K-stack mapping spaces and extension from smooth affine schemes

Assume the Axiom of Choice and compatible size universes. Use the source-native smooth Zariski stack category over a qcqs base $S$. Write ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{n,S}$ for the stack of rank-$n$ vector bundles, pointed by ${\mathcal{O}}^n$, and $B_{\mathrm{\infty }}={\mathrm{colim}}_n{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{n,S}$ under addition of a trivial line. Let $\mathcal{K}$ be the K-theory stack and let $\tilde{\mathcal{K}}=\mathrm{fib}(\mathrm{r}\mathrm{k}:\mathcal{K}\to \underset{‾}{\mathbb{Z}})$. The rank map has its trivial-bundle section; subtraction of this section gives rank reduction. Let $E$ be a homotopy-commutative oriented motivic ring satisfying PBF, with the coefficient compactness and multiplicativity hypotheses of the framework; source §2.2.6 includes the smooth stack category itself among these coefficients. For an $E$-module $M$ in the elementary-blowup-excision localization, put $Y={\mathrm{\Omega }}_{{\mathbb{P}}^1}^{\mathrm{\infty }}M$. Then the canonical reduced object map $u:B_{\mathrm{\infty }}\to \tilde{\mathcal{K}}$ induces $u^{\ast }:{\mathrm{Map}}_{\ast }(\tilde{\mathcal{K}},Y)\simeq {\mathrm{Map}}_{\ast }(B_{\mathrm{\infty }},Y)\mathrm{.}$ This is an equivalence of mapping spaces, natural in $M$; it can transport coherent diagrams already given in these mapping spaces. It asserts no higher coherent stabilization from a list of component classes alone. Elementary blowup excision means sending the blowup square of ${\mathbb{A}}_T^n$ at its zero section to a pullback square; for oriented spectra it is equivalent to PBF. Thus shifts of the actual oriented PBF coefficient spectrum are permitted here.

For a fixed commutative base ring $R$, let $j$ include smooth $R$-algebras into all $R$-algebras. On classical affine rings, the K-space ${\mathcal{K}}_{\mathrm{a}\mathrm{f}\mathrm{f}}$ and the fixed-rank stack ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n$ are left Kan extended along $j$. Precisely, if $L⊣j^{\ast }$ is the space-valued left Kan extension/restriction adjunction, its counits $L{j}^{\ast }{\mathcal{K}}_{\mathrm{a}\mathrm{f}\mathrm{f}}\to {\mathcal{K}}_{\mathrm{a}\mathrm{f}\mathrm{f}}$ and $L{j}^{\ast }{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n\to {\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_n$ are equivalences. The analogous assertion holds for every smooth algebraic stack with quasi-affine diagonal. All functors and adjunctions here retain higher coherences. The affine K-space is identified with the specified perfect K-space using the canonical full affine comparison, including components and bundle object maps.

For the classical-stack clause, an algebraic stack has the smooth-presentation and representable-diagonal meaning of the declared algebraic-stack definition. Smooth over $R$ means that a smooth atlas is smooth over $R$; quasi-affine diagonal means that every base change of the diagonal to a scheme is represented by a quasi-affine scheme over that scheme. If $(U,R_1,s,t,c)$ is a smooth groupoid in algebraic spaces, its quotient stack is algebraic and $U$ maps smoothly and surjectively to it. In particular a smooth affine group $G$ over a field $F$ has the smooth trivial-torsor atlas $\mathrm{Spec}F\to BG$, where $BG=[\mathrm{Spec}F\mathrm{/}G]$ is the stack of torsors and their isomorphisms. A $G$-torsor is locally isomorphic to $G$ with its translation action in the faithfully flat topology of locally finitely presented covers. The quasi-affine-diagonal hypothesis in the preceding Kan-extension clause must still be checked in each application.

Source-specific non-A¹ motivic spectrum framework and hypotheses

Nonconnective K theory of perfect complexes

Full affine K-space comparison and its sheafification

The Axiom of Choice

Algebraic stacks and their inertia stacks

The mapping-space equivalence, rank splitting and reduced plus description are imported from source record 0, at the displayed equivalences in the proof of Theorem 5.1.4. Taking fibers of evaluation at the trivial bundle gives the pointed version written here. Lemmas 3.3.4–3.3.5 give the excision qualification. These statements do not require the desired Chern map to be additive. Source record 1 supplies the affine left Kan extension assertions. Source record 2 supplies the smooth quotient atlas, with the classical algebraic-stack type fixed by the declared definition. Their hypotheses and reading limits are recorded separately; the specified perfect model is matched by the declared full affine supplier. No proof of these substantial imported results is claimed here. The assumed Choice and size bounds are those of the framework and the supplied affine comparison.