Actual supported Chern operation and its smooth comparison

Fix $i\ge 1$. Use the EM P¹-ring model of [F1] on the size-bounded smooth qcqs site over $X$, which includes the identity of $X$ and its quasi-compact opens. The universal bundle classes are the projective-relation classes of [F3]; their stability under $E\mapsto E\oplus \mathcal{O}$ follows from actual Whitney. Apply [F1] to obtain the pointed maps on the local perfect K-space model, and compose on derived sections with its canonical unit from the actual K-values. This gives the displayed maps $C_i$ and their actual restriction arrows; the target is the shifted Eilenberg spectrum in the Statement. Their values on bundles are the original classes. The full affine comparison and local-replacement unit, rather than an assertion of unbounded K-hyperdescent or global bundle representation, are the specified inputs.

On affine $X$, [F3] represents each perfect class as $[P]-[Q]$ for actual bundles. Apply [F4] to obtain a finite projective complement $Q^{\mathrm{\prime }}$ with $Q\oplus Q^{\mathrm{\prime }}\cong {\mathcal{O}}_X^N$. The same class is $[P\oplus Q^{\mathrm{\prime }}]-[{\mathcal{O}}_X^N]$. The completed universal construction in [F1] projects away from trivial rank, so its Chern value is that of the actual bundle $P\oplus Q^{\mathrm{\prime }}$. The actual Whitney identity gives $c_t(Q)c_t(Q^{\mathrm{\prime }})=c_t({\mathcal{O}}_X^N)=1$ and thus $c_t(P\oplus Q^{\mathrm{\prime }})=c_t(P)c_t(Q{)}^{-1}$. Its positive coefficients are precisely the prescribed virtual-series values. Varying ranks are handled on the finite clopen rank strata as in [F3]; no global bundle-difference representation on nonaffine schemes is used.

If $U$ is not quasi-compact, take the inverse limits of the natural operations over its quasi-compact open subsets, exactly as prescribed in [F1] and [F5]. These subsets are qcqs by quasi-separatedness of $X$. Since ${\mathrm{\Omega }}^{\mathrm{\infty }}$ preserves limits, the resulting source and target spaces are those of the declared extensions. Apply these natural pointed maps to the restriction arrow from $X$ to $U$ and take homotopy fibers over zero. The target is ${\mathrm{\Omega }}^{\mathrm{\infty }}(M_i(X\mathrm{/}U)[2i])$, whose ${\pi }_0$ is $H_x^{2i,i}(X)$ by [F5]. Taking ${\pi }_0$ defines $c_{i,x}^{\mathrm{f}\mathrm{i}\mathrm{b}}$. The natural fiber projections give the support-to-global square; compatible restrictions give open naturality. If $U$ is quasi-compact, [F5] also identifies the source with $K_x(X)$ and its support-forgetting map. This yields the asserted $c_{i,x}$ on the category-defined group. No additive spectrum map for the individual higher Chern operations is asserted.

Assume $U$ quasi-compact and let $W\subseteq X$ be a quasi-compact open containing $x$. Then $X=U\cup W$, and $U\cap W=W\setminus \{x\}$ is quasi-compact by quasi-separatedness. The K-localization/open-descent supplier in [F5] and the actual derived motivic descent in [F6] therefore apply to the same four qcqs opens. The conditional excision lemma of [F6] identifies the support fibers on $X$ and $W$; for K-theory these are also the category-defined support spectra by [F5]. Naturality of step 2.2 gives the commuting squares of supported Chern operations under these excision equivalences, compatible with forgetting support.

Under the additional classical-comparison hypotheses, every object of the stated smooth finite-presentation qcqs site over $V$ is smooth finite type over the perfect field $k$. The same holds for its quasi-compact opens, including $V\setminus \{x\}$, since $V$ is noetherian. Thus [F2] applies naturally to the coefficient restriction diagram on $(V,x)$. It preserves products and the actual Picard first class, so the supplied operation-transport square identifies the projective-relation universal classes, their completed pointed K-space maps, and their support fibers. By step 2.2 this is the restriction of the constructed operation; when $U$ is quasi-compact, step 3.1 makes it the asserted excision identification. In the article specialization, algebraic closedness implies perfectness, finite type over a field makes $X$ noetherian and $U$ quasi-compact, and the smooth locus near the stated point has an affine finite-type neighbourhood of pure dimension three. Hence the article's $i=3$ application satisfies all these hypotheses without narrowing the general qcqs construction. The same actual Picard operation is compared throughout; the name of a tautological line alone introduces no extra sign.