Math Forum Discussions

baez@math.removethis.ucr.andthis.edu

Posts: 413
Registered: 12/13/04

coherent sheaves versus vector bundles
Posted: Jul 12, 2006 2:27 AM

Plain Text

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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