^D $ [Reference DemarcheDem11a, Théorème 2.9]. The arrow on the second column is bijective (Kneser–Harder–Chernousov; see [Reference Platonov and RapinchukPR94, Theorems 6.4 and 6.6]). The other columns are obviously exact. Since $\varepsilon ' \in \mathrm {H}^1(k,\widehat {C})^D$ is mapped to , it comes from an element $c_{\mathbb {A}} = (c_v)_{v \in \Omega _k} \in \mathbb {P}^0(k,C)$ . Its image $\gamma _{\mathbb {A}}$ in $\mathbb {P}^1(k,G^{\operatorname {sc}})$ comes from a unique element $\gamma \in \mathrm {H}^1(k,G^{\operatorname {sc}})$ , whose image $\delta $ in $\mathrm {H}^1(k,G)$ lies in . Since $\operatorname {ab}^1(\delta ) = 0$ , we have $\delta = 1$ by Proposition 2.4 (iii). It follows that $\gamma $ comes from an element $c \in \mathrm {H}^0(k,C)$ . Its localization $\operatorname {loc}_\Omega (c) := (\operatorname {loc}_v(c))_{v \in \Omega _k} \in \mathbb {P}^0(k,C)$ has the same image in $\mathbb {P}^1(k,G^{\operatorname {sc}})$ as that of $c_{\mathbb {A}}$ , hence there exists $g_{\mathbb {A}} = (g_v)_{v \in \Omega _k} \in G(\mathbb {A}_k)$ such that $\operatorname {ab}^0(g_v) = \operatorname {loc}_v(c) - c_v$ for all $v \in \Omega _k$ . Since the first column is a complex, the image of $(\operatorname {ab}^0(g_v))_{v \in \Omega _k}$ in $\mathrm {H}^1(k,\widehat {C})^D$ is the same as that of $-c_{\mathbb {A}}$ , i.e., $-\varepsilon '$ . Let $y_{\mathbb {A}} \cdot g_{\mathbb {A}}:=(y_v \cdot g_v)_{v \in \Omega _k}$ . By virtue Lemma 3.5, we have, for all $\alpha ^1 \in \operatorname {Br}_{\operatorname {nr}}(Y)$ (noting that $\varepsilon = \varepsilon ' \circ \varphi ^1$ ),

We have found a lifting $y_{\mathbb {A}} \cdot g_{\mathbb {A}} \in Y(\mathbb {A}_k)^{\operatorname {Br}_{\operatorname {nr}}(Y)}$ of $x_{\mathbb {A}} \in X(\mathbb {A}_k)^A$ , and this concludes the proof of Theorem 3.4.

