Comments on: Line Bundles and the Picard Group http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/ Wed, 23 Jul 2008 07:51:46 +0000 http://wordpress.org/?v=MU By: Greg Stevenson http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/#comment-378 Greg Stevenson Mon, 14 Apr 2008 22:05:49 +0000 http://rigtriv.wordpress.com/?p=89#comment-378 I just thought I'd point out that I think the usual way to define an immersion is to give a factorization as an open immersion followed by a closed immersion, i.e. an open subscheme of a closed subscheme. It is true though that whenever this is the case that you can rewrite in the other order, as a closed immersion then an open one. However, the converse requires the morphism to be quasi-compact. Working with varieties and good maps though this isn't really an issue. Also, keep up the good work ;) I just thought I’d point out that I think the usual way to define an immersion is to give a factorization as an open immersion followed by a closed immersion, i.e. an open subscheme of a closed subscheme. It is true though that whenever this is the case that you can rewrite in the other order, as a closed immersion then an open one.

However, the converse requires the morphism to be quasi-compact. Working with varieties and good maps though this isn’t really an issue.

Also, keep up the good work ;)

]]> By: Charles http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/#comment-374 Charles Mon, 14 Apr 2008 17:06:01 +0000 http://rigtriv.wordpress.com/?p=89#comment-374 And this is what I get for attempting to write a bunch of posts quickly to get a bit ahead. It does need to be an immersion, which is just that for some open subset of $latex \mathbb{P}^n$ containing the image, the map induces a homeomorphism with the image (which is a closed subset of the open set) and the induced map on sheaves is surjective. And this is what I get for attempting to write a bunch of posts quickly to get a bit ahead. It does need to be an immersion, which is just that for some open subset of \mathbb{P}^n containing the image, the map induces a homeomorphism with the image (which is a closed subset of the open set) and the induced map on sheaves is surjective.

]]> By: Matt http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/#comment-373 Matt Mon, 14 Apr 2008 15:06:53 +0000 http://rigtriv.wordpress.com/?p=89#comment-373 I believe the usual definition of a very ample line bundle is that the map from X to P^n is an immersion, not simply that the map is everywhere defined... I believe the usual definition of a very ample line bundle is that the map from X to P^n is an immersion, not simply that the map is everywhere defined…

]]> By: Charles http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/#comment-372 Charles Mon, 14 Apr 2008 15:03:26 +0000 http://rigtriv.wordpress.com/?p=89#comment-372 Well, the biggest reason that I only want to talk about line bundles (and anything else, for that matter) up to isomorphism (or up to embedding, or whatever) is that I'm still learning about categories fibered in groupoids, which is a rather natural structure floating around here. As far as keeping track of isomorphisms, and talking at the categorical (or, dare I say, the higher categorical) level, it's important, of course, but not what I'm going for with this exposition. Once I have a better grip on stacks, I'm intending to talk about them a little, though not in this series. Well, the biggest reason that I only want to talk about line bundles (and anything else, for that matter) up to isomorphism (or up to embedding, or whatever) is that I’m still learning about categories fibered in groupoids, which is a rather natural structure floating around here. As far as keeping track of isomorphisms, and talking at the categorical (or, dare I say, the higher categorical) level, it’s important, of course, but not what I’m going for with this exposition. Once I have a better grip on stacks, I’m intending to talk about them a little, though not in this series.

]]> By: John Armstrong http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/#comment-370 John Armstrong Sat, 12 Apr 2008 02:49:39 +0000 http://rigtriv.wordpress.com/?p=89#comment-370 Right, Greg. I'm pretty sure the obvious direction goes through, but I'm partly fishing around to see if the nontrivial twisting can get you hung up, and partly trying to get people using the categorical-level language to talk about familiar objects. Right, Greg. I’m pretty sure the obvious direction goes through, but I’m partly fishing around to see if the nontrivial twisting can get you hung up, and partly trying to get people using the categorical-level language to talk about familiar objects.

