Filtered affine K-space continuity with the actual free-class arrow

First justify passage from spectral colimits to entire zeroth spaces within [F4]'s coherent enhancement. For any $I$-diagram of spectra write its reduced excisive functors as $E_i:S_{\ast }^{\mathrm{f}\mathrm{i}\mathrm{n}}\to S$, and form their colimit $H$ in the ambient functor infinity-category. By [F5], $H(T)\simeq {\mathrm{colim}}_i{E}_i(T)$ coherently in $T$. The value $H(\ast )$ is contractible because filtered space colimits preserve the terminal point. For every pushout square of finite pointed spaces, each $E_i$ gives a homotopy pullback square; [F5]'s finite-limit preservation makes its pointwise filtered colimit a homotopy pullback square too. Thus $H$ is reduced and excisive, hence lies in $\mathrm{S}\mathrm{p}$. Since this is a full subcategory, the ambient colimit universal property restricts to an equivalence ${\mathrm{Map}}_{\mathrm{S}\mathrm{p}}(H,Z)\simeq {\lim }_{i\in I^{\mathrm{o}\mathrm{p}}}{\mathrm{Map}}_{\mathrm{S}\mathrm{p}}(E_i,Z)$ for every spectrum $Z$. Therefore $H$, with its same coherent cocone, is the spectral colimit. Evaluation at $S^0$ now gives the canonical equivalence ${\mathrm{colim}}_i{\mathrm{\Omega }}^{\mathrm{\infty }}{E}_i\simeq {\mathrm{\Omega }}^{\mathrm{\infty }}({\mathrm{colim}}_i{E}_i)$ of entire spaces. Apply this to the connective spectra $K^c(A_i)$ of [F1]. To justify the index hypothesis of [F3], first check that $N(I)$ is filtered in [F6]'s sense. For a finite simplicial set $L$ and a map $L\to N(I)$, choose arrows from its finitely many vertex images to a common object of $I$. For each edge, the two resulting arrows from its initial vertex to that object can be made equal after postcomposition, by ordinary filteredness. There are only finitely many edge conditions; equalize them successively, retaining previous equalities. These compatible arrows give the cone $L^{▹}\to N(I)$, since higher simplices in an ordinary nerve are determined by their composable arrows. For empty $L$ use nonemptiness of $I$. Thus [F6] supplies a small directed poset $J$ and $u:J\to I$ with $N(u)$ cofinal. Its cocone criterion in the ordinary category of commutative unital rings gives the canonical isomorphism ${\mathrm{colim}}_{j\in J}{A}_{u(j)}\cong A$; in spectra it gives the canonical equivalence ${\mathrm{colim}}_{j\in J}{K}^c(A_{u(j)})\simeq {\mathrm{colim}}_{i\in I}{K}^c(A_i)$. Set $X_j=\mathrm{Spec}{A}_{u(j)}$. This is now a directed inverse system of affine qcqs schemes with affine transition maps. Appendix C.1 in [F3] identifies its inverse limit with $\mathrm{Spec}({\mathrm{colim}}_j{A}_{u(j)})=\mathrm{Spec}A$. Applied also in the opposite category of schemes, [F6]'s equivalence of ordinary cocone categories shows that this same scheme is the inverse limit of the original $I^{\mathrm{o}\mathrm{p}}$-diagram. Proposition 3.20.2 gives ${\mathrm{colim}}_j{K}^c(A_{u(j)})\simeq K^c(A)$. All these maps are induced by the same scalar-extension cocone; therefore the canonical $I$-indexed map ${\mathrm{colim}}_i{K}^c(A_i)\to K^c(A)$ is an equivalence as well. The space-colimit analogue is supplied by the same cofinality theorem, and [F4]'s comparison to degreewise simplicial colimits applies to both filtered categories $I$ and $J$. The strict-perfect affine comparison in [F3] matches this theory with [F1]'s bounded-projective theory and its scalar-extension maps. Evaluate this continuity equivalence and compose with [F1]'s natural equivalences ${\mathrm{\Omega }}^{\mathrm{\infty }}{K}^c(B)\simeq \mathcal{K}(B)$ for $B=A_i,A$. The comparison just proved and [F4]'s canonical simplicial homotopy-colimit comparison give exactly the canonical map in the Statement as a weak equivalence, including all components. This uses only connective K-spectrum continuity; it does not identify an ordinary strict prespectrum colimit with a coherent colimit.

An invertible $n$ by $n$ matrix is a pair $(g,h)$ with $gh=hg=1$; the inverse $h$ is unique. These are finite equations on finitely many entries. Thus [F2]'s finite-product and equalizer comparison identifies ${\mathrm{colim}}_{i}G{L}_n(A_i)$ with $G{L}_n(A)$, retaining multiplication and stabilization. Interchanging the two colimits gives the stable group comparison. In degree $q$ the group nerve is the $q$-fold product, so [F2] gives the asserted stable nerve isomorphism. Taking product with the constant $\mathbb{Z}$ also commutes with this colimit: the index category is nonempty and filtered, so every integer occurs and cannot change under a transition. By [F1] the free-class construction is natural under each ring map and preserves the same objects, automorphisms and sums. Its square with the canonical colimit comparison in step 1.1 therefore commutes in spaces. Reducing by the same trivial rank commutes with scalar extension, and translation by $m[A_i]$ goes to translation by $m[A]$, for negative $m$ as well by the grouplike K-space inverse. This proves every claimed compatibility without assuming freeness of arbitrary projectives.