2017/09/26 by Garkusha, Grigory
#Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.1709.09097
Given a perfect field of exponential characteristic e and a functor f:\mathcal A→\mathcal B between symmetric monoidal strict V-categories of correspondences satisfying the cancellation property such that the induced morphisms of complexes of Nisnevich sheaves f_*:\mathbb Z\mathcal A(q)[1/e]→\mathbb Z\mathcal B(q)[1/e], q≥ 0, are quasi-isomorphisms, it is proved that for every k-smooth algebraic variety X the morphisms of twisted motives of X with \mathbb Z[1/e]-coefficients M\mathcal A(X)(q)⊗\mathbb Z[1/e]→ M\mathcal B(X)(q)⊗\mathbb Z[1/e] are quasi-isomorphisms. Furthermore, it is shown that the induced functors between triangulated categories of motives DM\mathcal Aeff(k)[1/e]→ DM\mathcal Beff(k)[1/e], DM\mathcal A(k)[1/e]→ DM\mathcal B(k)[1/e] are equivalences. As an application, the Cor-, K0^⊕-, K0- and \mathbb K0-motives of smooth algebraic varieties with \mathbb Z[1/e]-coefficients are locally quasi-isomorphic to each other. Moreover, their triangulated categories of motives with \mathbb Z[1/e]-coefficients are shown to be equivalent. Another application is given for the bivariant motivic spectral sequence.