# Framed motives of algebraic varieties (after V. Voevodsky)

# Framed motives of algebraic varieties (after V. Voevodsky)

This is joint work with G .Garkusha. Using the machinery of framed sheaves developed by Voevodsky, a triangulated category of framed motives is introduced and studied. To any smooth algebraic variety X, the framed motive Mfr(X) is associated in that category. Theorem. The bispectrum (MfrX,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. (this result is an A1-homotopy analog of a theorem due to G.Segal). We also construct a compactly generated triangulated category of framed bispectra and show that it reconstructs the Morel--Voevodsky category SH(k). This machinery allows to recover in characteristic zero the celebrated theorem due to F. Morel stating that the stable π0,0(k)= the Grothendiek-Witt ring of the field k . Also this machinery makes approachable Serre finitness conjecture: rational πn,0(the reals)=0 if n is not zero.