External symmetric P¹ model, orientation criterion and fundamental stabilization

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

In the source’s non-A¹ theory over a qcqs derived scheme $S$, a symmetric lax P¹-spectrum is a module over the free commutative sphere algebra generated by P¹ in symmetric-sequence degree1 (Definition1.3.4). It is a P¹-spectrum precisely when its adjoint bonding maps are equivalences (Definition1.3.8). The localization is symmetric monoidal. A commutative algebra over that sphere algebra therefore gives a motivic ring spectrum when the Ω-condition holds. Coefficient stabilization has the induced adjunction with lax monoidal right adjoint Ω∞, and the P¹-shift evaluates at the next Ω-spectrum level (Notation2.2.7–8). A pointed Picard class restricting along $\mathcal{O}(1)$ to the P¹-suspended unit gives a linear orientation (Lemma3.1.4/Remark3.1.5). An orientable spectrum is fundamental. On fundamental spectra coefficient stabilization is an equivalence (Theorem2.4.5), and modules over an oriented ring spectrum form a stable category (Corollary2.4.8). The PBF is the actual finite sum of powers of that first class; Lemma3.2.3 gives the bundle version. These results are over arbitrary qcqs derived bases and require no A¹-invariance.

Import exactly these formal constructions and criteria. They permit the concrete model assembled in the local lemmas; they do not assert that any abstract bigraded oriented theory automatically satisfies the source’s stronger hypotheses.