Framed motives of algebraic varieties (after V. Voevodsky)

TL;DR

This paper develops the theory of framed motives for algebraic varieties using Voevodsky's framed correspondences, providing explicit models and computations that connect motivic homotopy theory with classical topological spectra.

Contribution

It introduces and studies framed motives of algebraic varieties, establishing their role as computational tools and connecting motivic spectra with classical topological spectra.

Findings

Framed motives serve as explicit computational tools for motivic spectra.

The bispectrum of a framed motive has the motivic homotopy type of the suspension bispectrum.

The framed motive of a point models the classical sphere spectrum in characteristic zero.

Abstract

Using the theory of framed correspondences developed by Voevodsky, we introduce and study framed motives of algebraic varieties. They are the major computational tool for constructing an explicit quasi-fibrant motivic replacement of the suspension $\mathbb P^1$-spectrum of any smooth scheme $X\in Sm/k$. Moreover, it is shown that the bispectrum $$(M_{fr}(X),M_{fr}(X)(1),M_{fr}(X)(2),\ldots),$$ each term of which is a twisted framed motive of $X$, has motivic homotopy type of the suspension bispectrum of $X$. Furthermore, an explicit computation of infinite $\mathbb P^1$-loop motivic spaces is given in terms of spaces with framed correspondences. We also introduce big framed motives of bispectra and show that they convert the classical Morel--Voevodsky motivic stable homotopy theory into an equivalent local theory of framed bispectra. As a topological application, it is proved that the framed motive $M_{fr}(pt)(pt)$ of the point $pt=Spec(k)$ evaluated at $pt$ is a quasi-fibrant model of the classical sphere spectrum whenever the base field $k$ is algebraically closed of characteristic zero.