Motivic splitting principle

PDF p. 8: Theorem 3.1 and following paragraph

Assume the Axiom of Choice. Let $k$ be the article's algebraically closed field, let $X$ be a quasi-compact quasi-separated $k$-scheme, and let $V$ be a vector bundle on $X$. There is a quasi-compact quasi-separated $k$-scheme $Y$ and a morphism $q:Y\to X$ with the following properties.

The finitely many nonempty rank strata $X_{\alpha }$ of $V$ are open and closed. The scheme $Y$ is the finite disjoint union of schemes $Y_{\alpha }$ over $X_{\alpha }$, and on each $Y_{\alpha }$ the bundle $q^{\ast }V$ has a finite filtration by locally direct-summand subbundles whose successive quotients are line bundles. The filtration has length equal to the rank on $X_{\alpha }$; at rank zero it is the zero filtration and $Y_{\alpha }=X_{\alpha }$ with the identity map.

For every $i\in \mathbb{Z}$ and $j\ge 0$, pullback $q^{\ast }:H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i,j}(X,R)⟶H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i,j}(Y,R)$ is injective for $R=\mathbb{Z}$ and for $R=\mathbb{Z}\mathrm{/}n$, $n>0$. With the explicit convention $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{i,j}(-,R)=0$ for $j<0$, this also holds for all integer weights.

Write $L_{\alpha ,s}$ for the line-quotient bundles on $Y_{\alpha }$ and $j_{\alpha }:Y_{\alpha }\hookrightarrow Y$ for the clopen inclusions. In the vector-bundle Grothendieck group, $[q^{\ast }V]={\sum }_{\alpha ,s}[j_{\alpha ,\ast }{L}_{\alpha ,s}],$ where $j_{\alpha ,\ast }{L}_{\alpha ,s}$ is extension by zero: it is $L_{\alpha ,s}$ on $Y_{\alpha }$ and zero on the other components. If $V$ has constant rank, $[q^{\ast }V]$ is a sum of global line-bundle classes on $Y$, with the empty sum in rank zero. In particular, equality of natural motivic Chern-class expressions of the same bidegree can be checked after pullback to this tower and hence on its rank components.

A vector bundle is a finite locally free ${\mathcal{O}}_X$-module, and a line bundle is one of rank one. A rank stratum is the locus where the locally constant rank has a specified value. Quasi-compact means every open cover has a finite subcover; quasi-separated means intersections of quasi-compact open subsets are quasi-compact. The vector-bundle Grothendieck group is generated by isomorphism classes of vector bundles subject to $[E]=[E^{\mathrm{\prime }}]+[E^{\mathrm{\prime }\mathrm{\prime }}]$ for short exact sequences $0\to E^{\mathrm{\prime }}\to E\to E^{\mathrm{\prime }\mathrm{\prime }}\to 0$. The motivic groups and functorial pullbacks in nonnegative weights are those of Motivic complexes and cohomology.