Let $X$ be a smooth scheme over $\mathbb{C}$.
A $O\_X$-algebra $A$ is called Azumaya algebra on $X$
if locally it's ismorphic to matrix algebra: ie for
every $p \in X$ there exist open $U \subset X$ with
$p \in U$ and $A \vert \_U \cong Mat\_{r}(O\_U) $
for some rank $ r >0 $.


Two Azumaya algebra
$A$ & $B$ are *equivalent* iff there exist two locally free $O\_X$-modules $E,F$ locally of finite rank
with $A \otimes End(E) \cong B \otimes End(F)$ (as Asumaya algebras).
We call $Br(X)$ as the Brauer group defined as
the set of isomorphism classes of Azumaya
algebras modulo the described equivalence relation.


We want to draw analogy of the isomorphism
$Pic(X) \cong H^1(X, O\_X^\*)$ and endow $H^(X,O\_X^\*)$ with a
intepretation classifying isomorphy classes
of intersting geometric objects as well.


In
<https://en.wikipedia.org/wiki/Brauer_group#The_Brauer_group_of_a_scheme>


is remarked that for quasi-compact scheme $X$ the torsion subgroup of
the étale cohomology group $H^2\_{et}(X, O\_X^\*)$ is called
the *cohomological Brauer group*.


*Question*: Is there an explicit way to relate $Br(X)$ with
torsion group of $H^2\_{et}(X, O\_X^\*)$. A morphism from $Br(X)$ to this group ? If yes,
how this construction work? Could anybody sketch the idea or give a reference
where this construction (if it exist) is explaned?



Cohomological Brauer group vs classical
Briefly, one uses the exact sequence
$$
H^{1}(X,GL\_{n})\rightarrow H^{1}(X,PGL\_{n})\rightarrow H^{2}(X,\mathbb{G}\_{m})
$$
(etale cohomology). The set $H^{1}(X,PGL\_{n})$ classifies the isomorphism classes of Azumaya
algebras of degree $n^{2}$ over $X$, the set $H^{1}(X,GL\_{n})$ classifies the
isomorphism classes of
locally free $\mathcal{O}\_{X}$-modules of rank $n$, and the first map sends $V$ to
$End(V)$. See the references in the comments.


