Timeline for "Direct" calculation of $K_0$ for surfaces, 3-folds
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 3, 2021 at 14:39 | comment | added | A. S. | Though this is quite old now but I realize what I was unable to express back then is that I was looking for the simple notion of a presentation of $K_0$ with generators and relations, outside of (probably) the cases of toric varieties and non-singular curves. | |
| Jul 30, 2017 at 23:05 | comment | added | Jason Starr | "I guess I want to see what the difference actually is, in this subgroup." Precisely what differences are you considering? | |
| Jul 30, 2017 at 15:18 | comment | added | A. S. | When two sheaves are equal in $K$-theory, their difference in the free abelian group generated by isomorphism classes of sheaves is in the subgroup generated by relations coming from exact sequences. I guess I want to see what the difference actually is, in this subgroup. | |
| Jul 30, 2017 at 15:07 | comment | added | Jason Starr | What do you mean by "exact sequences involved"? | |
| Jul 30, 2017 at 14:58 | comment | added | A. S. | Thank you both for your comments. I am actually familiar with the filtration, and the Riemann-Roch isomorphism, but neither are exactly what I am looking for. I guess what I would like is some way to see directly all the exact sequences involved. Unfortunately I am not sure how to do this or even ask a precise question since obviously there are a lot of exact sequences. | |
| Jul 30, 2017 at 6:14 | comment | added | abx | More generally, for a smooth projective variety the Chern character gives an isomorphism of $K_0(X)\otimes _{\mathbb{Z}}\mathbb{Q}$ with the Chow ring with rational coefficients. So modulo torsion this is the same problem as computing the Chow ring -- not so easy, but doable in some cases. | |
| Jul 30, 2017 at 4:36 | history | answered | Mohan | CC BY-SA 3.0 |