Flag detection over an arbitrary field by prime-field restriction

Let $F\subseteq k$ be the prime subfield. Forgetting to $F$ changes neither the underlying qcqs scheme, its vector bundles and projective bundles, nor the value of its restricted motivic complex in [F1]. For a positive constant-rank bundle $G$, an invertible quotient of $G$ dualizes to a locally split line subbundle of $G^{\vee }$, and dualizing again is inverse. Thus the source projective bundle for $G^{\vee }$ is the required quotient-projective bundle for $G$, with the same tautological $\mathcal{O}(1)$. Apply [F2] to $G^{\vee }$. Projection onto the index-zero summand of the inverse equivalence is a left inverse to the actual quotient-projective pullback. Thus that pullback is injective on cohomology in every degree and nonnegative weight.

For constant ranks the tower in [F3] is finite. Applying step 1.1 at each stage and composing the injective homomorphisms proves detection for that tower. Rank-zero cases contribute the identity tower.

For varying ranks write $X={\coprod }_{\alpha }{X}_{\alpha }$ for the finite clopen common rank partition and $Y={\coprod }_{\alpha }{Y}_{\alpha }$ for its towers. Derived Nisnevich descent in [F1] for this disjoint open cover identifies the complexes on $X$ and $Y$ with the finite products on their pieces; their empty intersections contribute zero, since derived sheaf descent for the empty cover gives the zero complex on the empty scheme. Taking cohomology commutes with these finite products. Therefore $q^{\ast }$ is the product of the injective maps of step 2.1 and is injective. The additional negative-weight assertion follows only from the expressly adopted zero extension.