Framed motives of algebraic varieties (after V. Voevodsky)

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.