Framed motives of relative motivic spheres

The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2]. Based on the theory, framed motives are introduced and studied in [GP1]. These are Nisnivich sheaves of $S^1$-spectra and the major computational tool of [GP1]. The aim of this paper is to show the following result which is essential in proving the main theorem of [GP1]: given an infinite perfect base field $k$, any $k$-smooth scheme $X$ and any $n\geq 1$, the map of simplicial pointed Nisnevich sheaves $(-,\mathbb{A}^1//\mathbb G_m)^{\wedge n}_+\to T^n$ induces a Nisnevich local level weak equivalence of $S^1$-spectra $$M_{fr}(X\times (\mathbb{A}^1// \mathbb G_m)^{\wedge n})\to M_{fr}(X\times T^n).$$ Moreover, it is proven that the sequence of $S^1$-spectra $$M_{fr}(X \times T^n \times \mathbb G_m) \to M_{fr}(X \times T^n \times\mathbb A^1) \to M_{fr}(X \times T^{n+1})$$ is locally a homotopy cofiber sequence in the Nisnevich topology. Another important result of this paper shows that homology of framed motives is computed as linear framed motives in the sense of [GP1]. This computation is crucial for the whole machinery of framed motives [GP1].