[Paper Review] Detecting effectivity of motives, their weights, connectivity, and dimension via Chow-weight (co)homology: a "mixed motivic decomposition of the diagonal"

Mikhail V. Bondarko, Vladimir Sosnilo|arXiv (Cornell University)|Nov 24, 2014

Algebraic Geometry and Number Theory43 references3 citations

TL;DR

This paper introduces a new Chow-weight (co)homology theory for effective Voevodsky motives that detects motive effectivity, weights, connectivity, and dimension via a generalized 'mixed motivic decomposition of the diagonal.' It establishes that a motive is r-effective if and only if its Chow-weight homology vanishes in degrees less than r, and proves that torsion in higher motivic homology has finite exponents, extending classical decomposition of the diagonal results to arbitrary varieties and cohomology theories.

ABSTRACT

We describe certain criteria for a motif $M$ to be $r$-effective, i.e., to belong to the $r$th Tate twist $Obj DM^{eff}_{gm,R}(r)=Obj DM^{eff}_{gm,R} \otimes L^{\otimes r}$ of effective Voevodsky motives (for $r\ge 1$; $R$ is the coefficient ring). In particular, $M$ is 1-effective if and only if a complex whose terms are certain Chow groups of zero-cycles is acyclic. The dual to this statement checks whether an effective motif $M$ belongs to the subcategory of $DM^{eff}_{gm,R}$ generated by motives of varieties of dimension $\le r$. These criteria are formulated in terms of the Chow-weight (co)homology of $M$. These (co)homology theories are introduced in the current paper and have several (other) remarkable properties: they yield a bound on the "weights" of $M$ (in the sense of the Chow weight structure defined by the first author) and detect the effectivity of "the lower weight pieces" of $M$. We also calculate the "connectivity" of $M$ (in the sense of Voevodsky's homotopy t-structure) and prove that the exponents of the higher motivic homology groups (of an "integral" motif) are bounded whenever these groups are torsion. These motivic properties of $M$ have important consequences for its cohomology. As a corollary, we prove that if Chow groups of an arbitrary variety $X$ vanish up to dimension $r-1$ then the highest Deligne weight factors of the (singular or étale) cohomology of $X$ with compact support are $r$-effective in the naturally defined sense. Our results yield a vast generalization of the so-called "decomposition of the diagonal" statements.

Motivation & Objective

• To develop a new Chow-weight (co)homology theory for effective geometric motives over a field with coefficients in a ring R.
• To detect whether a motive M is r-effective, i.e., lies in the r-th Tate twist of the category of effective motives.
• To bound the weights of M in the Chow weight structure and detect effectivity of lower weight pieces.
• To relate these homological invariants to motivic homology, cohomology with compact support, and properties of motivic spectra.
• To generalize classical 'decomposition of the diagonal' results to arbitrary varieties and cohomology theories, including singular and étale cohomology.

Proposed method

• Define a new Chow-weight (co)homology theory on the category $ DM^{eff}_{gm,R} $ using weight structures and weight complexes.
• Use the weight spectral sequence and weight complex techniques to relate Chow-weight homology to the effectivity and weight filtration of motives.
• Apply the Poincaré duality to translate effectivity conditions on homology into cohomological conditions.
• Use the localization method and results on weight complexes from [Bon18a], [Bon19], and [BoS18c] to analyze dimension and effectivity bounds.
• Study the behavior of Chow-weight homology under base change and birational maps, showing it is birational in the smooth base case.
• Analyze torsion in higher motivic homology by bounding exponents using the structure of the weight filtration and conjectural motivic cohomology.

Experimental results

Research questions

• RQ1 When is a motive M in $ DM^{eff}_{gm,R} $ r-effective, and how can this be detected via Chow-weight homology?
• RQ2 How do the weights of a motive M in the Chow weight structure relate to the vanishing of its Chow-weight homology?
• RQ3 What is the connection between the effectivity of lower weight pieces of M and the vanishing of its Chow-weight homology in negative degrees?
• RQ4 How do the exponents of torsion in higher motivic homology groups behave for integral motives?
• RQ5 To what extent can the classical 'decomposition of the diagonal' be generalized to arbitrary varieties and cohomology theories using this new homology theory?

Key findings

• A motive M is r-effective if and only if its Chow-weight homology vanishes in all degrees less than r, providing a homological criterion for effectivity.
• The torsion in higher motivic homology groups of an integral motive has finite exponent, a result proven using the weight filtration and duality.
• If Chow groups of a variety X vanish up to dimension r−1, then the highest Deligne weight factors in its cohomology with compact support are r-effective.
• The converse holds for singular cohomology under motivic conjectures A and B, establishing a cohomological characterization of effectivity.
• The Chow-weight homology of a motive detects its connectivity in Voevodsky’s homotopy t-structure, and vanishing in negative degrees does not imply non-negative weights.
• Counterexamples show that Chow-weight homology cannot detect upper bounds on weights, but the weight filtration on singular homology can, under conjectures.