In several textbooks ("The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn, "Calcul differentiel et classes caracteristiques..." by Angeniol and Lejeune-Jalabert) it is mentioned that the trace of the p-th atiyah class equals the p-th chern class or the p-th component of the chern character. I could not find a reference where this statement is proven. Thanks for any help.



trace of the atiyah class equals chern class
It is more an approach to the definition of the Chern character than a fact that needs to be proven.


An old reference that uses the language of *twisted cochains* is "The trace map and characteristic classes for coherent sheaves", by O'Brian, Toledo, and Tong, Amer. J. Math. 103 (1981), pp. 225–252 (MR 82f:32021). They use this construction in further papers to prove a Riemann-Roch theorem in Hodge cohomology.


A more modern reference is Caldararu's "The Mukai pairing, I: the Hochschild structure" (arXiv:math/0308079), see also arXiv:math/0308080 and arXiv:0707.2052. Here, the Chern character is presented in the language of derived categories and Fourier-Mukai transforms.


