One can switch between the Chern character and Chern class on vector bundles via Newton-Girard formulas, e.g. https://en.wikipedia.org/wiki/Chern_class#The_Chern_character.
One can define chern classes/characters of coherent sheaves by taking locally free resolutions and using additivity/multiplicitivity.
Suppose one has the chern class of a (not necessarily locally free) coherent sheaf. Is its chern character still given by the formula in the wiki link above? Does assuming that the sheaf is torsion free help?
This question came up because I want to compute the chern class of the tensor product of a torsion free sheaf with a line bundle (where the chern classes of each factor are known).