Comparing motives of smooth algebraic varieties

Grigory Garkusha

Published 2017-09-26Version 1

Given a perfect field of exponential characteristic $e$ and a functor $f:\mathcal A\to\mathcal B$ between symmetric monoidal strict $V$-categories of correspondences satisfying the cancellation property such that the induced morphisms of complexes of Nisnevich sheaves $$f_*:\mathbb Z_{\mathcal A}(q)[1/e]\to\mathbb Z_{\mathcal B}(q)[1/e],\quad q\geq 0,$$ are quasi-isomorphisms locally in the Nisnevich topology, it is proved that for every $k$-smooth algebraic variety $X$ the morphisms of twisted motives of $X$ with $\mathbb Z[1/e]$-coefficients $$M_{\mathcal A}(X)(q)\otimes\mathbb Z[1/e]\to M_{\mathcal B}(X)(q)\otimes\mathbb Z[1/e]$$ are quasi-isomorphisms locally in the Nisnevich topology. Furthermore, it is shown that the induced functors between triangulated categories of motives $$DM_{\mathcal A}^{eff}(k)[1/e]\to DM_{\mathcal B}^{eff}(k)[1/e],\quad DM_{\mathcal A}(k)[1/e]\to DM_{\mathcal B}(k)[1/e]$$ are equivalences. As an application, the Cor-, $K_0^\oplus$-, $K_0$- and $\mathbb K_0$-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.