Smooth Affine Model for the Framed Correspondences Spectrum

Morel–Voevodsky’s unstable pointed motivic homotopy category H_●(k) over an infinite perfect field is considered. For a smooth affine scheme Y over k, a smooth ind-scheme F_l(Y) and an open subscheme E_l(Y) are constructed for all l > 0, so that the motivic space F_l(Y)/E_l(Y) is equivalent in H_●(k) to the motivic space ${\Omega}_{{\mathrm{\mathbb{P}}}^1}^{\infty}\sum \limits_{{\mathrm{\mathbb{P}}}^1}^{\infty}\left(Y\times {T}^l\right),\kern0.5em T=\left({\mathbbm{A}}^1/{\mathbbm{A}}^1-0\right),\kern0.5em l>0$. The construction is not functorial on the category of affine schemes but is functorial on the category of so-called framed schemes constructed for this purpose.