= Chern class axioms in the Chow ring
{c}
{title2=$c_i(E)\in\operatorname{CH}^i(X)$}
For a <vector bundle> $E$ of rank $r$ on a smooth <algebraic variety>, algebraic <Chern classes> have $c_0(E)=1$, vanish for $i>r$, commute with pullback, and obey the <Whitney sum formula for Chern classes> for every short exact sequence of vector bundles. A <line bundle> $\mathcal O(D)$ has total class $1+[D]$. The projective-bundle splitting construction makes pullback injective and supplies a filtration by line bundles, so these axioms determine all classes as elementary symmetric functions of their first classes. In particular, $c_i(E^*)=(-1)^ic_i(E)$ and $c_1(L\otimes M)=c_1(L)+c_1(M)$ in the ordinary <Chow ring>.
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