Framed motives of algebraic varieties (after V. Voevodsky)

Abstract

A new approach to stable motivic homotopy theory is given. It is based on Voevodsky’s theory of framed correspondences. Using the theory, framed motives of algebraic varieties are introduced and studied in the paper. They are the major computational tool for constructing an explicit quasi-fibrant motivic replacement of the suspensionP1\mathbb P^1-spectrum of any smooth schemeX∈Sm/kX\in Sm/k. Moreover, it is shown that the bispectrum(Mfr(X),Mfr(X)(1),Mfr(X)(2),…),\begin{equation*} (M_{fr}(X),M_{fr}(X)(1),M_{fr}(X)(2),\ldots ), \end{equation*}each term of which is a twisted framed motive ofXX, has the motivic homotopy type of the suspension bispectrum ofXX. Furthermore, an explicit computation of infiniteP1\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 motiveMfr(pt)(pt)M_{fr}(pt)(pt)of the pointpt=Spec⁡kpt=\operatorname {Spec} kevaluated atptptis a quasi-fibrant model of the classical sphere spectrum whenever the base fieldkkis algebraically closed of characteristic zero.