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

Use the graded algebra $A_n$ from [F1], with trivial symmetric-group actions on each level and the induced coherent symmetric products. Equivalently restrict its grading along the symmetric monoid $B\mathrm{\Sigma }\mathbb{N}\to \mathbb{N}$. The class $b:{\mathrm{\Sigma }}^{\mathrm{\infty }}({\mathbb{P}}^1,\mathrm{\infty })\to A_1$ defines an algebra map from the free graded sphere algebra of [F2]. Thus $A$ is a commutative lax P¹-spectrum in Zariski sheaves of ordinary spectra; its bonding maps are multiplication by this very class.

The adjoint map $A_n(Y)\to \mathrm{fib}(A_{n+1}({\mathbb{P}}_Y^1)\to A_{n+1}(Y))$ is the second summand of the actual P¹ formula in [F1], hence an equivalence. The fiber uses the infinity section, on which $b$ is zero with the trivial-line nullhomotopy supplied by its pointed Picard map. Consequently $A$ is already an Ω-P¹-spectrum, so its levels need no replacement by unrelated objects.

Apply the lax monoidal right adjoint Ω∞ of coefficient stabilization in [F2]. It preserves the adjoint bonding equivalences and gives the stated ring spectrum ${\mathcal{E}}_S$ with levels ${\mathrm{\Omega }}^{\mathrm{\infty }}{A}_n$. By the shift/evaluation rule, ${\mathcal{E}}_S^{2i,i}(Y)={\pi }_0{A}_i(Y)=H^{2i}(\mathbb{Z}(i{)}_{\mathrm{m}\mathrm{o}\mathrm{t}}(Y))$. The maps arise from the actual graded products, so this identification preserves their multiplication and naturality. All constructions restrict from all qcqs schemes over the prime field, so they apply when $S$ is singular.