Proof. Let $G^{\operatorname {u}}$ be the unipotent radical of G and $G^{\operatorname {red}} := G/G^{\operatorname {u}}$ (which is connected reductive). Letting $Z:=Y/G^{\operatorname {u}}$ , we obtain a torsor $g: Z \to X$ under $G^{\operatorname {red}}$ and a torsor $h: Y \to Z$ under $G^{\operatorname {u}}$ . Now, [Reference SansucSan81, Lemme 1.13] tells us that $\mathrm {H}^1(K, (G^{\operatorname {u}})^{\sigma }) = 1$ for any field extension $K/k$ and any cocycle $\sigma \in \mathrm {Z}^1(k,G)$ and that the map $\mathrm {H}^1(k,G) \to \mathrm {H}^1(k,G^{\operatorname {red}})$ is bijective. For every $[\tau ] \in \mathrm {H}^1(k,G^{\operatorname {red}})$ , there is a unique class $[\sigma ] \in \mathrm {H}^1(k,G)$ lifting $[\tau ]$ , hence the torsor $f^{\sigma }: Y^{\sigma } \to X$ has a factorization $Y^{\sigma } \xrightarrow {h^\sigma } Z^\tau \xrightarrow {g^\tau } X$ . Since $h^{\sigma }$ is a torsor under $(G^{\operatorname {u}})^{\sigma }$ and $\mathrm {H}^1(k(Z^\tau ), (G^{\operatorname {u}})^{\sigma }) = 1$ , it is a stable birational equivalence, hence induces an isomorphism $(h^{\sigma })^\ast : \operatorname {Br}_{\operatorname {nr}}(Z^\tau ) \xrightarrow {\cong } \operatorname {Br}_{\operatorname {nr}}(Y^{\sigma })$ . In particular, A is equal to the preimage of $\operatorname {Br}_{\operatorname {nr}}(Z)$ by the map $g^\ast : \operatorname {Br}(X) \to \operatorname {Br}(Z)$ . In view of Theorem 3.4, it remains to show that $h^{\sigma }(Y^{\sigma }(\mathbb {A}_k)^{\operatorname {Br}_{\operatorname {nr}}(Y^{\sigma })}) = Z^\tau (\mathbb {A}_k)^{\operatorname {Br}_{\operatorname {nr}}(Z^\tau )}$ . The inclusion ‘ $\subseteq $ ’ is obvious. Conversely, let $z_{\mathbb {A}} \in Z^\tau (\mathbb {A}_k)^{\operatorname {Br}_{\operatorname {nr}}(Z^\tau )}$ . Viewing that $\mathrm {H}^1(k_v,(G^{\operatorname {u}})^{\sigma }) = 1$ for all $v \in \Omega _k$ , we may lift $z_{\mathbb {A}}$ to a family $y_{\mathbb {A}} \in Y^{\sigma }(\mathbb {A}_k)$ . Since $\operatorname {Br}_{\operatorname {nr}}(Z^\tau ) \cong \operatorname {Br}_{\operatorname {nr}}(Y^{\sigma })$ , we have $y_{\mathbb {A}} \in Y^{\sigma }(\mathbb {A}_k)^{\operatorname {Br}_{\operatorname {nr}}(Y^{\sigma })}$ by functoriality of the pairing (BM_𝔸 ). The theorem is proved.

Proof.

1. (i) Combine [Reference Colliot-Thélène and SkorobogatovCTS13, Lemma 1.3] and [Reference DebarreDeb01, Corollary 4.18].

2. (ii) Let $X^c$ be a smooth compactification of X (so that $\operatorname {Br}_{\operatorname {nr}}(X) = \operatorname {Br}(X^c)$ ) and apply [Reference Colliot-Thélène and SkorobogatovCTS00, Proposition 1.1] (which is a consequence of Harari’s formal lemma) to the inclusion $X \subseteq X^c$ . This result says that $\overline {X(\mathbb {A}_k)^A} = X^c(k_\Omega )^{\operatorname {Br}_{\operatorname {nr}}(X)}$ in $X^c(k_\Omega )$ , a fortiori $\overline {X(\mathbb {A}_k)^A} = X(k_\Omega )^{\operatorname {Br}_{\operatorname {nr}}(X)}$ in $X(k_\Omega )$ .

Proof. The inclusion ‘ $\supseteq $ ’ is obvious by functoriality and continuity of the pairing (BM_nr ), so let us show the inclusion ‘ $\subseteq $ ’. Let $A \subseteq \operatorname {Br}(X)$ denote the preimage of $\operatorname {Br}_{\operatorname {nr}}(Y)$ by the map $f^\ast : \operatorname {Br}(X) \to \operatorname {Br}(Y)$ . By Theorem 3.9, we have

