Brauer Group III

We continue with Part II of our discussion of the Brauer group. In 1970, Manin explained how elements of $\mathrm{Br}(X)$ produces obstructions to local-global principles on $X$ in ⟦cite:Man71⟧ . On the other hand, it was shown that a torsor under an algebraic group $G$ over $X$ could give rise to an obstruction. In this post, we will explain these obstructions, and all related concepts. We will still be following ⟦cite:Poo17⟧ , mostly chapter 8.

Obstructions from functors

Let $K$ be a global field and $F:\mathbf{Sch}^{\mathrm{op}}\rightarrow\mathbf{Set}$ be a functor. Let $A\in F(L)$ for a $K$-algebra $L$, there is a map $\mathrm{ev}_A:X(L)\rightarrow F(L)$, by sending a point $x\in X(L)$ to the image of $A$ of the map $F(X)\rightarrow F(L)$ induced by $x$. Let $X(\mathbb{A}_K)^A$ be the adelic points of $X$ whose image under $\mathrm{ev}_A$ is contained in the image of $F(K)\rightarrow F(\mathbb{A}_K)$. Define $X(\mathbb{A}_K)^F=\bigcap_{A\in F(K)} X(\mathbb{A}_K)^A\supseteq X(K)$. The diagram

commutes.

If $X(\mathbb{A}_K)\ne \emptyset$ but $X(\mathbb{A}_K)^F=\emptyset$, we say that there is an obstruction to the local-global principle on $X$ coming from $F$, in which case $X$ has no $K$-rational points. Take $F=\mathrm{Br}$, then the set $X(\mathbb{A}_K)^{\mathrm{Br}}$ is called the Brauer-Manin set of $X$, and the obstruction is called the Brauer-Manin obstruction. Take $F=H^1(-,G)$ for an algebraic group $G$, then the obstruction is called the descent obstruction coming from $G$.

We have $X(\mathbb{A}_K)\subseteq X(\prod_{v}K_v)=\prod_{v}X(K_v)$ (if $X$ is proper then all three are the same). We have a similar commutative diagram.

Define $X(\prod_v K_v)^F=\bigcap_{A\in F(K)} X(\prod_v K_v)^A\subseteq X(\prod_v K_v)$ similarly.

If $X(\prod_v K_v)^F\subsetneq X(\prod_v K_v)$, we say that there is an obstruction to the local-global principle on $X$ coming from $F$, in which case $X(K)$ is not dense in $X(\prod_v K_v)$.

References

[Poo17]Bjorn Poonen. Rational Points on Varieties. American Mathematical Society. 2017.
[Man71]Y. I. Manin. Le groupe de Brauer-Grothendieck en géométrie diophantienne. In Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1. pp. 401--411. Gauthier-Villars. 1971.