Comparing motives of smooth algebraic varieties

TL;DR

This paper proves that under certain conditions, motives of smooth algebraic varieties are equivalent across different categories, leading to quasi-isomorphisms and equivalences of their associated triangulated categories.

Contribution

It establishes the equivalence of motives and triangulated categories for smooth algebraic varieties across various correspondence categories with inverted characteristic, unifying their motivic frameworks.

Findings

Motives of smooth varieties are quasi-isomorphic across different correspondence categories.

Triangulated categories of motives are equivalent after inverting the characteristic.

Motives like Cor-, $K_0^igoplus$, $K_0$, and $bK_0$ are locally quasi-isomorphic with $bZ[1/e]$-coefficients.

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, 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. 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.