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John Baez science forum Guru Wannabe
Joined: 01 May 2005
Posts: 220

Posted: Wed Jul 12, 2006 6:27 am Post subject:
coherent sheaves versus vector bundles



Some guy on the street told me that if you have a smooth complex algebraic
variety X, you can not only get its Ktheory by forming the Grothendieck
group of all vector bundles over X  you can also get a kind of dual theory,
its "Khomology", by forming the Grothendieck group of all coherent
sheaves of vector spaces over X.
There's supposed to be some nice dual pairing between these....
Is this right so far? Where's a nice easy place to learn this stuff?
But what I'm really wondering now is something else! To what extent
can you get all coherent sheaves of vector spaces over X from
vector bundles supported on subvarieties of X?
I think these give you coherent sheaves as follows: if we have
a subvariety S with inclusion map
i: S > X
we can take a vector bundle E over S, take its sheaf of sections
and get a coherent sheaf over S, and push this forward along i
to get a coherent sheaf over X.
Is every coherent sheaf on X a direct sum of coherent sheaves of
this form? Or maybe built out of them by extensions, or something?
Or maybe I need to say "subscheme" instead of "subvariety"?
I'm probably being overoptimistic. But don't just tell me the bad news
 "you're wrong"  tell me some good news too.

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David Madore science forum beginner
Joined: 23 Feb 2005
Posts: 24

Posted: Wed Jul 12, 2006 10:31 pm Post subject:
Re: coherent sheaves versus vector bundles



John Baez in litteris <e924oa$mgf$1@glue.ucr.edu> scripsit:
Quote:  Is this right so far?

Essentially yes, at least as far as I understand.
Quote:  Where's a nice easy place to learn this stuff?

You can learn some things in Fulton's book *Intersection Theory*,
though I won't promise it's very readable (could be worse, though).
Quote:  But what I'm really wondering now is something else! To what extent
can you get all coherent sheaves of vector spaces over X from
vector bundles supported on subvarieties of X?

If X is smooth  as you assumed  then the K^0 (Grothendieck group of
vector bundles) and the K_0 (Grothendieck group of coherent sheaves)
coincide. You don't need to take use vector bundles on subvarieties
at all. See loc. cit., 15.1 and B.3: the point is that any coherent
sheaf on a smooth algebraic variety has a finite resolution by locally
free sheaves (i.e., [sheaves of sections of] vector bundles).

David A. Madore
(david.madore@ens.fr,
http://www.dma.ens.fr/~madore/ ) 

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John Baez science forum Guru Wannabe
Joined: 01 May 2005
Posts: 220

Posted: Tue Jul 18, 2006 11:41 am Post subject:
Re: coherent sheaves versus vector bundles



In article <e93t7c$2lo$1.repost@nef.ens.fr>,
David Madore <david.madore@ens.fr> wrote:
Quote:  John Baez in litteris <e924oa$mgf$1@glue.ucr.edu> scripsit:
Where's a nice easy place to learn this stuff?
You can learn some things in Fulton's book *Intersection Theory*,
though I won't promise it's very readable (could be worse, though).

Thanks! Your faint praise reminds me of this reference I just
read in the Wikipedia article on large countable ordinals:
W. Pohlers, Proof theory, ISBN 0387518428 (for Veblen hierarchy
and some impredicative ordinals). This is probably the most readable
book on large countable ordinals (which is not saying much).
:)
Quote:  But what I'm really wondering now is something else! To what extent
can you get all coherent sheaves of vector spaces over X from
vector bundles supported on subvarieties of X?
If X is smooth  as you assumed  then the K^0 (Grothendieck group of
vector bundles) and the K_0 (Grothendieck group of coherent sheaves)
coincide. You don't need to take use vector bundles on subvarieties
at all. See loc. cit., 15.1 and B.3: the point is that any coherent
sheaf on a smooth algebraic variety has a finite resolution by locally
free sheaves (i.e., [sheaves of sections of] vector bundles).

Thanks! I think I understand this, thanks to an email from Kevin
Buzzard.
I also got an interesting reply from my pal Minhyong Kim:
Quote:  I think Madore's answer addressed most of your question,
and is the relevant view for Ktheory.
But I thought I'd add a remark in case it's of interest.
It is true that you can build up cohorent sheaves
the way you described using pushforward from
subschemes (it is important to use subschemes
here if you want all coherent sheaves). There's
a nice description of the process in Mumford's book
`Curves on Algebraic Surfaces' and the key phrase
is `flattening stratifications' where the main point
is that you can stratify you scheme in a universal
way so that the restriction of your sheaf to the strata
is a bundle.


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Esperanto its official language before the Germans invaded?
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