[Paper Review] Relations between the Chow motive and the noncommutative motive of a smooth projective variety

Marcello Bernardara, Gonçalo Tabuada|arXiv (Cornell University)|Mar 13, 2013

Algebraic Geometry and Number Theory23 references3 citations

TL;DR

This paper establishes precise relations between Chow motives and noncommutative motives of smooth projective varieties over a field. It proves that when the noncommutative motive is of unit type (a direct sum of copies of the unit), the Chow motive is of Lefschetz type—answering a key question about the converse of a known implication. The result holds under mild conditions on the coefficient ring, including when R is a principal ideal domain or contains 1/(2d)!.

ABSTRACT

In this note we relate the notions of Lefschetz type, decomposability, and isomorphism, on Chow motives with the notions of unit type, decomposability, and isomorphism, on noncommutative motives. Examples, counter-examples, and applications are also described.

Motivation & Objective

• To determine whether the implication from Chow motive of Lefschetz type to noncommutative motive of unit type has a converse.
• To investigate the relationship between the decomposability of Chow motives and noncommutative motives.
• To examine whether isomorphism of Chow motives implies isomorphism of noncommutative motives, and vice versa.
• To provide conditions under which noncommutative motives of unit type imply Chow motives of Lefschetz type.
• To establish a precise link between the algebraic structure of motives and geometric invariants such as K-theory and Chow rings.

Proposed method

• Uses the universal additive invariant functor $ U(-)_{R} $ from dg categories to noncommutative motives to relate algebraic invariants of perfect complexes on smooth projective varieties.
• Applies the fully faithful functor $ heta $ and the projection $ heta: ext{Chow}(k)_{R} o ext{Chow}(k)_{R}/_{-igotimes R(1)} $ to compare endomorphism rings of motives.
• Employs semi-orthogonal decompositions of $ \mathrm{perf}(X) $ to derive motivic decompositions in the noncommutative setting.
• Relies on the Chern character isomorphism $ K_0(X)_{R[1/(2d)!]} \to \bigoplus_{i} CH^i(X)_{R[1/(2d)!]} $ to establish isomorphisms in the localized category.
• Uses the symmetric monoidal structure of $ U(-)_{R} $ to relate tensor products of noncommutative motives to direct sums.
• Applies results from algebraic K-theory and cyclic homology to show that under mild assumptions (e.g., $ R $ a PID or $ 1/(2d)! \in R $), unit-type noncommutative motives imply Lefschetz-type Chow motives.

Experimental results

Research questions

• RQ1 Does the implication from Chow motive of Lefschetz type to noncommutative motive of unit type admit a converse?
• RQ2 How are the decomposability properties of Chow motives and noncommutative motives related?
• RQ3 Does isomorphism of Chow motives imply isomorphism of noncommutative motives, and vice versa?
• RQ4 Under what conditions does a noncommutative motive of unit type imply that the corresponding Chow motive is of Lefschetz type?
• RQ5 What is the precise relationship between the endomorphism rings of Chow motives and noncommutative motives in the localized category $ \text{Chow}(k)_{R[1/(2d)!]} $?

Key findings

• The paper proves that if the noncommutative motive $ U(\mathrm{perf}_{\mathsf{dg}}(X))_{R} $ is of unit type (a direct sum of copies of the unit motive), then the Chow motive $ M(X)_{R} $ is of Lefschetz type, under the assumption that $ \mathbb{Z} \subseteq R $ and every finitely generated projective $ R[1/(2d)!] $-module is free.
• This converse implication holds when $ R $ is a principal ideal domain or contains $ 1/(2d)! $, providing a precise condition under which noncommutative unit-type implies Chow Lefschetz-type.
• The endomorphism ring of the noncommutative motive is isomorphic to that of the Chow motive in the localized category $ \text{Chow}(k)_{R[1/(2d)!]} $, via the fully faithful functor $ \theta $, establishing a key algebraic link.
• The Chern character induces an isomorphism $ K_0(X)_{R[1/(2d)!]} \to \bigoplus_{i=0}^{d} CH^i(X)_{R[1/(2d)!]} $, which is essential for proving the isomorphism of motives in the localized setting.
• The motivic decomposition $ U(\mathrm{perf}_{\mathsf{dg}}(X))_{R} \simeq \bigoplus_{i=0}^{d-1} U(\underline{A}^{igotimes i})_{R} $ arises from a semi-orthogonal decomposition of $ \mathrm{perf}(X) $, and this lifts to a Chow motive decomposition under the given conditions.
• The implication $ M(X)_{R} \simeq M(Y)_{R} \Rightarrow U(\mathrm{perf}_{\mathsf{dg}}(X))_{R} \simeq U(\mathrm{perf}_{\mathsf{dg}}(Y))_{R} $ holds in the localized category $ \text{Chow}(k)_{R[1/(2d)!]} $, under the assumption that $ 1/(2d)! \in R $.