External actual universal-class detection on Vect and Grass

In the coefficient convention of [F3], mapping homotopy degree $u$ into the weight-$i$ target is cohomological bidegree $(2i-u,i)$. A Chern monomial of weight $w$ therefore has coefficient ${\mathcal{E}}_S^{2(i-w)-u,i-w}(S)$. This group is zero whenever $w>i$, for every $u$, in particular $u=0,1$. The vanishing is about the actual represented model by [F3]. The general coring formula in [F1] remains valid, but negative-weight vanishing for an arbitrary coring is not being inferred from vanishing over $S$.

Apply the finite basis formula in [F1] to $G(n,N)$, and successively to $G(n,N){\times }_{S}G(m,M)$ using the same actual class maps. The latter mapping groups in homotopy degree $u$ have a finite additive basis $c^{\alpha }{d}^{\beta }$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le N-n$ and $\mathrm{\mid }\beta \mathrm{\mid }\le M-m$; their coefficients are ${\mathcal{E}}_S^{2(i-w)-u,i-w}(S)$ for $w=w(\alpha )+w(\beta )$. For any surviving monomial, step 1.1 gives $w\le i$, hence $\mathrm{\mid }\alpha \mathrm{\mid }\le i$, $\mathrm{\mid }\beta \mathrm{\mid }\le i$, and every variable index occurring is at most $i$. Consequently $N-n\ge i$ and $M-m\ge i$ include every potentially nonzero monomial. The one-factor assertion follows by omitting $\beta$. These are additive basis statements; no quotient-ring presentation of the finite Grassmannian is presumed.

Increasing $N$ or $M$ restricts each universal quotient bundle to the previous one, so the pullback preserves all these basis monomials. In the ambient ranges of step 2.1 it is an isomorphism in mapping homotopy degrees zero and one. Under rank stabilization $V\mapsto V\oplus \mathcal{O}$, the old Chern classes are preserved and the new top class pulls back to zero: this follows from the source's Whitney formula and the pointed orientation in [F1]–[F2]. Once $n,m\ge \max \{1,i\}$, every newly introduced variable has weight greater than $i$ and has zero coefficient by step 1.1. Thus the rank restriction maps are also isomorphisms in both homotopy degrees. Applying the actual comparison in [F1] gives the same stabilization on the classifying stacks and their product.

Take a cofinal countable diagonal in the finite approximation indices, for example $n=m=k$ and $N=M=2k$, followed from $k\ge \max \{1,i\}$. The mapping-spectrum tower has eventually isomorphic ${\pi }_0$ and ${\pi }_1$ by step 3.1; the same holds for the rank tower of classifying stacks. In the sequential homotopy-limit model, the possible kernel of ${\pi }_0\mathrm{holim}\to \lim {\pi }_0$ is the cokernel of $1-\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}$ on the product of the ${\pi }_1$ groups. On the constant tail identify their transition maps with identity. Given a sequence $(a_k)$, set $b_{k_0}=0$ and recursively $b_{k+1}=b_k-a_k$; then $(1-\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t})(b{)}_k=a_k$. Thus this cokernel, equivalently ${\lim }^1{\pi }_1$, is zero. The bounded-weight universal classes are jointly detected by the finite approximations with no phantom term. Nothing in this argument requires a Noetherian or dimension bound on $S$.