External formal universal Chern maps and supported construction

PDF p. 17: Corollary6.9 supported factor calculation

Assume the Axiom of Choice and the fixed site and universe bounds. For every qcqs scheme $X$ over a field and every integer $i\ge 1$, the specified natural stable pointed universal bundle maps $c_{i,n}:BG{L}_n\to K(D_i,2i)$ extend by the formal Gillet completion construction to pointed operations on the local perfect K-space model. Here $D_i$ denotes the weight-$i$ coefficient complex, and $K(D_i,2i)={\mathrm{\Omega }}^{\mathrm{\infty }}(D_i[2i])$ is its derived Eilenberg target. The construction uses the local equivalence $\mathrm{\Omega }BQ\mathcal{P}\simeq \underset{‾}{\mathbb{Z}}\times {\mathbb{Z}}_{\mathrm{\infty }}BGL$, projection away from trivial rank, completion of the specified stable class, and ${\mathbb{Z}}_{\mathrm{\infty }}K(D_i,2i)\simeq K(D_i,2i)$. It is natural in the coefficient maps and restrictions and retains the original universal bundle classes.

For the actual EM complexes, the supplied non-${\mathbb{A}}^1$ ${\mathbb{P}}^1$-model and its actual orientation provide these projective-relation classes on ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{n,X}$, with their stability. The natural full affine comparison, including degree zero, identifies the sheaf Q-space model with the local sheaf model of the specified perfect K-space. Compose with the canonical perfect K-space local-replacement unit to obtain the actual operations on global perfect K-spaces and their restriction arrows. This applies on the small site of opens of $X$ and on its smooth big site, with Zariski or Nisnevich topology and the same stalkwise local homotopy convention for the source and derived Eilenberg targets. The affine-basis comparison is valid before and after this local replacement; no identification of ordinary and hypercomplete localizations is required. It requires neither global bundle-difference representatives for perfect $K_0$ nor ${\mathbb{A}}^1$-invariance of EM.

For every closed support $Z\subseteq X$, taking the homotopy fibers of these pointed restriction arrows gives the supported operations and their forget-support squares. For a non-quasi-compact open complement, extend the actual K-spaces and coefficient targets by the inverse limits over its quasi-compact open subsets, using the declared support convention; the compatible operations extend by the same limits. When the complement is quasi-compact, identification with category-defined supported perfect K-theory uses the separate localization input when that interpretation is needed. No such category comparison is inferred for an arbitrary open complement. The index-zero convention is separately $C_0=1$ as an unpointed constant. It is the constant term of the total Chern series, not a pointed map to the additive coefficient target; no index-zero support-fiber operation is supplied by this pointed construction.

An actual non-A¹ P¹-spectrum for the supplied EM complexes

Actual EM orientation and projective bundle maps on that model

Full affine K-space comparison and its sheafification

Only the formal construction at the stated locators is imported from Asakura–Sato. Their §3.1 and Appendix A Theorem A.1 have an explicit resolution hypothesis for motivic admissibility. Neither that theorem nor a resolution hypothesis for the EM complexes is used here. The class inputs and full-space affine comparison are supplied separately. No general determinant coherence theorem is needed for this formal construction beyond the actual Picard orientation already in the EM supplier.

The Asakura–Sato 2019 primary PDF read here has SHA256 6736b9710dd4d8d5fd89dfa2863a72cf90fc274ba47a45651fc7024209b4dcbf. The Annala–Iwasa April 14, 2025 primary PDF has SHA256 ad36a2ed6a7e8b5815c94b19100b2431f2533fb538c57843bdbeef3d1b400261. The live source URLs were independently opened; actual reading is limited to the passages specified above.