Framed motives of smooth affine pairs

Abstract

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category SH ( k ) in terms of Voevodsky's framed correspondences. In particular, the motivically fibrant Ω-resolution in positive degrees of the motivic suspension spectrum Σ P 1 ∞ X + , where X + = X ⨿ ⁎ , for a smooth scheme X ∈ 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 ( A l × X ) / ( ( A l − 0 ) × X ) , X ∈ 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 ⊂ X . The result gives an explicit motivically fibrant Ω-resolution in positive degrees for the motivic suspension spectrum Σ P 1 ∞ ( X + / U + ) of the quotient-sheaf X + / U + .