I apologize in advance for the vagueness of my question, but I am looking for sources (if they exist) where $K_0$ (the Grothendieck group of coherent sheaves) is computed "by hand" for some low-dimensional varieties (besides nonsingular curves, which is an exercise in Hartshorne).

To try to explain further, here is an example of what I am not looking for: in the case of toric surfaces, one can obtain the $T$-equivariant $K$-theory by localization, and then mod out the equivariance to obtain regular $K$-theory. This does not give me enough understanding about which relations between sheaves are really at play.


For smooth projective surfaces (over $\mathbb{C}$ say) one has filtration $F^1K_0(X)\subset K_0(X)$ with the quotient being $\mathbb{Z}$, $F^2K_0(X)\subset F^1K_0(X)$, with the quotient being Picard group of $X$ and $F^2K_0(X)$ isomorphic to the the Chow group of zero cycles mod rational equivalences, $A_0(X)$. The latter can be calculated in some instances, for example, if $X$ is rational, it is $\mathbb{Z}$.

  • 1
    $\begingroup$ 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. $\endgroup$ – abx Jul 30 '17 at 6:14
  • $\begingroup$ 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. $\endgroup$ – A. S. Jul 30 '17 at 14:58
  • $\begingroup$ What do you mean by "exact sequences involved"? $\endgroup$ – Jason Starr Jul 30 '17 at 15:07
  • $\begingroup$ 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. $\endgroup$ – A. S. Jul 30 '17 at 15:18
  • $\begingroup$ "I guess I want to see what the difference actually is, in this subgroup." Precisely what differences are you considering? $\endgroup$ – Jason Starr Jul 30 '17 at 23:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.