Chow motives versus noncommutative motives

AbstractThe main goal of this paper is to deduce (from a recent resolution of singularities result of Gabber) the following fact: (effective) Chow motives with ℤ[1/p]-coefficients over a perfect field k of characteristic p generate the category DMeffgm[1/p] (of effective geometric Voevodsky’s motives with ℤ[1/p]-coefficients). It follows that DMeffgm[1/p] can be endowed with a Chow weight structure wChow whose heart is Choweff[1/p] (weight structures were introduced in a preceding paper, where the existence of wChow for DMeffgmℚ was also proved). As shown in previous papers, this statement immediately yields the existence of a conservative weight complex functor DMeffgm[1/p]→Kb (Choweff [1/p]) (which induces an isomorphism on K0-groups), as well as the existence of canonical and functorial (Chow)-weight spectral sequences and weight filtrations for any cohomology theory on DMeffgm[1/p] . We also mention a certain Chow t-structure for DMeff−[1/p] and relate it with unramified cohomology.

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Noncommutative motives of Azumaya algebras

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s147474801400005x ◽

2014 ◽

Vol 14 (2) ◽

pp. 379-403 ◽

Cited By ~ 11

Author(s):

Gonçalo Tabuada ◽

Michel Van den Bergh

Keyword(s):

Commutative Ring ◽

Global Dimension ◽

Azumaya Algebras ◽

Finite Global Dimension ◽

Finite Dimensional ◽

Noncommutative Motives ◽

Finite Dimensional Algebras ◽

The Way

AbstractLet $k$ be a base commutative ring, $R$ a commutative ring of coefficients, $X$ a quasi-compact quasi-separated $k$-scheme, and $A$ a sheaf of Azumaya algebras over $X$ of rank $r$. Under the assumption that $1/r\in R$, we prove that the noncommutative motives with $R$-coefficients of $X$ and $A$ are isomorphic. As an application, we conclude that a similar isomorphism holds for every $R$-linear additive invariant. This leads to several computations. Along the way we show that, in the case of finite-dimensional algebras of finite global dimension, all additive invariants are nilinvariant.

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The Chow Motives of Relative Fulton-Macpherson Space

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15479 ◽

2013 ◽

Vol 113 (1) ◽

pp. 20

Author(s):

Fumitoshi Sato

Keyword(s):

Projective Variety ◽

Chow Groups ◽

Configuration Spaces ◽

Chow Motives

Suppose that $X$ is a complex nonsingular projective variety and $D$ is a smooth divisor. Compactifications of configuration spaces of distinct and non-distinct $n$ points in $X$ away from $D$ were constructed by the author and B. Kim in "A generalization of Fulton-MacPherson configuration spaces" by using the method of wonderful compactification. In this paper, we give explicit presentations of Chow motives and Chow groups of these configuration spaces.

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