Chern classes of a vector bundle on a smooth scheme

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth-immersion and homological suppliers. Let $k$ be a field and let $X$ be a smooth equidimensional scheme of finite type over $k$ of dimension $n$ (Smooth morphism of schemes, Relative dimension of a smooth morphism at a point), with Chow ring $A^{\ast }(X)$ (The intersection product and Chow ring of a smooth scheme). Let $\mathcal{E}$ be a finite locally free ${\mathcal{O}}_X$-module of rank $r\ge 1$ (Locally free sheaves of finite rank), with projective bundle $\pi :\mathbb{P}(\mathcal{E})\to X$ and $\xi =c_1(\mathcal{O}(1))$ (The projective bundle formula for Chow groups).

Definition. By the projective bundle formula, ${\xi }^r\in A^r(\mathbb{P}(\mathcal{E}))$ has a unique expression ${\xi }^r+{\sum }_{i=1}^r(-1{)}^i{c}_i(\mathcal{E}){\xi }^{r-i}=0\text{in }{A}^{\ast }(\mathbb{P}(\mathcal{E})),$ with $c_i(\mathcal{E})\in A^i(X)$ pulled back from $X$; the classes $c_i(\mathcal{E})$ are the Chern classes of $\mathcal{E}$. Set $c_0(\mathcal{E}):=1\in A^0(X)$, $c_i(\mathcal{E}):=0$ for $i>r$, and call $c(\mathcal{E}):=1+c_1(\mathcal{E})+\cdots +c_r(\mathcal{E})$ the total Chern class.

For the zero bundle define $c(0)=1$ and $c_i(0)=0$ for $i>0$, as for the rank-zero operational classes; no projective bundle of rank zero is used.

Basic properties. (i) (Normalization) For an invertible sheaf $\mathcal{L}$: $c(\mathcal{L})=1+c_1(\mathcal{L})$ with $c_1(\mathcal{L})\in A^1(X)$ the first Chern class of Intersection with an invertible sheaf and the first Chern class; in particular $c_1({\mathcal{O}}_X(D))=[D]$ for an effective Cartier divisor $D$. (ii) (Naturality) For a morphism $f:X^{\mathrm{\prime }}\to X$ of smooth equidimensional $k$-schemes, $c_i(f^{\ast }\mathcal{E})=f^{\ast }{c}_i(\mathcal{E})$. (iii) (Vanishing) $c_i(\mathcal{E})=0$ for $i>\mathrm{rank}\mathcal{E}$ and $c_i(\mathcal{E})\in A^i(X)$. (iv) (Direct sums of line bundles) If $\mathcal{E}\cong {\mathcal{L}}_1\oplus \cdots \oplus {\mathcal{L}}_r$ then $c(\mathcal{E})={\prod }_{i=1}^r(1+c_1({\mathcal{L}}_i))$. (v) (Top class) $c_r(\mathcal{E})\cap [X]=[Z(s)]$ for a regular section $s$ of $\mathcal{E}$ with $\dim Z(s)=n-r$; in particular $c_r(\mathcal{E})$ is the class of the zero locus of a regular section.

Well-definedness. The classes are defined by applying the operational Chern operators of Operational Chern classes and the Whitney formula to the fundamental class $[X]$ and using the operational-ring isomorphism of The intersection product and Chow ring of a smooth scheme, which identifies the operational action with multiplication: their defining projective bundle relation is exactly the displayed relation, and the basis part of The projective bundle formula for Chow groups gives existence and uniqueness of the coefficients. Normalization, naturality under arbitrary morphisms of smooth schemes, rank vanishing, the Whitney and line-summand formulas and the regular-section formula (including its generic multiplicities and the case of a non-reduced zero scheme) are already established for the operational classes; restriction of operational classes is the graph pullback under the operational isomorphism, by Naturality of the Chow ring and the projection formula, so they become the claimed ring identities. Only the injectivity of projective-bundle pullback is used, not injectivity of arbitrary $f^{\ast }$.