]]> By: Greg Stevenson http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/#comment-369 Greg Stevenson Sat, 12 Apr 2008 01:45:08 +0000 http://rigtriv.wordpress.com/?p=89#comment-369 If you consider the category of presheaves of modules over your given scheme, then you can define the braiding to be the usual switching of the factors in the tensor product over open sets. This gives a symmetric monoidal structure on presheaves which sheafifies. I'm not really an expert on this sort of stuff, but I think this argument should work... in particular the naturality will be fine. If you consider the category of presheaves of modules over your given scheme, then you can define the braiding to be the usual switching of the factors in the tensor product over open sets. This gives a symmetric monoidal structure on presheaves which sheafifies. I’m not really an expert on this sort of stuff, but I think this argument should work… in particular the naturality will be fine.

]]> By: John Armstrong http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/#comment-368 John Armstrong Fri, 11 Apr 2008 16:17:51 +0000 http://rigtriv.wordpress.com/?p=89#comment-368 Well, what you need is a specific morphism $latex \beta_{L,M}:L\otimes M\to M\otimes L$ for each pair of sheaves $latex L$ and $latex M$, not just the statement that they're isomorphic. You also need to pick these isomorphisms so that if you have morphisms $latex f:L\to L'$ and $latex g:M\to M'$, then the "naturality" condition holds: $latex \beta_{L',M'}\circ(f\otimes g)=(g\otimes f)\circ\beta_{L,M}$. Now the question is whether these isomorphisms are <em>their own</em> inverses. You know that $latex \beta_{L,M}^{-1}$ exists, but is it equal to $latex \beta_{M,L}$? Well, what you need is a specific morphism \beta_{L,M}:L\otimes M\to M\otimes L for each pair of sheaves L and M, not just the statement that they’re isomorphic. You also need to pick these isomorphisms so that if you have morphisms f:L\to L' and g:M\to M', then the “naturality” condition holds: \beta_{L',M'}\circ(f\otimes g)=(g\otimes f)\circ\beta_{L,M}.

Now the question is whether these isomorphisms are their own inverses. You know that \beta_{L,M}^{-1} exists, but is it equal to \beta_{M,L}?

]]> By: Vishal Lama http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/#comment-367 Vishal Lama Fri, 11 Apr 2008 15:16:29 +0000 http://rigtriv.wordpress.com/?p=89#comment-367 <em>You’re decategorifying!</em> Is that a sin?! :-) (Ok, it's supposed to be a joke!) You’re decategorifying!

Is that a sin?! :-) (Ok, it’s supposed to be a joke!)

]]> By: Charles http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/#comment-366 Charles Fri, 11 Apr 2008 15:08:46 +0000 http://rigtriv.wordpress.com/?p=89#comment-366 I, sadly, don't know enough about braided monoidal categories to answer that. I, sadly, don’t know enough about braided monoidal categories to answer that.

]]> By: John Armstrong http://rigtriv.wordpress.com/2008/04/11/line-bundles-and-the-picard-group/#comment-364 John Armstrong Fri, 11 Apr 2008 14:58:45 +0000 http://rigtriv.wordpress.com/?p=89#comment-364 Note that those axioms are all written in terms of isomorphisms. You're passing from the collection of invertible sheaves to the collection of <em>isomorphism classes</em> when you define the Picard group. You're decategorifying! What you're implicitly doing is taking the full subcategory of invertible sheaves and showing that it's a monoidal (and more) subcategory, which is actually groupal (everything has a monoidal inverse up to isomorphism). All the coherence conditions should hold because they do for the whole category of sheaves. The one thing I'm not sure on (though I think it's true) is if the monoidal product on sheaves is symmetric, or is it only braided? Either way this subcategory decategorifies to an abelian group. Note that those axioms are all written in terms of isomorphisms. You’re passing from the collection of invertible sheaves to the collection of isomorphism classes when you define the Picard group. You’re decategorifying!

What you’re implicitly doing is taking the full subcategory of invertible sheaves and showing that it’s a monoidal (and more) subcategory, which is actually groupal (everything has a monoidal inverse up to isomorphism). All the coherence conditions should hold because they do for the whole category of sheaves.

The one thing I’m not sure on (though I think it’s true) is if the monoidal product on sheaves is symmetric, or is it only braided? Either way this subcategory decategorifies to an abelian group.

]]>