The Brauer-Manin obstruction on algebraic stacks

Let $k$ be a number field, $R$ a commutative $k$-algebra, $\mathcal{X}$ an algebraic stack over $k$. The $R$-points of $\mathcal{X}$ is the set of isomorphism classes of objects in the groupoid $\operatorname{Hom}_{{\mathbb{C}\text{hp}}/k}(R,\mathcal{X})$ denoted by $\mathcal{X}(R)$. Let $\mathcal{X}(\mathbf{A}_{k})$ (resp. $\mathcal{X}(k)$) be the adèlic points (resp. rational points) of $\mathcal{X}$.

For a stable functor $F:({\mathbb{C}\text{hp}}/k)^{\text{op}}\rightarrow{Set}$ and $A\in F(\mathcal{X})$, as the classical case for varieties in Poonen [Poo17, 8.1.1], one may also define the obstruction given by $A$ to be the subset $\mathcal{X}(\mathbf{A}_{k})^{A}$ of $\mathcal{X}(\mathbf{A}_{k})$ whose elements are characterized by

(which is well-defined by stability of $F$), and the $F$-set (or $F$-obstruction) to be the subset $\mathcal{X}(\mathbf{A}_{k})^{F}$ whose elements are characterized by

Then we have the map induced by $q$,

Anyway, by abusing notation we may let $\mathcal{X}(k)$ be its image in $\mathcal{X}(\mathbf{A}_{k})$. Thus we may write $\mathcal{X}(k)\subset\mathcal{X}(\mathbf{A}_{k})^{F}$.

For any $\mathcal{T}\in{\mathbb{C}\text{hp}}/S$ where $S$ is a scheme, denote by $\mathcal{T}_{\text{\'{e}t}}$ (resp. $\mathcal{T}_{\text{fppf}}$) the site with topology inherited from the big étale site $({Sch}/S)_{\text{\'{e}t}}$ (resp. the big fppf site $({Sch}/S)_{\text{fppf}}$), cf. [Sta22, Tags 067N and 06TP].

Let $\tau\in\{\text{\'{e}t},\text{fppf}\}$ and $f:{\mathcal{T}^{\prime}}\rightarrow\mathcal{T}$ be a $1$-morphism in ${\mathbb{C}\text{hp}}/S$. Then compositing with $f$ gives the functor $f^{*}:{PSh}(\mathcal{T})\rightarrow{PSh}(\mathcal{T}^{\prime})$, which has a right adjoint $f_{*}$. Both $f^{*}$ and $f_{*}$ send $\tau$-sheaves to $\tau$-sheaves, and define a morphism of topoi

which is also a pair of adjoint functors with ${Sh}$ replaced by ${Ab}$ (or ${SGrp}$ the category of sheaves of groups) and is compatible with the classical case when $\mathcal{T}$ and $\mathcal{T}^{\prime}$ are represented by $S$-schemes, cf. [Sta22, Tags 06TS and 075J].

We have an adjoint pair

between derived categories of complexes of Cartesian$\tau$-sheaves, c.f. [Sta22, Tag 073S].

We also have

Then we have the cohomology functors

We will write $H_{\tau}^{i}(\mathcal{T},-)=H^{i}(\mathcal{T}_{\tau},-)$. For $T\in{Sch}/S$, $H_{\tau}^{i}(T,-)$ recovers the $\tau$-cohomology for schemes over $S$. See [Sta22, Tag 075F].

Now we consider the case where $S=k$ a number field, and fix $\mathscr{G}\in{Ab}(({Sch}/k)_{\tau})$ and let $\mathscr{G}_{\mathcal{T}}$ be the pullback of $\mathscr{G}$ to $\mathcal{T}$. Then the functor $H^{i}(-,\mathscr{G}_{-}):({\mathbb{C}\text{hp}}/k)^{\text{op}}\rightarrow{Set}$ is stable (c.f. [Lv21, 2.29]). If there is no other confusion, we simply write $\mathscr{G}\in{Ab}(\mathcal{T}_{\tau})$ for $\mathscr{G}_{\mathcal{T}}$. Thus we can take $F=H^{i}(-,\mathscr{G})$ in 2.1 and obtain $\mathcal{X}(\mathbf{A}_{k})^{H^{i}(\mathcal{X},\mathscr{G})}$, the $H^{i}(-,\mathscr{G})$-set.

Fix an base scheme $S$. Let $\mathcal{X}\in{\mathbb{C}\text{hp}}/S$ and $\mathscr{G}$ an sheaf of group on $\mathcal{X}_{\tau}$, where $\tau\in\{\text{\'{e}t},\text{fppf}\}$.

By [Ols16, Ex. 2.H], we may assume $\mathcal{X}$ is an object of $\mathcal{X}_{\tau}$. Then there is a one-to-one correspondence (c.f. Giraud [Gir71, III.3.6.5 (5)]) of pointed sets

In the case of $\mathscr{G}\in{Ab}(\mathcal{X}_{\tau})$, we have a bijection (c.f. [Gir71, IV.3.4.2 (i)])

which maps the neutral element to $0$ and is functorial both in $\mathcal{X}$ and $\mathscr{G}$.

