This question may be very elementary, sorry but I would like to ask this question.
Brauer group of local ring $(R,m)$ is defined to be a group of equivalent classes of Azumaya algebra over $R$. Local ring $R$ is Azumaya algebra if only if there exists some $n$ and $A \cong R^n$ and $A/mA$ is central simple algebra over $R/m$, and equivalent relation is given by $A~B$ defined as there exists some $m,n$ such that $A \otimes M_m(R)\cong B \otimes M_n(R)$.
Let $K$ be a number field. $X/K$ be an algebraic variety over $K$. Let $O_{X,P}$ be a local ring of $X$ at $P$. Let $Br(X)=\bigcap_{P \in X(\overline{K})} Br(O_{X,P})$ be a Brauer group of $X$.
Then, why $Br(Spec(K))=Br(K)$ holds ?
The latter is $Br(0)$ I think, and I wonder such a equation holds.
Relatedwith this one, https://mathoverflow.net/questions/445304/what-is-a-definition-of-ap-v-in-the-definition-of-brauer-manin-obstruction