2016/01/25 by Ananyevskiy, Alexey, Garkusha, Grigory, Panin, Ivan
#14F05 #14F42 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.1601.06642
The machinery of framed (pre)sheaves was developed by Voevodsky [V1]. Based on the theory, framed motives of algebraic varieties are introduced and studied in [GP1]. An analog of Voevodsky's Cancellation Theorem [V1] is proved in this paper for framed motives stating that a natural map of framed S1-spectra Mfr(X)(n)→\underline\textrmHom(\mathbb 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 [GP1] computing fibrant resolutions of suspension \mathbb P1-spectra Σ^∞\mathbb P1X+ 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 groups \mathbb ZF(Δ^\bullet × X,Y) → \mathbb ZF((Δ^\bullet × X)\wedge (\mathbb Gm,1),Y\wedge (\mathbb Gm,1)), X,Y∈ Sm/k, is a quasi-isomorphism, where \mathbb ZF(X,Y) is the group of stable linear framed correspondences in the sense of [GP1].