Framed Motives of Relative Motivic Spheres

The category of framed correspondences Fr∗(k), framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [Voe]. Based on the theory, framed motives are introduced and studied in [GP1]. The aim of this paper is to prove the following results stated in [GP1, 9.3]: for any k-smooth scheme X and any n > 1 the map of simplicial pointed sheaves (−,A⌋Gm) + → T n induces a Nisnevich local level weak equivalence of S1spectra M f r(X × (A ⌋Gm) ∧n)→ M f r(X ×T n) and the sequence of S1-spectra M f r(X ×T n ×Gm)→ M f r(X ×T n ×A1)→ M f r(X ×T n+1) is locally a homotopy cofiber sequence in the Nisnevich topology.