# Comparing motives of smooth algebraic varieties

Research paper by **Grigory Garkusha**

Indexed on: **26 Sep '17**Published on: **26 Sep '17**Published in: **arXiv - Mathematics - Algebraic Geometry**

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#### Abstract

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.