Detection on a simultaneous flag tower

In the constant-rank abstract case, apply [F1] to the finite bundle family and [F2] to every stage. Their finite composite is injective in every integer bidegree; if there are no stages the map is the identity and remains injective.

Apply [F7] to the global locally split filtrations from [F1]. Telescoping the short-exact-sequence relations gives $[q^{\ast }{E}_i]={\sum }_s[L_{i,s}]$, with no summands for rank zero. This uses only those vector-bundle relations and asserts neither a globally split extension nor cohomological detection from a K-group equality.

In the abstract varying-rank case, let ${\iota }_{\alpha }:X_{\alpha }\hookrightarrow X$ and $j_{\alpha }:Y_{\alpha }\hookrightarrow Y$ be the inclusions. The construction satisfies $q{j}_{\alpha }={\iota }_{\alpha }{q}_{\alpha }$. All the bundles on $X_{\alpha }$ have constant ranks, so step 1.1 gives injectivity of each $q_{\alpha }^{\ast }$.

If $q^{\ast }x=0$, then functoriality gives $q_{\alpha }^{\ast }{\iota }_{\alpha }^{\ast }x=j_{\alpha }^{\ast }{q}^{\ast }x=0$. Step 3.1 yields ${\iota }_{\alpha }^{\ast }x=0$ for every $\alpha$. The stated joint-restriction hypothesis then implies $x=0$. This proves conditional varying-rank detection without adding an unproved axiom to the abstract contract.

For the actual motivic theory, take one positive-constant-rank quotient-projective stage $\pi :\mathbb{P}(G)\to S$ of a component tower. The source in [F5] uses line subbundles. Dualizing a locally split invertible quotient of the finite locally free $G$ gives a line subbundle of $G^{\vee }$, and dualizing again recovers the quotient. Hence the represented projective bundles are canonically identified over $S$, with the same projection and the source's dual tautological line equal to the quotient $\mathcal{O}(1)$. Apply [F5] to $G^{\vee }$, of the same positive constant rank. Its index-zero summand is $\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(S)$ and its map is precisely ${\pi }^{\ast }$. The inverse of the derived equivalence followed by projection onto that summand is a retraction. Thus ${\pi }^{\ast }$ is injective on every integral cohomology degree $i$, for $j\ge 0$.

Applying the finite-coefficient cone to that equivalence and using [F6] gives the same zero-summand retraction and injectivity for $R=\mathbb{Z}\mathrm{/}n$. Compose these injections through every finite component tower; identity stages at rank zero preserve the conclusion. Joint restriction to the common rank strata is injective by [F4]. The functorial argument of step 4.1 therefore proves the asserted actual motivic injectivity for both coefficient choices. The optional negative-weight convention gives zero source and target groups and so extends it trivially when that convention is explicitly used.

Finally [F7] gives all stated varying-rank class equalities, with supported extensions in the global formula, and explains why they cannot in general be called sums of global line-bundle classes. If $q^{\ast }u=q^{\ast }v$ in any of the detecting cases, functorial additive pullback gives $q^{\ast }(u-v)=0$, and the relevant injection from step 1.1, step 4.1 or step 6.1 implies $u-v=0$. Naturality identifies pulled-back expressions with those formed from the pulled-back bundles, which proves the final identity-detection assertion and completes the proof.