Additivity, naturality and the splitting principle for Chern classes

Whitney additivity. By [L1] the operational classes of $\mathcal{E}$ satisfy $c(\mathcal{E})=c({\mathcal{E}}^{\mathrm{\prime }})c({\mathcal{E}}^{\mathrm{\prime }\mathrm{\prime }})$: after simultaneous flag pullback the two filtrations concatenate to a line filtration of the middle bundle, whose Chern product is determined by successive Cartier hyperplanes; injectivity of projective-bundle pullback descends the identity; under the operational isomorphism of [L2] this identity of operators is the ring identity $c(\mathcal{E})=c({\mathcal{E}}^{\mathrm{\prime }})c({\mathcal{E}}^{\mathrm{\prime }\mathrm{\prime }})$ in $A^{\ast }(X)$.

Naturality. For a flat morphism $f:X^{\mathrm{\prime }}\to X$ of smooth equidimensional finite type $k$-schemes, the pullback $f^{\ast }$ on the Chow ring agrees with flat pullback by [L2], and flat pullback commutes with the operational cap actions by [L1]; applying this to the definitions $c_i(\mathcal{E})=c_i(\mathcal{E})\cap [X]$ and using the projection formula of [L2] gives $f^{\ast }{c}_i(\mathcal{E})=c_i(f^{\ast }\mathcal{E})$.

The splitting principle. Since $X$ is quasi-compact, the locally constant rank of $\mathcal{E}$ has finite image; its rank loci $X_r$ are open and closed. Chern classes, cycle groups and rational equivalence decompose over this finite disjoint union, so [L1] applies on every constant-rank locus (and on a common refinement of the three rank decompositions for Whitney additivity). If $X$ is empty, its Chow ring is zero and take the identity with empty filtrations. Otherwise let $R$ be the largest occurring rank. If $R\le 1$, take $f=\mathrm{id}$, with the empty filtration on $X_0$ and the one-step filtration on $X_1$. Otherwise build a tower indexed by $j=R,R-1,\dots ,2$. Over a locus of original rank $r\ge j$, projectivize the current kernel bundle of rank $j$ and replace it by the kernel of its tautological line quotient. Over a locus with $r<j$, projectivize the trivial bundle of rank $j$ and leave the pulled-back $\mathcal{E}$ unchanged. At each stage these bundles glue across the open-and-closed loci to a bundle of constant rank $j$, so every projection is smooth of constant relative dimension $j-1$ and has injective flat pullback by [L3]. The resulting $X^{\mathrm{\prime }}$ is smooth equidimensional of dimension $\dim X+R(R-1)\mathrm{/}2$. On each inverse image of $X_r$, the successive tautological kernels, reversed in order, yield a filtration with exactly $r$ invertible quotients. Repeated Whitney additivity gives $f^{\ast }c(\mathcal{E}){\mathrm{\mid }}_{f^{-1}{X}_r}={\prod }_{i=1}^r(1+c_1({\mathcal{L}}_i))$; the composite pullback is injective, so polynomial identities verified on every rank locus descend to $X$.

Consequences. The dual of the filtration has successive quotients ${\mathcal{L}}_i^{\vee }$ with $c_1({\mathcal{L}}_i^{\vee })=-c_1({\mathcal{L}}_i)$ by [L4], so multiplying ${\prod }_i(1-c_1({\mathcal{L}}_i))$ gives $c_i({\mathcal{E}}^{\vee })=(-1{)}^i{c}_i(\mathcal{E})$ after injectivity; the tensor identity for line bundles follows by multiplying rational sections and adding their Cartier divisors by [L4]; the vanishing $c_i(\mathcal{E})=0$ for $i>\mathrm{rank}\mathcal{E}$ is the defining relation of [L1], and the multiplicativity for exact sequences is step 1.1.