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Let $K$ be a field, we know elements in $H^2(K,\mathbb{G}_m)=\mathrm{Br}(K)$ can be represented by division algebras over $K$.

Do we have some description of elements in $H^2(K,GL_n)$ for $n>1$? They should give elements in $\mathrm{Br}(K)$ via taking determinant.

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    $\begingroup$ What is $H^2(K,GL_n)$ for you? $\endgroup$
    – abx
    Jul 13, 2018 at 16:52
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    $\begingroup$ @QiaochuYuan Right, sorry I asked as stupid question...but for $\mathbb{G}_m$, cohomology class in $H^2(K,\mathbb{G}_m)$ represents a gerbe, which is a stack locally isomorphic to $B\mathbb{G}_m$, do we have similar notion for stacks locally isomorphic to $BGL_n$? $\endgroup$
    – user39380
    Jul 13, 2018 at 18:00
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    $\begingroup$ Giraud worked out a theory of non-abelian $H^2$ in his book, which is possibly what you are after. $\endgroup$ Jul 13, 2018 at 20:00
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    $\begingroup$ @Qixiao: stacks locally isomorphic to $BG$ are classified by $H^1(-, \text{Aut}(BG))$ where $\text{Aut}(BG)$ is a $2$-group with $\pi_0$ given by $\text{Out}(G)$ and $\pi_1$ given by $Z(G)$. This is cohomology but with coefficients in a $2$-group rather than a group, and calling it nonabelian $H^2$ is in my opinion highly misleading. Among other things, it does not specialize to abelian $H^2$ when $G$ is abelian: $H^2(-, A)$ for $A$ abelian is given by Picard stacks (symmetric monoidal grouplike stacks) locally isomorphic to $BA$. $\endgroup$ Jul 13, 2018 at 20:28
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    $\begingroup$ Nonabelian $H^2$ in the Galois-cohomological setting was developed by Springer. See my preprint for references and definitions. $\endgroup$ Jul 20, 2018 at 16:29

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