Framed motives of smooth affine pairs

Abstract

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

The deduction of the theorem 1 from the cone theorem for is corrected

References in corpus (1)

• Strictly homotopy invariance of Nisnevich sheaves with GW-transfers

Cited by in corpus (4)

• Modules over algebraic cobordism
• Framed correspondences and the zeroth stable motivic homotopy group in odd characteristic
• Notes on motivic infinite loop space theory
• The homomorphism of presheaves ${\mathrm{K}}_{\ast }^{\mathrm{MW}}\to {\pi }_s^{\ast ,\ast }$ over a base