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I am trying to emerge from my complete ignorance about intersection theory. I have a bias toward sheaves, so I like the idea of doing intersection theory with the K-group of coherent sheaves. From what I understand the graded (by codimension) K-group GK is isomorphic to the Chow group A after tensoring with the rational numbers (and I probably need to say smooth variety).

I know of two/three places where to learn this stuff: Fulton, Lang RR algebra; the book chapter by Henri Gillet titled k-theory and intersection theroy; the set of lecture notes by Pedro Sancho de Salas http://matematicas.unex.es/~sancho/ (the link is for his webpage, which contains a few goodies).

However, the numerical group (the quotient of A modulo numerical equivalence) doesn't seem to be mentioned. The codimension one group is usually called the Neron-Severi group and is mentioned in some books, the whole group is briefly (if i recall correctly) mentioned in Fulton's book on intersection theory.

Are there any good references for the numerical group treated from the point of view of the K-group?

Is it true that GN (the graded K-group of coherent sheaves modulo numerical equivalence) is isomorphic to N, the Chow group modulo numerical equivalence) without having to tensor by the rationals?

p.s. I say two elements of K are numerically equivalent if they $\chi(-,F)$ gives the same answer for any sheaf F.

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  • $\begingroup$ the 'algebraic k-theory' tag might be a bit pretentious. please get rid of it if you feel it is not appropriate. (I do not understand higher k-groups, at all) $\endgroup$
    – Yosemite Sam
    Commented Mar 19, 2012 at 18:44
  • $\begingroup$ $K_0$ is also $K$-theory $\endgroup$
    – Fernando Muro
    Commented Mar 19, 2012 at 20:18
  • $\begingroup$ albeit a very small bit of it ;) $\endgroup$
    – Yosemite Sam
    Commented Mar 19, 2012 at 21:02
  • $\begingroup$ Try the following: $K_{0}(X)$ is filtered by the dimension of support, and the statement that can be found in Fulton is that the associated graded tensor $Q$ is isomorphic to Chow tensor $Q$. Under some hypotheses (smooth and proper are sufficient, but probably something weaker would work if formulated correctly), you can put a `Mukai form' on Chow so that the Mukai vector map $\v:Ch(?)\sqrt{td(X)}$ respects the pairing (basically, this needs Riemann-Roch). $\endgroup$
    – Chris Brav
    Commented Mar 19, 2012 at 21:33
  • $\begingroup$ Then I suppose that you get an isometry between the associated graded of $K_{0}(X)$ tensor $Q$ and Chow tensor $Q$, and so numerical groups should match up. I haven't thought this through carefully, though. $\endgroup$
    – Chris Brav
    Commented Mar 19, 2012 at 21:34

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