coherent sheaves versus vector bundles

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

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

John Baez in litteris <e924oa$mgf$1@glue.ucr.edu> scripsit:

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

In article <e93t7c$2lo$1.repost@nef.ens.fr>,
David Madore <david.madore@ens.fr> wrote:

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