Whitney identities and virtual Chern arithmetic

PDF p. 8: Actual Chern and Whitney identities; PDF p. 11: Total virtual Chern class

Assume the Axiom of Choice and the compatible universe and site-size bounds of the supplied motivic constructions. Let $k$ be any field. For every qcqs $X\mathrm{/}k$ and exact sequence $0\to V^{\mathrm{\prime }}\to V\to V^{\mathrm{\prime }\mathrm{\prime }}\to 0$ of finite locally free sheaves, $c_t(V)=c_t(V^{\mathrm{\prime }})c_t(V^{\mathrm{\prime }\mathrm{\prime }})$. In particular $c_t(V\oplus W)=c_t(V)c_t(W)$, $c_1(V\oplus W)=c_1(V)+c_1(W)$ and $c_2(V\oplus W)=c_2(V)+c_1(V)c_1(W)+c_2(W)$. A line has no higher classes, $c_1(L\otimes M)=c_1(L)+c_1(M)$, and on a flag filtration all coefficients are elementary symmetric polynomials in its line roots. The resulting representative-independent virtual series satisfy $c_t(-W)=1-c_1(W)t+(c_1(W{)}^2-c_2(W))t^2+(-c_1(W{)}^3+2c_1(W)c_2(W)-c_3(W))t^3+\cdots$. These statements concern $K_{\mathrm{v}\mathrm{b}}^0$; only the completed affine comparison or locally Noetherian vector-bundle $K^0$ convention may be used for further group identifications.