Timeline for Canonical Sheaf of Projective Space
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 21, 2014 at 2:20 | comment | added | bananastack | sorry for writing the answer in a hurry, I was bored and playing around with my phone while waiting for a friend in my car (I should have also added that det of any line bundle is the line bundle itself, but you already figure that out). A good summary of these facts for sheaves can be found in Goertz-Wedhorn (a marvellous book -- we are ALL waiting for Volume II!!!) section (7.20) and the relevant exercise is 7.30. | |
| Jul 21, 2014 at 0:42 | comment | added | Rene Schipperus | @GHfromMO I think the way to show that is to use local freeness to split and get our equation locally then put put it back together into sheaves. | |
| Jul 21, 2014 at 0:39 | comment | added | GH from MO | @ReneSchipperus: According to Hartshorne, the determinant relation holds for any exact sequence of locally free sheaves. | |
| Jul 21, 2014 at 0:19 | comment | added | Rene Schipperus | @GHfromMO One last question, does this require the sequence to split ? | |
| Jul 21, 2014 at 0:10 | comment | added | GH from MO | @ReneSchipperus: Yes, this is what I had in mind. | |
| Jul 21, 2014 at 0:04 | comment | added | Rene Schipperus | @GHfromMO For example if you just have vector spaces of dimension $n,m,k$ and $N=M\oplus K$ then $\wedge^n N =(M\oplus K)^n=\wedge^m M \otimes \wedge^k K$ is obvious. | |
| Jul 21, 2014 at 0:00 | vote | accept | Rene Schipperus | ||
| Jul 21, 2014 at 0:00 | comment | added | Rene Schipperus | @GHfromMO OK thanks ill have a look at that now. | |
| Jul 20, 2014 at 23:57 | comment | added | GH from MO | @ReneSchipperus: I am no expert, but I think this should follow locally by linear algebra. I found a semi-reference in Hartshorne's book: part (d) of Exercise 5.16 on Page 128. | |
| Jul 20, 2014 at 23:54 | comment | added | Rene Schipperus | @GHfromMO I see. I was taking all to the same power. Is there a reference for this fact ? | |
| Jul 20, 2014 at 23:49 | comment | added | GH from MO | @ReneSchipperus: Hartshorne says in Example 8.20.1 of his book Algebraic geometry that "we take the highest exterior powers of the exact sequence (8.13)". This is the same as what the response says above. | |
| Jul 20, 2014 at 23:46 | review | First posts | |||
| Jul 20, 2014 at 23:56 | |||||
| Jul 20, 2014 at 23:44 | comment | added | Rene Schipperus | So in the case of $n=1$ I would take the second exterior product of the middle term ? I was thinking I should take the $n$ th exterior product of all terms. | |
| Jul 20, 2014 at 23:38 | history | answered | user56234 | CC BY-SA 3.0 |