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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2014
   • (edited May 22nd 2014)
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   Author: Urs
   Format: MarkdownItexOver in the _[thread on "Picard infinity-stack"](http://nforum.mathforge.org/discussion/5968/picard-infinitystack/?Focus=47317#Comment_47317)_ we turned to discussion of _[[Brauer stack]]_. Just for completeness I should probably make this a separate thread here: I had created _[[Brauer stack]]_ for the moment only with the following Idea-section *** 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). (?)

   Over in the thread on “Picard infinity-stack” we turned to discussion of Brauer stack. Just for completeness I should probably make this a separate thread here: I had created Brauer stack for the moment only with the following Idea-section

   ---

   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). (?)

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 22nd 2014
   • (edited May 22nd 2014)
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   Author: David_Corfield
   Format: MarkdownItexDo you have a general way to refer to later members of the sequences Picard X, Brauer X, ..., for X = group, stack, etc.?

   Do you have a general way to refer to later members of the sequences Picard X, Brauer X, …, for X = group, stack, etc.?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI would just call them _moduli $\infty$-stacks of line $n$-bundles_. I think this is a typical case of a field with a very long history, where concepts did not develop top-down in the most straightforward general abstract way, but bottom-up by a long history of feeling one's way. That's why there is all this terminology for what abstractly is really one single concept. But I really would like to get all the conections to the trational story sorted out. As hilbertthm90 also said: while the words "Brauer stack" seem not to be in the literature, it seems almost inconceivable that nobody considered this, but probably under yet another unsuggestive name! :-)

   I would just call them moduli $\infty$-stacks of line $n$-bundles. I think this is a typical case of a field with a very long history, where concepts did not develop top-down in the most straightforward general abstract way, but bottom-up by a long history of feeling one’s way. That’s why there is all this terminology for what abstractly is really one single concept. But I really would like to get all the conections to the trational story sorted out. As hilbertthm90 also said: while the words “Brauer stack” seem not to be in the literature, it seems almost inconceivable that nobody considered this, but probably under yet another unsuggestive name! :-)

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 22nd 2014
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   Author: David_Corfield
   Format: MarkdownItexA very cursory search didn't give me an answer to your question, but I see [here](http://staff.science.uva.nl/~heinloth/brauer.pdf) the Brauer group concept (or some 'bigger' version) can be applied not just to schemes but to algebraic stacks. Maybe this is implicit in your use of 'mapping stack'.

   A very cursory search didn’t give me an answer to your question, but I see here the Brauer group concept (or some ’bigger’ version) can be applied not just to schemes but to algebraic stacks. Maybe this is implicit in your use of ’mapping stack’.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2014
   • (edited May 22nd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo the thing is that these articles (as far as I am aware) discuss the Brauer group as a [[discrete group]], not as a group scheme/group stack, hence they do not equip the Brauer group which its evident geometry. The mapping stack construction takes care of that canonically, and the "bigger Brauer group" is just the group of global points in the 0-truncation of the "Brauer stack" (without remembering the geometry that holds them together). For the Picard case this difference is well-recognized and reflected in the dichotomy [[Picard group]] $\leftrightarrow$ [[Picard scheme]]/[[Picard stack]] (no geometry on the left, but geometry on the right). Here I am after this with-geometry version of the Brauer group.

   So the thing is that these articles (as far as I am aware) discuss the Brauer group as a discrete group, not as a group scheme/group stack, hence they do not equip the Brauer group which its evident geometry. The mapping stack construction takes care of that canonically, and the “bigger Brauer group” is just the group of global points in the 0-truncation of the “Brauer stack” (without remembering the geometry that holds them together).

   For the Picard case this difference is well-recognized and reflected in the dichotomy Picard group $\leftrightarrow$ Picard scheme/Picard stack (no geometry on the left, but geometry on the right). Here I am after this with-geometry version of the Brauer group.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 22nd 2014
   • (edited May 22nd 2014)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexYes, I was just asking (implicitly) whether you should say that you can form the Brauer stack of a _stack_. At the moment it's all in terms of schemes. I guess the generality is there at [[Picard stack]].

   Yes, I was just asking (implicitly) whether you should say that you can form the Brauer stack of a stack. At the moment it’s all in terms of schemes. I guess the generality is there at Picard stack.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMay 22nd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexOh, I see. Yes, let $X$ be any spectral derived higher whatsoever $\infty$-stack, then $[X, \mathbf{B}^2 \mathbb{G}_m]$ is its Brauer $\infty$-stack. I'll edit to make that clearer in the entry.

   Oh, I see. Yes, let $X$ be any spectral derived higher whatsoever $\infty$-stack, then $[X, \mathbf{B}^2 \mathbb{G}_m]$ is its Brauer $\infty$-stack.

   I’ll edit to make that clearer in the entry.