Framed motives of smooth affine pairs

Abstract

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular the motivically fibrant $\Omega$-resolution in positive degrees of the motivic suspension spectrum $\Sigma_{\mathbb P^1}^\infty X_+$, where $X_+=X\amalg *$, for a smooth scheme $X\in \mathrm{Sm}_k$ over an infinite perfect field $k$, is computed. The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $(\mathbb A^l\times X,(\mathbb A^l-0)\times X)$, $X\in \mathrm{Sm}_k$, 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 $\Omega$-resolution in positive degrees for the motivic suspension spectrum $\Sigma_{\mathbb P^1}^\infty (X_+/U_+)$ of the factor-sheaf $X_+/U_+$.