Okay, thanks. Regarding non-torsion: I guess that is called the “bigger Brauer group”.

]]>Yes. I rarely think in these terms, but this is what I had in mind. It has just always seemed “obvious” that considering the relation between the Picard and Brauer groups, there would be a natural generalization to the stack version (I had in mind defining the psuedofunctor explicitly $U \mapsto$ groupoid of Azumaya algebras over U or something). The mapping stack seems much more clean though.

]]>Thanks for feedback,I was wondering about this. So I have now written at *Brauer stack* the following, please let me know if you’d agree with that or else what you would want to change or add:

It is traditional to speak, for a suitable scheme $X$, of its *Picard group* and of its *Brauer group*. Moreover, it is a classical fact that under suitable conditions the former admits itself a canonical geometric structure that makes it the *Picard scheme* of $X$. Still well known, if maybe less commonly highlighted, is that this is just the 0-truncation of the Picard stack of $X$, which is simply the mapping stack $[X, \mathbf{B}\mathbb{G}_m]$ into the delooping of the multiplicative group. In this form this applies immediately also to more general context such as E-∞ geometry ("spectral geometry") and gives a concept of *Picard ∞-stack* ("derived Picard stack").
Given this and the relation of the Brauer group to étale cohomology it is clear that the Brauer group similarly arises as the torsion subgroup of the 0-truncation of the ∞-stack which *ought* to be called the **Brauer stack**, given as the mapping stack

into the second delooping of the multiplicative group (modulating line 2-bundles). Indeed, just as the Picard stack turns under Lie integration (evaluation on infinitesimally thickened points) and 0-truncation into what is commonly called the *formal Picard group*, so this Brauer $\infty$-stack similarly gives what is commonly called the *formal Brauer group*.

However, while therefore the terminology "Brauer stack" is the evident continuation of a traditional pattern (which in the other direction continues with the group of units and the mapping scheme $[X,\mathbb{G}_,]$), it seems that this terminology has never been introduced in the literature (at time of this writing). (?)

]]>I see you created “Brauer stack” as well. I’ve wanted to use this term several times, but couldn’t find it anywhere so I’ve always been worried that it exists somewhere under a different name.

]]>I started a separate page for *Picard stack* (which used to be just a redirect to *Picard scheme*), stated the general nonsense idea with a pointer to Lurie’s thesis, where this essentially appears.

(BWT, where in the DAG series did this end up? I forget.)

Of course the upshot is that it’s simply the internal hom/mapping stack $\mathbf{Pic}(X) = [X,\mathbf{B}\mathbb{G}_m]$. I have a question here: it seems clear that the higher versions $[X, \mathbf{B}^k \mathbb{G}_m]$ want to be called the higher intermediate Jacobians (their deformation theory at 0 are the Artin-Mazur formal groups). Why does nobody say this? (Or if they do, where?)

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