Thus, it remains to show that $X(k_\Omega )^{\operatorname {Br}_{\operatorname {nr}}(X)} = \overline {X(k_\Omega )^A}$ . In view of Lemma 3.10 (ii), we are reduced to show that the group $A/\operatorname {Br}(k)$ is finite. Since X is rationally connected, it follows from the Graber–Harris–Starr theorem [Reference Graber, Harris and StarrGHS03] that Y is also rationally connected (see [Reference DebarreDeb03, Corollaire 3.1]). By Lemma 3.10 (i), the group $\operatorname {Br}_{\operatorname {nr}}(Y)/\operatorname {Br}(k)$ is finite. Thus, to establish the finiteness of $A/\operatorname {Br}(k)$ , it is enough to show that $\operatorname {Ker}(\operatorname {Br}(X) \to \operatorname {Br} (Y))$ is finite. Using Sansuc’s units-Picard-Brauer exact sequence [Reference SansucSan81, (6.10.1)], we are reduced to prove the finiteness of $\operatorname {Pic}(G)$ . The same sequence tells us that this property is stable under extensions of groups. Hence, we may deal separately with the following cases.

• • If G is unipotent, then $\operatorname {Pic}(G) = 0$ since the k-variety G is isomorphic to an affine space.

• • If G is a torus, then $\operatorname {Pic}(G) = \mathrm {H}^1(k,\widehat {G})$ by [Reference SansucSan81, Lemme 6.9 (ii)]. Let $K/k$ be a finite Galois extension splitting G, then $\mathrm {H}^1(K,\widehat {G}) = 0$ , so $\mathrm {H}^1(k,\widehat {G}) = \mathrm {H}^1(\operatorname {Gal}(K/k),\widehat {G})$ by the inflation-restriction sequence. This latter is torsion and finitely generated, hence finite.

• • If G is semisimple, by [Reference SansucSan81, Lemme 6.9 (iii)], $\operatorname {Pic}(G)$ injects into the character module of the algebraic fundamental group $\pi _1(\bar {G}) = \operatorname {Ker}(\bar {G}^{\operatorname {sc}} \to \bar {G})$ , which is finite.

Proof. By Lemma 2.1 (iii), two X-torsors of descent type $\lambda $ differ by twisting by a Galois cycle. Hence, deducing the present theorem from Theorem 3.11 amounts to showing that torsors of descent type $\lambda $ exist whenever $X(k_\Omega )^{\operatorname {Br}_{\operatorname {nr}}(X)} \neq \varnothing $ . To this end, the condition is in fact sufficient, where the subgroup was defined in (1.2). Indeed, this has been proved in [Reference LinhLin25, Proposition 3.10].

Proof. Let G be a connected linear algebraic group over k and let X be a homogeneous space of G. Fix a $\bar {k}$ -point of X and let $\bar {H} \subseteq \bar {G}$ be its stabilizer, which we assume to be connected. As in [Reference Harari and SkorobogatovHS02, Proposition 3.3], the subgroup $E_X \subseteq \operatorname {SAut}(\bar {G}/X)$ consisting of semilinear s-automorphisms (for some $s \in \Gamma _k$ ) for which for all $x \in X(\bar {k})$ and $g \in G(\bar {k})$ fits in an exact sequence

of topological groups, which defines a k-kernel $L_X$ as well as a descent type $\lambda = [\bar {G},E_X]$ on X bound by $L_X$ . The torsors $Y \to X$ of descent type $\lambda $ (if they exist) are left torsors under G (they are k-forms of the variety G twisted by cocycles with values in $\bar {H}(\bar {k})$ ). According to [Reference SansucSan81, Corollaires 8.7 and 8.13], these torsors satisfy $\mathbf {(BMHP)}$ and $\mathbf {(BMWA)}$ . In fact, Sansuc’s original result concerning the property $\mathbf {(BMHP)}$ requires the absence of $E_8$ factors, since his proof used the result of Kneser–Harder (see Théorème 4.2 in loc. cit.) which excluded the $E_8$ case. This problem, however, was later resolved by Chernousov [Reference ChernousovChe89] (see also [Reference Platonov and RapinchukPR94, Theorems 6.4 and 6.6]). It follows from Theorem 3.12 that the properties $\mathbf {(BMHP)}$ and $\mathbf {(BMWA)}$ hold for X.