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 $X(\mathbb{A}_K)^F=\bigcap_{A\in F(K)} X(\mathbb{A}_K)^A\supseteq X(K)$.

Unfinished.

References