Since in particular a sheaf $\mathscr{Y}$ on $\mathcal{X}_{\tau}$ is a stack over $\mathcal{X}_{\tau}$, one obtain a morphism of stacks $\mathscr{Y}\rightarrow\mathcal{X}$ over $S$ (c.f. [Sta22, Tag 09WX]). If $f:\mathcal{Z}\rightarrow\mathcal{X}$ is a morphism of algebraic $S$-stacks, then obviously the pullback $\mathscr{Y}_{\mathcal{Z}}=\mathcal{Z}\times_{\mathcal{X}}\mathscr{Y}$ is also a $\mathscr{G}$-torsor. It is the pullback sheaf of $\mathscr{Y}$ under $f^{*}:{Sh}(\mathcal{X}_{\tau})\rightarrow{Sh}(\mathcal{Z}_{\tau})$. This operation corresponds to the cohomological pullback

In the following, if not otherwise explained, a $\mathscr{G}$-torsor over $\mathcal{X}$ means a $\mathscr{G}$-torsor over $\mathcal{X}_{\text{fppf}}$.

For an algebraic stack $\mathcal{X}$ over $k$ and a linear $k$-group $G$, contracted products (resp. twists) of $G$-torsors over $\mathcal{X}$ can also be defined. See [Lv21, 3.24 (resp. 3.30)].

Let $f:\mathcal{Y}\xrightarrow{G}\mathcal{X}$ be a torsor. Since the functor $\check{H}_{\text{fppf}}^{1}(-,G):({\mathbb{C}\text{hp}}/k)^{\text{op}}\rightarrow{Set}$ is also stable by [Lv21, 3.14], we may define $\mathcal{X}(\mathbf{A}_{k})^{f}\subseteq\mathcal{X}(\mathbf{A}_{k})$ to be the obstruction set given by $[\mathcal{Y}]$ (c.f. (2.2)). It contains $\mathcal{X}(k)$ and satisfies

where $f_{\sigma}:\mathcal{Y}_{\sigma}\rightarrow\mathcal{X}\in\mathsf{Tors}(\mathcal{X},G_{\sigma})$ is the twist of $\mathcal{Y}$ by $\sigma$. See [Lv21, 3.20] for a proof.

Again, if $\mathcal{X}$ is a $k$-scheme, the definitions and facts of this subsection coincide with classical ones.

Sansuc [San81, Prop. 6.10] gave an exact sequence, and now we generalize it to quotient stacks.

In general for a torsor $\mathcal{X}\xrightarrow{G}\mathcal{Y}$ where $\mathcal{Y}\in{\mathbb{C}\text{hp}}/k$ and $G$ a smooth $k$-group, let $\rho,\ p_{2}:\mathcal{X}\times_{k}G\rightarrow\mathcal{X}$ be as in Theorem 3.1, and $\mathfrak{p}_{1}:\mathcal{X}\times_{k}G\rightarrow G$ be the projection to $G$. Following [Cao18, Def. 3.1], we may also define the invariant Brauer subgroup to be

The next statement extends [Cao18, Thm. 3.10].

Consider

where $D(k)$ is the derived category of complexes of $k$-modules. Let $S$ be a $k$-group of multiplicative type. Then its Cartier dual $\hat{S}$ is a finitely generated $k$-module. Let $\operatorname{KD}^{\prime}(\mathcal{X})$ be the cone of $\mathbf{G}_{m}[1]\rightarrow Rp_{*}\mathbf{G}_{m}[1]$ in $D(k)$.

Let $a\in H^{1}(k,\hat{S})$. As shown in the above proof, for $\mathcal{X}$ being an algebraic stack over $k$, the diagram in [Lv22, Rkm. 2.7 (2)] is still correct. We shall use a part of this diagram

Let $f:\mathcal{Y}\rightarrow\mathcal{X}$ be an $S$-torsor. Define $\lambda=\chi([f])$, and

Suppose that $\mathcal{X}$ is of finite type over $k$. Then there exists a finite set of places $S$, such that $\mathcal{X}$ extends to an integral model $\mathscr{X}$ of $\mathcal{X}$ over $\mathscr{O}_{k,S}$ (that is, a $\mathscr{O}_{k,S}$-stack $\mathscr{X}$ such that $\mathscr{X}\times_{\mathscr{O}_{k,S}}k\xrightarrow{\approx}\mathcal{X}$, and this can be shown by looking at a smooth groupoid presentation of $\mathcal{X}$ ). Moreover, by [Chr20, Sec. 13], we have

In particular, any $x\in\mathcal{X}(\mathbf{A}_{k})$ can be viewed as $(x_{v})_{v}\in\prod_{v\in S^{\prime}}\mathscr{X}(k_{v})\times\prod_{v\not\in S^{\prime}}\mathscr{X}(\mathscr{O}_{v})$ for some finite $S^{\prime}\supseteq S$. Then one can imitate the proof of [Poo17, 8.2.1], to show that the Brauer-Manin pairing for $\mathcal{X}$

is well-defined, and the Brauer-Manin set $\mathcal{X}(\mathbf{A}_{k})^{\operatorname{Br}}$ coincides with the classical definition using $\langle-,-\rangle_{\text{BM}}$ since $\operatorname{Br}\mathbf{A}_{k}\rightarrow\bigoplus_{v}\operatorname{Br}k_{v}$ is an isomorphism.

Define

where $\mathcal{X}_{v}=\mathcal{X}\times_{k}k_{v}$. For $A\in\cyrille B\mathcal{X}$ and $(x_{v})\in\mathcal{X}(\mathbf{A}_{k})$, one checks that

does not depend on the choice of $(x_{v})$. Assuming $\mathcal{X}(\mathbf{A}_{k})\neq\emptyset$, we obtain a well-defined map $i=\langle(x_{v}),-\rangle_{\text{BM}}:\cyrille B\mathcal{X}\rightarrow\mathbb{Q}/\mathbb{Z}$. This notation will be used in Subsection 7.2.

Since for a (resp. Commutative) $k$ group $G$, $\check{H}_{\text{fppf}}^{1}(-,G)$ (resp. $H_{\text{\'{e}t}}^{2}(-,G)$) is stable, the following definitions also make sense.