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I am trying to read the book “Fourier-Mukai transforms in algebraic geometry”. Around the end of the page 126 of this book, it is written that the Chern character is from $K(X)$ (Grothendieck group) to $H^*(X,\mathbb{Q})$. However, according to Fulton’s intersection theory book or Hartshorne’s algebraic geometry book, the Chern character is from $K(X)$ to $A(X)$ (Chow ring). Now, my question is that how is this possible? Indeed, how can we look at the Chern character with codomain $H^*(X,\mathbb{Q})$? One answer to this question could be that we consider the composition with cycle map, but then, if we do this, why do we again have Grothendieck-Riemann-Roch theorem or indeed Theorem 5.26, page 127, “Fourier-Mukai transforms in algebraic geometry”?

Pouya Layeghi
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    Compose $\operatorname{ch}:K(X) \to A(X)$ with the cycle class map $A(X) \to H^*(X,\mathbb Q)$. – Tabes Bridges Sep 22 '21 at 22:52
  • @TabesBridges Thanks for your comment. I guess I found this at the first time when I was searching for it and I forgot. Now, my question is that why if we do this, then again we have Grothendieck-Riemann-Roch (Indeed Thm 5.26 in “Fourier-Mukai transforms in algebraic geometry”? – Pouya Layeghi Sep 22 '21 at 23:01
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    I'm not sure I follow your question. – Tabes Bridges Sep 22 '21 at 23:10
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    Because intersection of cycles factors through the cycle map and intersection in cohomology. – Sasha Sep 23 '21 at 04:23
  • @TabesBridges I was looking at Fultons book and I noticed that as it is stated in Fulton, cycle map goes to Borel-Moore homology and what I am asking is the sheaf cohomology with rational coefficients. How can I relate these to each other? – Pouya Layeghi Sep 23 '21 at 06:37
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    @PouyaLayeghi Borel Moore homology and standard cohomology are the same if the space is smooth. – Roland Sep 23 '21 at 06:48
  • @Roland Thanks. May you give a reference for this? – Pouya Layeghi Sep 23 '21 at 06:49
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    I don't have a reference. It is mentioned in the wikipedia article on Borel-Moore homology and there are several sources. I guess they all talk about it, I mean this Poincare duality is really an important property of Borel-Moore homology. – Roland Sep 23 '21 at 06:53

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