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1. • CommentRowNumber1.
   • CommentAuthorhilbertthm90
   • CommentTimeAug 10th 2011
   • PermaLink
   Author: hilbertthm90
   Format: MarkdownItexThere seemed to be several pages pointing to a blank [[Picard group]], so I created it.

   There seemed to be several pages pointing to a blank Picard group, so I created it.

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeAug 10th 2011
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   Author: Tim_Porter
   Format: MarkdownItexThat looks good. Have you any references to more modern sources where the theory is developed in other directions than the classical ones?

   That looks good. Have you any references to more modern sources where the theory is developed in other directions than the classical ones?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeAug 10th 2011
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   Author: Urs
   Format: MarkdownItexI have added the following paragraphs right at the beginning to indicate the more general notion: > Generally, given a [[monoidal category]] $(C, \otimes)$, its _Picard group_ is the group of [[isomorphism class]]es of [[object]]s that have an [[inverse]] under the [[tensor product]] -- the [[line object]]s. Or rather, more naturally (before [[decategorification]]), it is the maximal [[2-group]] inside a monoidal category. > More specifically, what is often meant by _Picard group_ is the special case of this general notion where $C$ is a category of [[vector bundle]]s or more generally [[quasicoherent sheaves]] or similar. In this case the Picard group consists of the [[line bundle]]s or similar.

   I have added the following paragraphs right at the beginning to indicate the more general notion:

   Generally, given a monoidal category $(C, \otimes)$, its Picard group is the group of isomorphism classes of objects that have an inverse under the tensor product – the line objects. Or rather, more naturally (before decategorification), it is the maximal 2-group inside a monoidal category.

   More specifically, what is often meant by Picard group is the special case of this general notion where $C$ is a category of vector bundles or more generally quasicoherent sheaves or similar. In this case the Picard group consists of the line bundles or similar.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeAug 10th 2011
   • (edited Aug 10th 2011)
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   Author: zskoda
   Format: MarkdownItexI changed the wording a bit emphasizing the full (and quite standard in my experience) phrase _Picard category of $(C,\otimes)$_. I think vague phrase like "general case" is then not needed any more. So one of the sentences above became: > In geometry, by _Picard group_ one usually means the Picard group of the monoidal category of [[vector bundle]]s... We should make some clarification about the sense of the inverse here -- it should be up to isomorphism, that is not with respect to a fixed unit object but any.

   I changed the wording a bit emphasizing the full (and quite standard in my experience) phrase Picard category of $(C,\otimes)$. I think vague phrase like “general case” is then not needed any more. So one of the sentences above became:

   In geometry, by Picard group one usually means the Picard group of the monoidal category of vector bundles…

   We should make some clarification about the sense of the inverse here – it should be up to isomorphism, that is not with respect to a fixed unit object but any.

5. • CommentRowNumber5.
   • CommentAuthorhilbertthm90
   • CommentTimeAug 10th 2011
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   Author: hilbertthm90
   Format: MarkdownItex2. Tim: I just wrote what came to mind, so I didn't look at any references (I just threw Hartshorne in since I knew it was in there). The book FGA Explained has a great chapter on the Picard functor/Picard scheme, but it would be nice to have a source on the internet. That reminds me, do you think that "Picard scheme" and "Picard stack" should just be another section in Picard group, or should they be their own page? I might start typing something about these today.

   1. Tim: I just wrote what came to mind, so I didn’t look at any references (I just threw Hartshorne in since I knew it was in there). The book FGA Explained has a great chapter on the Picard functor/Picard scheme, but it would be nice to have a source on the internet.

   That reminds me, do you think that “Picard scheme” and “Picard stack” should just be another section in Picard group, or should they be their own page? I might start typing something about these today.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeAug 10th 2011
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   Author: zskoda
   Format: MarkdownItexI think it should be a separate page. Picard scheme is a big topic and specific to algebraic geometry. Picard group is used also in other contexts.

   I think it should be a separate page. Picard scheme is a big topic and specific to algebraic geometry. Picard group is used also in other contexts.

7. • CommentRowNumber7.
   • CommentAuthorzskoda
   • CommentTimeAug 10th 2011
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   Author: zskoda
   Format: MarkdownItexI have created a stub for [[Picard scheme]] with redirect for the special case, the [[Picard variety]], so you can now fill more material in.

   I have created a stub for Picard scheme with redirect for the special case, the Picard variety, so you can now fill more material in.

8. • CommentRowNumber8.
   • CommentAuthorTim_Porter
   • CommentTimeAug 10th 2011
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI agree with Zoran. It would be best to have separate entries for them as Zoran has created.

