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.