Posted by Chris Schommer-Pries in Algebraic Geometry, Algebraic Topology, Characteristic Classes, Chromatic Homotopy, Spectral Sequences.
Now that we’ve all gotten over the excitement surrounding the new iPad, I wanted to talk about something else which I actually find very exciting (unlike the iPad). This semester Jacob Lurie is giving a course on Chromatic Homotopy Theory. This is a beautiful picture which relates algebraic topology and algebraic geometry. Hopefully with Jacob at the helm we’ll also see the derived/higher categorical perspective creeping in. This seems like a great opportunity the learn this material “in my heart”, as my old undergraduate advisor used to say.
And with most of our principal bloggers distracted by MathOverflow, it also seems like a good time to experiment with new media. So here’s the plan so far:
During lectures I’m going to be live-TeXing notes, which I’ll flush out and post to my website. (Jacob’s also posting his own notes!)
In addition, I’ll try to post blog articles (like this one) about related topics or topics I find interesting/confusing.
There might be a little MO action thrown in for fun.
The offshoot is that today I want to talk a little about chromatic homotopy and about the Atiyah-Hirzebruch Spectral Sequence.
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Posted by David Speyer in Algebraic Geometry.
This post is about a computation every algebraic geometry student should do, but that none of my courses covered. Let 
be a smooth, projective curve over 
. Then 
is a one dimensional -vector space. If you’ve read Hartshorne III.7 carefully, you’ll remember that there is a canonical isomorphism: 
. Explicitly, let 
and 
be two points of ; consider the open cover 
of and let 
be a holomorphic 
-form on 
. Let 
be the cocycle 
. Then 
sends the cocycle to the residue, at , of . (A good question which I might ask on a qual one day: Why is it OK that this is asymmetric in and ?)
On the other hand, suppose that 
. Then is isomorphic to 
. An element of is a 
-closed 
form, modulo -exact -forms. But, because only has two real dimensions, this simplifies: Every 
form is a -form and every -form is closed because there are no 
-forms. So an element of is a -form modulo -exact -forms. It turns out that this is the same as a -form modulo 
-exact two forms. In other words, is the same as the deRham cohomology group 
that we learn about in differential geometry. And we know a canonical map 
: Take the integral!
The point of this post is to compute the relation between and 
. I invite you to try it yourself, then meet me on the other side to see if we got the same answer.
UPDATE: I claimed earlier that this was easy to show 
for curves. As Akhil Mathew points out, it seems to only be easy to show that there is a well defined surjection 
. Since I only need the map to exist, I’ll leave it at that for now.
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