On the triangulated category of framed motives $\text{DFr}_{-}^{eff}(k)$

TL;DR

This paper constructs a new triangulated category of framed motives over a field, showing it aligns with classical motivic categories and retains symmetric monoidal structure, facilitating explicit calculations in motivic homotopy theory.

Contribution

It introduces the triangulated category of framed motives $ ext{D} ext{Fr}_{-}^{eff}(k)$ and proves its equivalence with a subcategory of classical motivic spectra, preserving monoidal structures.

Findings

$ ext{D} ext{Fr}_{-}^{eff}(k)$ is equivalent to $ ext{SH}^{eff}_{-}(k)$

The new category is naturally symmetric monoidal

The construction simplifies explicit calculations in motivic homotopy theory

Abstract

The category of framed correspondences $Fr_*(k)$ was invented by Voevodsky in his notes in order to give another framework for SH(k) more amenable to explicit calculations. Based on that notes and on their JAMS paper Garkusha and the author introduced in a very recent paper a triangulated category of framed bispectra $\text{SH}_{nis}^{fr}(k)$. It is shown in the latter paper that $\text{SH}_{nis}^{fr}(k)$ recovers classical Morel-Voevodsky triangulated category of bispectra $\text{SH}(k)$. For any infinite perfect field $k$ a triangulated category of $\mathbb{F}\text{r}$-motives $\text{D}\mathbb{F}\text{r}_{-}^{eff}(k)$ is constructed in the style of the Voevodsky construction of the category $\text{DM}_-^{eff}(k)$. In our approach the Voevodsky category of Nisnevich sheaves with transfers is replaced with the category of $\mathbb{F}\text{r}$-modules. To each smooth $k$-variety $X$ the $\mathbb{F}\text{r}$-motive $\text{M}_{\mathbb{F}\text{r}}(X)$ is associated in the category $\text{D}\mathbb{F}\text{r}_{-}^{eff}(k)$. We identify the triangulated category $\text{D}\mathbb{F}\text{r}_{-}^{eff}(k)$ with the full triangulated subcategory $\text{SH}^{eff}_{-}(k)$ of the classical Morel-Voevodsky triangulated category $\text{SH}^{eff}(k)$ of effective motivic bispectra. Moreover, the triangulated category $\text{D}\mathbb{F}\text{r}_{-}^{eff}(k)$ is naturally symmetric monoidal. The mentioned identification of the triangulated categories respects the symmetric monoidal structures on both sides.