Cohomological Brauer group vs classical

Question

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?

Answer 1

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.