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Let $X$ be a quasicompact quasiseparated scheme over a field $k$. The connection between Azumaya algebras over $X$ and $\mathbb G_{\mathrm{m}}$-gerbes over $X$ is well-known: there exists an injection of groups $$\mathrm{Br}(X)\to \mathrm{H}^2(X,\mathbb G_{\mathrm{m}})$$ where the lhs is the group of Azumaya algebras on $X$ up to Morita equivalence.

In Derived Azumaya algebras and generators for twisted derived categories, Bertrand Toën extended the lhs to the derived Brauer group, i.e. the group of derived Azumaya algebras up to Morita equivalence. In this setting there is a surjection $$\operatorname{dBr}(X)\to \operatorname{H}^2(X,\mathbb G_{\mathrm{m}})$$ which however is an isomorphism when $X$ is normal.

Since $\mathrm{H}^2(X,\mathbb G_{\mathrm{m}})$ classifies $\mathbb G_{\mathrm{m}}$-gerbes over $X$, we conclude that $\mathbb G_{\mathrm{m}}$-gerbes over $X$ are the same thing as derived Azumaya algebras over $X$ up to Morita equivalence, at least in the case when $X$ is normal.

I like to point out (but perhaps this picture is conceptually less relevant than what I think) that in this procedure the higher categorical structure inherent to the notion of gerbe (namely, the fact that it is a stack in groupoids over $X$, and not just in sets) is converted into a different kind of "structure", namely an algebra structure on something living at a lower categorical level, i.e. the associated (derived) Azumaya algebra (which is now an ordinary sheaf of $k$-algebras). I stress the "algebra structure" part and not the "derived" part since there are, after all, nontrivial gerbes whose associated Azumaya algebra is classical.

Now my question is: if we let $G$ be an arbitrary reductive group over $k$ (the field of complex numbers is a perfectly nice example for my purposes), there is a well-defined notion of $G$-gerbe (see Giraud, Cohomologie non abélienne, and Breen, On the classification of 2-gerbes and 2-stacks). Is there any reinterpretation of this notion in terms of "generalized" (derived) Azumaya algebras, following the same principle of converting higher categorical structures into additional algebraic structures on categorically "lower" objects?

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    $\begingroup$ For $G$ a commutative group scheme, you can certainly consider the set of pairs of a central extension of reductive group schemes, $1\to G\to H\to K\to 1$ together with a $K$-torsor. If you want to hew closer to your original example, you can insist that $K$ is semisimple of type $A_n$. In the case of the Brauer group, you even insist that $K$ is $\textbf{PGL}_n$ for some integer $n$, so that a $K$-torsor is equivalent to a rank $n$ Azumaya algebra. None of this works, of course, when $G$ is not commutative. $\endgroup$
    – Jason Starr
    Commented Apr 19, 2022 at 10:39
  • $\begingroup$ @JasonStarr so you are suggesting that to any $G$-gerbe on $X$, $G$ commutative, one can associate such a central extension and a $K$-torsor? $\endgroup$
    – W.Rether
    Commented Apr 19, 2022 at 16:23
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    $\begingroup$ Yes, I am suggesting that for those schemes $S$ such that the "cohomological Brauer group" equals the "Brauer group" (e.g., the ones studied by Gabber and de Jong), then every $G$-gerbe for a group $G$ of multiplicative type should arise from a $K$-torsor and a central extension of $K$ by $G$. I am not certain I know how to prove that though (it might follow from Gabber - de Jong using Weil restriction relative to a cover that "splits" the multiplicative group scheme $G$). $\endgroup$
    – Jason Starr
    Commented Apr 19, 2022 at 16:27
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    $\begingroup$ Of course then that raises exactly the question in your post: if $G$ is a group of multiplicative type over a scheme $S$ that is more general than those in the Gabber -- de Jong work, what is the correct extension of "derived Azumaya algebras" that will lead to a surjective map to the cohomological Brauer group. $\endgroup$
    – Jason Starr
    Commented Apr 19, 2022 at 16:29

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