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We describe certain criteria for a motif $M$ to be $r$-effective, i.e., tobelong 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 thecoefficient ring). In particular, $M$ is 1-effective if and only if a complexwhose terms are certain Chow groups of zero-cycles is acyclic.  The dual to this statement checks whether an effective motif $M$ belongs tothe subcategory of $DM^{eff}_{gm,R}$ generated by motives of varieties ofdimension $\le r$. These criteria are formulated in terms of the Chow-weight(co)homology of $M$. These (co)homology theories are introduced in the currentpaper 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 thefirst author) and detect the effectivity of "the lower weight pieces" of $M$.We also calculate the "connectivity" of $M$ (in the sense of Voevodsky'shomotopy t-structure) and prove that the exponents of the higher motivichomology groups (of an "integral" motif) are bounded whenever these groups aretorsion. These motivic properties of $M$ have important consequences for itscohomology. As a corollary, we prove that if Chow groups of an arbitraryvariety $X$ vanish up to dimension $r-1$ then the highest Deligne weightfactors of the (singular or \'etale) 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 ofthe diagonal" statements.

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