c h ( ϕ ! ( F ) ) = ϕ ∗ ( c h ( F ) ⋅ t d ( T ϕ ) ) ∈ C h o w ( Y ) {\displaystyle \mathrm {ch} (\phi _{!}({\mathcal {F}}))=\phi _{*}(\mathrm {ch} ({\mathcal {F}})\cdot \mathrm {td} (T_{\phi }))\in \mathrm {Chow} (Y)}
dim K e r ( D ) − dim c o K e r ( D ) = ⟨ c h ( D ) , t d ( X ) ⟩ {\displaystyle \dim \mathrm {Ker} (D)-\dim \mathrm {coKer} (D)=\langle \mathrm {ch} (D),\mathrm {td} (X)\rangle }