Framed motives of relative motivic spheres

TL;DR

This paper proves key equivalences and sequences in the theory of framed motives related to motivic spheres, facilitating computations and advancing the foundational framework established by Voevodsky and further developed in prior work.

Contribution

It establishes Nisnevich local weak equivalences and homotopy cofiber sequences for framed motives, and shows that their homology can be computed as linear framed motives, enhancing the computational framework.

Findings

Proves a Nisnevich local level weak equivalence involving framed motives and motivic spheres.

Establishes a homotopy cofiber sequence of $S^1$-spectra in the context of framed motives.

Shows that homology of framed motives equals that of linear framed motives, simplifying calculations.

Abstract

The category of framed correspondences $Fr_*(k)$ and framed sheaves were invented by Voevodsky in his unpublished notes [V2].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].