Additivity of the first Chern class

Question

I have a highly elementary question: is the first Chern class additive? More specifically, given a short exact sequence of coherent sheaves on a nonsingular curve $X$ $$ 0 \to \mathscr F'\to \mathscr F \to \mathscr F''\to 0, $$ does it necessarily hold that $c_1(\mathscr F)=c_1(\mathscr F')+c_1(\mathscr F'')$? I know that the tensor product formula $c_1(\mathscr F\otimes \mathscr F')=\operatorname{rank}(\mathscr F')\cdot c_1(\mathscr F) + \operatorname{rank}(\mathscr F)c_1(\mathscr F'),$ but don't know how to relate it to that. ${}$ ${}$

Answer 1

First we restrict to the case of vector bundles (locally free sheaves), then this follows from the Whitney sum formula, which states

If $E \to X$ is a vector bundle, let $c_t(E) = \sum_{i=0}^{\infty} c_i(E) t^i$ denote the Chern power series of $E$. Then for any short exact sequence $$ 0 \to E' \to E \to E'' \to 0, $$ we have the equality $c_t(E) = c_t(E') c_t(E'')$.

A reference is Ravi Vakil's notes on intersection theory. An immediate consequence of the Whitney sum formula is $$ c_1(E) = c_0(E'') c_1(E') + c_1(E'')c_0(E') = c_1(E') + c_1(E''), $$ since the zeroth Chern classes are $c_0(E') = c_0(E'') = 1$.

Now for the general case. Given a coherent sheaf, it admits a resolution of length $\dim X$ by locally free sheaves (this is Exercise 6.9 of section III.6 of Hartshorne). Its Chern classes are defined by forcing the Whitney sum formula to hold, so we will also have that $c_1(\mathscr{F}) = c_1(\mathscr{F}') + c_1(\mathscr{F}'')$ for a short exact sequence $0 \to \mathscr{F}' \to \mathscr{F} \to \mathscr{F}'' \to 0$ of coherent sheaves.