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

Abstract

We describe certain criteria for a motif $M$ to be $r$-effective, i.e., to belong to the $r$th Tate twist $ObjD{M}_{gm,R}^{eff}(r)=ObjD{M}_{gm,R}^{eff}\otimes L^{\otimes r}$ 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 $D{M}_{gm,R}^{eff}$ 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.

Several minor corrections made

References in corpus (4)

• Birational motives, II: Triangulated birational motives
• Gersten weight structures for motivic homotopy categories; direct summands of cohomology of function fields and coniveau spectral sequences
• On morphisms killing weights, weight complexes, and Eilenberg-Maclane (co)homology of spectra
• On perverse homotopy $t$-structures, coniveau spectral sequences, cycle modules, and relative Gersten weight structures

Cited by in corpus (2)

• Conservativity of realizations implies that numerical motives are Kimura-finite and motivic zeta functions are rational
• Dimensional homotopy t-structure in motivic homotopy theory