# Cohomological Brauer group vs classical

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.

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?

• From the same wikipedia page you link: "The Brauer group is always a subgroup of the cohomological Brauer group. Gabber showed that the Brauer group is equal to the cohomological Brauer group for any scheme with an ample line bundle (for example, any quasi-projective scheme over a commutative ring)." The map from the Brauer group to the cohomological Brauer group is described, for fields, a little higher up on that Wikipedia page. The idea in etale cohomology is similar. Full details are given in Milne's etale cohomology group, for example. Jul 11, 2020 at 20:58
• See section 3.3 of wwwf.imperial.ac.uk/~anskor/brauer.pdf
– RP_
Jul 11, 2020 at 21:03
• This is explained in great details in the Bourbaki lectures of Grothendieck, reprinted in "10 exposés sur la cohomologie des sch\'emas" (North-Holland).
– abx
Jul 12, 2020 at 5:28

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.