Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{S}\mathbf{H}(k)$ in terms of Voevodsky’s framed correspondences. In particular the motivically fibrant $\mathrm{\Omega }$-resolution in positive degrees of the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\mathrm{\infty }}{X}_{+}$, where $X_{+}=X⨿\ast$, for a smooth scheme $X\in {\mathrm{S}\mathrm{m}}_k$ over an infinite perfect field $k$, is computed. The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $({\mathbb{A}}^l\times X,({\mathbb{A}}^l-0)\times X)$, $X\in $\mathrm{S}\mathrm{m}$k$, is one of ingredients in the theory. In the article we extend this result to the case of a pair $(X,U)$ given by a smooth affine variety $X$ over $k$ and an open subscheme $U\subset X$. The result gives the explicit motivically fibrant $\mathrm{\Omega }$-resolution in positive degrees for the motivic suspension spectrum $\Sigma{\mathbb P^1}^\infty (X_+/U_+)$ofthefactor-sheaf$X_+/U_+$.