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Using the theory of framed correspondences developed by Voevodsky [22], 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 P 1 -spectrum of any smooth scheme X ∈ Sm/k. Moreover, it is shown that the bispectrumeach 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 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 f r (pt)(pt) of the point pt = Spec k evaluated at pt is a quasifibrant model of the classical sphere spectrum whenever the base field k is algebraically closed of characteristic zero.

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A kind of motivic algebra of spectral categories and modules over them is developed to introduce K-motives of algebraic varieties. As an application, bivariant algebraic K-theory K(X, Y ) as well as bivariant motivic cohomology groups H p,q (X, Y, Z) are defined and studied. We use Grayson's machinery [12] to produce the Grayson motivic spectral sequence connecting bivariant K-theory to bivariant motivic cohomology. It is shown that the spectral sequence is naturally realized in the triangulated category of K-motives constructed in the paper. It is also shown that ordinary algebraic K-theory is represented by the K-motive of the point.

The category of framed correspondences Fr * (k), framed presheaves and framed sheaves were invented by Voevodsky in his unpublished notes [12]. Based on the theory, framed motives are introduced and studied in [7]. The main aim of this paper is to prove that for any A 1 -invariant quasi-stable radditive framed presheaf of Abelian groups F , the associated Nisnevich sheaf F nis is A 1 -invariant whenever the base field k is infinite of characteristic different from 2. Moreover, if the base field k is infinite perfect of characteristic different from 2, then every A 1 -invariant quasi-stable Nisnevich framed sheaf of Abelian groups is strictly A 1 -invariant and quasi-stable. Furthermore, the same statements are true in characteristic 2 if we also assume that the A 1 -invariant quasi-stable radditive framed presheaf of Abelian groups F is a presheaf of Z[1/2]-modules.This result and the paper are inspired by Voevodsky's paper [13].