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coherent sheaves versus vector bundles
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John Baez
science forum Guru Wannabe


Joined: 01 May 2005
Posts: 220

PostPosted: Wed Jul 12, 2006 6:27 am    Post subject: coherent sheaves versus vector bundles Reply with quote

Some guy on the street told me that if you have a smooth complex algebraic
variety X, you can not only get its K-theory by forming the Grothendieck
group of all vector bundles over X - you can also get a kind of dual theory,
its "K-homology", 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

PostPosted: Wed Jul 12, 2006 10:31 pm    Post subject: Re: coherent sheaves versus vector bundles Reply with quote

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

PostPosted: Tue Jul 18, 2006 11:41 am    Post subject: Re: coherent sheaves versus vector bundles Reply with quote

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 K-theory.
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 push-forward from
sub-schemes (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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