Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
There seemed to be several pages pointing to a blank Picard group, so I created it.
That looks good. Have you any references to more modern sources where the theory is developed in other directions than the classical ones?
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.
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.
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.
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.
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.
I agree with Zoran. It would be best to have separate entries for them as Zoran has created.
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.
1 to 9 of 9