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 Ω-resolution in positive degrees of the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }{X}_{+}$, where $X_{+}=X⨿⁎$, 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 an explicit motivically fibrant Ω-resolution in positive degrees for the motivic suspension spectrum ${\mathrm{\Sigma }}_{{\mathbb{P}}^1}^{\infty }(X_{+}/U_{+})$ of the quotient-sheaf $X_{+}/U_{+}$.

Garkusha 和 Panin 的框架动机理论给出了稳定动机同伦范畴的计算 $\mathbf{上海}(克)$就 Voevodsky 的框架对应而言。特别是，动机悬浮谱的正度数的动机纤维 Ω 分辨率${\mathrm{\Sigma }}_{{\mathbb{磷}}^1}^{\infty }{X}_{+}$， 在哪里 $X_{+}=X⨿⁎$，对于一个平滑的方案 $X\in {\mathrm{SM}}_{克}$在无限完美域k上计算。

Garkusha、Neshitov 和 Panin 对相关动机领域框架动机的计算 $({\mathbb{一种}}^{升}\times X)/(({\mathbb{一种}}^{升}-0)\times X)$, $X\in {\mathrm{SM}}_{克}$, 是理论的组成部分之一。在文章中，我们将此结果扩展到一对的情况$(X,你)$由平滑的仿射各种给定的X超过ķ和打开subscheme$你\subset X$.

结果给出了动机悬浮谱的正度数的明确动机纤维 Ω 分辨率 ${\mathrm{\Sigma }}_{{\mathbb{磷}}^1}^{\infty }(X_{+}/{你}_{+})$ 商层的 $X_{+}/{你}_{+}$.