Cancellation theorem for framed motives of algebraic varieties

Abstract

The machinery of framed (pre)sheaves was developed by Voevodsky [17]. Based on the theory, framed motives of algebraic varieties are introduced and studied in [5]. An analog of Voevodsky's Cancellation Theorem [18] is proved in this paper for framed motives stating that a natural map of framed S1-spectraMfr(X)(n)→Hom_(G,Mfr(X)(n+1)),n⩾0, is a schemewise stable equivalence, where Mfr(X)(n) is the nth twisted framed motive of X. This result is also necessary for the proof of the main theorem of [5] computing fibrant resolutions of suspension P1-spectra ΣP1∞X+ with X a smooth algebraic variety.The Cancellation Theorem for framed motives is reduced to the Cancellation Theorem for linear framed motives stating that the natural map of complexes of abelian groupsZF(Δ•×X,Y)→ZF((Δ•×X)∧(Gm,1),Y∧(Gm,1)),X,Y∈Sm/k, is a quasi-isomorphism, where ZF(X,Y) is the group of stable linear framed correspondences in the sense of [5].