Search   Memberlist   Usergroups
 Page 1 of 1 [3 Posts]
Author Message
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>,

 Quote: John Baez in litteris 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.

------------------------------------------------------------------
Puzzle 27: Which neutral territory in Europe flirted with making
Esperanto its official language before the Germans invaded?

If you get stuck try my puzzle page at:

http://math.ucr.edu/home/baez/puzzles/
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).

--
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 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. ---------------------------------------------------------------------- Puzzle 25: Why do zookeepers give orangutans and other endangered species birth control pills? If you get stuck, try: http://math.ucr.edu/home/baez/puzzles/

 Display posts from previous: All Posts1 Day7 Days2 Weeks1 Month3 Months6 Months1 Year Oldest FirstNewest First
 Page 1 of 1 [3 Posts]
 The time now is Sat Nov 17, 2018 7:17 pm | All times are GMT
 Jump to: Select a forum-------------------Forum index|___Science and Technology    |___Math    |   |___Research    |   |___num-analysis    |   |___Symbolic    |   |___Combinatorics    |   |___Probability    |   |   |___Prediction    |   |       |   |___Undergraduate    |   |___Recreational    |       |___Physics    |   |___Research    |   |___New Theories    |   |___Acoustics    |   |___Electromagnetics    |   |___Strings    |   |___Particle    |   |___Fusion    |   |___Relativity    |       |___Chem    |   |___Analytical    |   |___Electrochem    |   |   |___Battery    |   |       |   |___Coatings    |       |___Engineering        |___Control        |___Mechanics        |___Chemical

 Topic Author Forum Replies Last Post Similar Topics Infinitesimal generator of a vector field Julien Santini Math 0 Fri Jul 21, 2006 8:01 am Vector field flow problem - help? Daniel Nierro Math 1 Wed Jul 19, 2006 10:28 am Obtaining a morphism of sheaves from homotopy data categorist Research 0 Mon Jul 17, 2006 8:15 pm ? VECTOR AND MATRIX MANAPLUATION Cheng Cosine Math 0 Mon Jul 03, 2006 6:49 pm Actual Reality Versus Observed Reality. Len Gaasenbeek Relativity 5 Sun Jul 02, 2006 7:37 pm

Other DeniX Solutions sites: Electronics forum |  Medicine forum |  Unix/Linux blog |  Unix/Linux documentation |  Unix/Linux forums  |  send newsletters