2014/09/15 by Garkusha, Grigory, Panin, Ivan
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.1409.4372
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 P1-spectrum of any smooth scheme X∈ Sm/k. Moreover, it is shown that the bispectrum (Mfr(X),Mfr(X)(1),Mfr(X)(2),…), 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 P1-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 Mfr(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.