We know that Chern class (or simply, let us focus on the first Chern class) could be defined for vector bundles over a variety $X$. I would like to ask is there a definition of Chern class for general coherent sheaves? For example, maybe using cohomological methods or resolutions? If there do exists such a notion, for vector bundles $\mathcal{L}$ and $\mathcal{M}$, suppose they are both subbundles of another vector bundle $\mathcal{H}$. Can we compute $c_1(\mathcal{L}+\mathcal{M})$ and $c_1(\mathcal{L}\cap \mathcal{M})$?
Thanks for your kind help!