   I agree with Zoran. It would be best to have separate entries for them as Zoran has created.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMay 15th 2014
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   Author: Urs
   Format: MarkdownItexAt _[[Picard group]]_ I have tried to clarify a bit more, or at least highlight more clearly, the general abstract concepts versus the specific meaning in locally ringed geometry. I have expanded the Idea-section and structured the Definition-section accordingly. Right now the Idea-section reads like this: *** Fully generally, a _Picard group_ is an [[abelian group]] defined for a [[symmetric monoidal category]] as the group of [[isomorphism classes]] of [[objects]] which are [[invertible object|invertible]] with respect to the [[tensor product]]. Traditionally though one speaks in the context of [[geometry]] of the _Picard group_ $Pic(X)$ of some kind of [[space]] and by default means the invertible objects in some monoidal category of something like [[vector bundles]] over $X$. Specifically for $X$ a [[ringed topos]] (in particular a [[ringed space]]), then the [[monoidal category]] to be understood is that of locally free [[module sheaves]] over the [[structure sheaf]] and hence the Picard group in this case is that of locally free sheaves of $\mathcal{O}_X$-modules of [[rank]] $1$ (i.e. the [[line bundles]]). Specifically in [[complex geometry]] these objects on a [[complex manifold]] $X$ are [[holomorphic vector bundles]] and hence in this case the Picard group of a $X$ is that of isomorphism classes of [[holomorphic line bundles]]. This case has an obvious genralization to [[schemes]] in [[algebraic geometry]], and in much of the literature a _Picard group_ is meant to be a Picard group of $\mathbb{G}_m$-[[torsors]] over a given [[scheme]]. In this (and other) geometric situations, the Picard group naturally inherits geometric structure itself and equipped with that it is then called the _[[Picard scheme]]_, see there for more. Not [[decategorification|decategorifying]] by passing to isomorphism classes instead yields the concept of [[Picard 2-group]] and geometrically that of [[Picard stack]], see there for more.

   At Picard group I have tried to clarify a bit more, or at least highlight more clearly, the general abstract concepts versus the specific meaning in locally ringed geometry.

   I have expanded the Idea-section and structured the Definition-section accordingly.

   Right now the Idea-section reads like this:

   ---

   Fully generally, a Picard group is an abelian group defined for a symmetric monoidal category as the group of isomorphism classes of objects which are invertible with respect to the tensor product.

   Traditionally though one speaks in the context of geometry of the Picard group $Pic(X)$ of some kind of space and by default means the invertible objects in some monoidal category of something like vector bundles over $X$. Specifically for $X$ a ringed topos (in particular a ringed space), then the monoidal category to be understood is that of locally free module sheaves over the structure sheaf and hence the Picard group in this case is that of locally free sheaves of $\mathcal{O}_X$-modules of rank $1$ (i.e. the line bundles).

   Specifically in complex geometry these objects on a complex manifold $X$ are holomorphic vector bundles and hence in this case the Picard group of a $X$ is that of isomorphism classes of holomorphic line bundles. This case has an obvious genralization to schemes in algebraic geometry, and in much of the literature a Picard group is meant to be a Picard group of $\mathbb{G}_m$-torsors over a given scheme. In this (and other) geometric situations, the Picard group naturally inherits geometric structure itself and equipped with that it is then called the Picard scheme, see there for more.

   Not decategorifying by passing to isomorphism classes instead yields the concept of Picard 2-group and geometrically that of Picard stack, see there for more.

10. • CommentRowNumber10.
    • CommentAuthornLab edit announcer
    • CommentTimeAug 7th 2023
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    Author: nLab edit announcer
    Format: MarkdownItexAdded fact that the ring of integers $R$ of an algebraic number field is a unique factorization if and only if its Picard group is trivial, and added the reference from which the fact came from: * [[John Baez]], *Hoàng Xuân Sính’s thesis: categorifying group theory* ([pdf](https://math.ucr.edu/home/baez/sinh.pdf)) Anonymaus <a href="https://ncatlab.org/nlab/revision/diff/Picard+group/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/Picard+group/16">v16</a>, <a href="https://ncatlab.org/nlab/show/Picard+group">current</a>

    Added fact that the ring of integers $R$ of an algebraic number field is a unique factorization if and only if its Picard group is trivial, and added the reference from which the fact came from:

    • John Baez, Hoàng Xuân Sính’s thesis: categorifying group theory (pdf)

    Anonymaus

    diff, v16, current