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I am looking for the formal definition of Chern classes for torsion free sheaves, at least on the projective spaces. There are several books that do made the defintion for the vector bundle case, and the Ph.D. thesis of H.I. Green "Chern Classes for Coherent Sheaves" define Chern classes in terms of the singular cohomology of some smooth variety, and prove that his definition coincides with the classical definition for locally free sheaves.

The point is that I was not able to prove that Green's definition for Chern classes is the unique way of assigning classes in the Chow ring of the variety, satisfying Functoriality, Whitney’s formula, and the condition on line bundles. (See for instance Theorem 5.3 of the book https://www.amazon.com.br/3264-All-That-Algebraic-Geometry/dp/1107602726). The difficult part is to prove the unicity.

Is there a reference for it?

Thanks in advance.

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    $\begingroup$ "I am looking for the formal definition of Chern classes for torsion free sheaves, at least on the projective spaces." The K-group of projective space $\mathbb{P}^n$ is generated as a free Abelian group by the classes of $\mathcal{O}(-d)$, for $d=1,\dots,n$. Thus, for every Abelian group $A$ and for every formula $\chi$ defined on coherent sheaves that is additive for short exact sequences, i.e., for $A= 1 + t\mathbb{Z}[[t]]$ under multiplication and for $\chi$ equal to the total Chern class, the formula is uniquely determined by its values on $\mathcal{O}(-d)$. $\endgroup$
    – Jason Starr
    Commented Jul 6, 2018 at 19:47
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    $\begingroup$ Similarly, for a smooth, projective $k$-scheme $X$, the natural map from the K-group of locally free $\mathcal{O}_X$_modules to the K-group of coherent $\mathcal{O}_X$-modules is an isomorphism of modules. Thus, as for projective space, $\chi$ is uniquely determined by its values on locally free $\mathcal{O}_X$-modules. All of this is described in the beautiful notes of Manin, "Lectures on the K-functor in algebraic geometry". $\endgroup$
    – Jason Starr
    Commented Jul 6, 2018 at 19:49
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    $\begingroup$ Thank you very much Jason Starr, that is exactly what I was looking for. $\endgroup$
    – User43029
    Commented Jul 7, 2018 at 20:45

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The fact that Green's definition is the unique one is 'answered' in https://hal.archives-ouvertes.fr/tel-00460334 . That is, the note after Corollary 1.7 (p. 44) says that this isn't known, because Grothendieck-Riemann-Roch for closed immersions is not known, but it actually follows from the exact argument used to prove Proposition 3.1 in https://arxiv.org/pdf/0910.5384.pdf. For more details about Green's construction of Chern classes, see my upcoming thesis ;-)

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