External actual EM graded algebra, Picard class and complex PBF

PDF p. 8: Actual Chern and Whitney identities; PDF p. 11: Total virtual Chern class

Assume the Axiom of Choice and the fixed compatible site and universe bounds of the supplied constructions. On qcqs schemes over a prime field, $A_n=\mathbb{Z}(n{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}[2n]$ forms a natural unital commutative graded algebra in spectra. Put $A_n=0$ for $n<0$. There is the actual natural complex map $R{\mathrm{\Gamma }}_{\mathrm{N}\mathrm{i}\mathrm{s}}({\mathbb{G}}_m)[-1]\to \mathbb{Z}(1{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}$ of Definition5.2; after shifting by2 and taking Ω∞ this is the pointed Picard class. Its restriction to $({\mathbb{P}}^1,\mathrm{\infty })$ is $b=c_1(\mathcal{O}(1))$ with the trivial-line nullhomotopy at infinity. Multiplication by its powers and pullback give the actual complex equivalences ${⨁}_{m=0}^d\mathbb{Z}(j-m{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(Y)[-2m]\stackrel{\sum c_1(\mathcal{O}(1){)}^m{\pi }^{\ast }}{\to }\mathbb{Z}(j{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}({\mathbb{P}}_Y^d)\mathrm{.}$ These hold for $j\ge 0$, and also for $j<0$ with all terms zero. The general bundle version holds with the stated line/quotient convention conversion by dual bundle. All structures restrict naturally to smooth schemes over any qcqs $S\mathrm{/}k$.

Import the stated EM constructions and actual maps, with the completed projective-bundle supplier’s quotient convention. No A¹-invariance is imported. The Picard map is a map of complexes and hence supplies its coherent pointed class, rather than a choice made from an identification of groups.