Additivity and multiplicativity of the Chern character and Todd class

Additivity of the Chern character. Work on the common finite open-and-closed refinement of the rank loci of $\mathcal{E}$, ${\mathcal{E}}^{\mathrm{\prime }}$, ${\mathcal{E}}^{\mathrm{\prime }\mathrm{\prime }}$ and $\mathcal{F}$. Chow rings and the claimed identities decompose over this refinement. On each locus all ranks are constant, including rank zero with an empty root list. By [L1] there is a flag bundle $f:X^{\mathrm{\prime }}\to X$ such that $f^{\ast }{\mathcal{E}}^{\mathrm{\prime }}$ and $f^{\ast }{\mathcal{E}}^{\mathrm{\prime }\mathrm{\prime }}$ have filtrations with invertible quotients, and the roots of $f^{\ast }\mathcal{E}$ are the union of the roots of $f^{\ast }{\mathcal{E}}^{\mathrm{\prime }}$ and of $f^{\ast }{\mathcal{E}}^{\mathrm{\prime }\mathrm{\prime }}$: the successive quotients of a filtration of $f^{\ast }\mathcal{E}$ refine to the union of the two filtrations. The degree-zero equality is $\mathrm{rank}\mathcal{E}=\mathrm{rank}{\mathcal{E}}^{\mathrm{\prime }}+\mathrm{rank}{\mathcal{E}}^{\mathrm{\prime }\mathrm{\prime }}$. Since the sum of $e^x$ over the union of the two root lists is the sum over the separate lists, $\mathrm{ch}(f^{\ast }\mathcal{E})=\mathrm{ch}(f^{\ast }{\mathcal{E}}^{\mathrm{\prime }})+\mathrm{ch}(f^{\ast }{\mathcal{E}}^{\mathrm{\prime }\mathrm{\prime }})$; applying $f^{\ast }$ to the expressions in rank and Chern classes (pullback preserves rank) and using injectivity of $f^{\ast }$ on $A^{\ast }(X{)}_{\mathbb{Q}}$ gives additivity for $\mathcal{E}$. Hence $\mathrm{ch}$ respects the exact-sequence relations and descends to the group completion $K^0(X)\to A^{\ast }(X{)}_{\mathbb{Q}}$.

Multiplicativity of the Todd class. On each rank locus with the same flag bundle, the roots of $f^{\ast }\mathcal{E}$ are the union of those of $f^{\ast }{\mathcal{E}}^{\mathrm{\prime }}$ and $f^{\ast }{\mathcal{E}}^{\mathrm{\prime }\mathrm{\prime }}$, and the product ${\prod }_j{x}_j\mathrm{/}(1-e^{-x_j})$ over the union is the product of the two partial products; injectivity of $f^{\ast }$ descends the identity $\mathrm{td}(\mathcal{E})=\mathrm{td}({\mathcal{E}}^{\mathrm{\prime }})\mathrm{td}({\mathcal{E}}^{\mathrm{\prime }\mathrm{\prime }})$. Thus $\mathrm{td}$ is multiplicative on exact sequences and factors through $K^0(X)$; its values are units because the degree-zero component is $1$, so it extends to virtual classes by inversion of the full Todd unit, using the finite geometric series in its nilpotent positive-degree part.

Tensor products and duality. On each common rank locus, splitting both $\mathcal{E}$ and $\mathcal{F}$ by a common flag bundle, the roots of the tensor product are the pairwise sums $x_i+y_j$ and $e^{x_i+y_j}=e^{x_i}{e}^{y_j}$, so $\mathrm{ch}(\mathcal{E}\otimes \mathcal{F})=\mathrm{ch}(\mathcal{E})\mathrm{ch}(\mathcal{F})$ after descent; the dual has roots $-x_j$, and the degree-$m$ part of $e^{-x}$ is $(-1{)}^m$ times that of $e^x$, giving ${\mathrm{ch}}_m({\mathcal{E}}^{\vee })=(-1{)}^m{\mathrm{ch}}_m(\mathcal{E})$ in the graded sense, with no ungraded assertion; $\mathrm{ch}({\mathcal{O}}_X)=1$ is the single-root case with root $0$. Consequently $\mathrm{ch}$ is a unital ring homomorphism on $K^0(X)$ with the vector-bundle tensor product.

Naturality and the tangent bundle. Pullback preserves the rank function, since it takes a local trivialization $\mathcal{E}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U^r$ to one of rank $r$ on $f^{-1}(U)$. By [L2] it also preserves the Chern classes, and by [L3] the components of the Chern character are universal polynomials in rank and Chern classes, while those of the Todd class are universal polynomials in the Chern classes; hence $f^{\ast }\mathrm{ch}(\mathcal{E})=\mathrm{ch}(f^{\ast }\mathcal{E})$ and $f^{\ast }\mathrm{td}(\mathcal{E})=\mathrm{td}(f^{\ast }\mathcal{E})$ for flat $f$. For the product $X{\times }_k{X}^{\mathrm{\prime }}$ the tangent bundle is the direct sum of the two pullbacks of the tangent bundles, so the Todd class multiplies: $\mathrm{td}(T_{X{\times }_k{X}^{\mathrm{\prime }}})={\mathrm{pr}}_1^{\ast }\mathrm{td}(T_X){\mathrm{pr}}_2^{\ast }\mathrm{td}(T_{X^{\mathrm{\prime }}})$.

Line bundles and coherent classes. For a direct sum of line bundles the identities are the definitions with $x_j=c_1({\mathcal{L}}_j)$; the series truncate because $A^m(X)=0$ above the dimension. If $X$ is quasi-projective, the finite-resolution isomorphism $K^0(X)=K_0(X)$ transports the additive and multiplicative structure, the product on coherent classes being the alternating Tor product; the ordinary tensor product of two arbitrary coherent sheaves is not assumed to be additive in exact sequences, and Todd values of virtual classes use inverses of the full Todd units.