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The category of framed correspondences $Fr_*(k)$ and framed sheaves wereinvented by Voevodsky in his unpublished notes [V2]. Based on the theory,framed motives are introduced and studied in [GP1]. These are Nisnivich sheavesof $S^1$-spectra and the major computational tool of [GP1]. The aim of thispaper is to show the following result which is essential in proving the maintheorem of [GP1]: given an infinite perfect base field $k$, any $k$-smoothscheme $X$ 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 locallevel weak equivalence of $S^1$-spectra  $$M_{fr}(X\times (\mathbb{A}^1// \mathbb G_m)^{\wedge n})\to M_{fr}(X\timesT^n).$$ 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 cofibersequence in the Nisnevich topology. Another important result of this papershows that homology of framed motives is computed as linear framed motives inthe sense of [GP1]. This computation is crucial for the whole machinery offramed motives [GP1].

Framed motives of relative motivic spheres