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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2014
   • (edited May 22nd 2014)
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   Author: Urs
   Format: MarkdownItexI 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?)

   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?)

2. • CommentRowNumber2.
   • CommentAuthorhilbertthm90
   • CommentTimeMay 22nd 2014
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   Author: hilbertthm90
   Format: MarkdownItexI 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 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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2014
   • (edited May 22nd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks 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-truncated|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](Brauer+group#RelationToEtaleCohomology) 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]] $$ \mathbf{Br}(X) \coloneqq [X,\mathbf{B}^2 \mathbb{G}_m] $$ 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). (?)

   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

   $$\mathbf{Br}(X) \coloneqq [X,\mathbf{B}^2 \mathbb{G}_m]$$

   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). (?)

4. • CommentRowNumber4.
   • CommentAuthorhilbertthm90
   • CommentTimeMay 22nd 2014
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   Author: hilbertthm90
   Format: MarkdownItexYes. 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2014
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   Author: Urs
   Format: MarkdownItexOkay, thanks. Regarding non-torsion: I guess that is called the "bigger Brauer group".

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