On relative $K$-motives, weights for them, and negative $K$-groups

Abstract

We study certain triangulated categories of $K$-motives $DK(-)$ over a wide class of base schemes, and define certain "weights" for them. We relate the weights of particular $K$-motives to (negative) homotopy invariant $K$-groups (tensored by $\mathbb{Z}[S^{-1}]$ for $S$ being the set of "non-invertible primes") ${\mathcal{K}}_{\ast }(-)$. Our results yield a (new) result on the vanishing of ${\mathcal{K}}_i(-)$ and of relative $\mathcal{K}$-groups for $i$ being "too negative"; this statement is closely related to a question of Ch. Weibel. We also prove that ${\mathcal{K}}_i(-)$ for $i<0$ is "supported in codimension $-i$". Moreover, we establish several criteria for bounding (below) the weights of $f_{\ast }(1_Y)$ ($1_Y$ is the tensor unit in $DK(Y)$); this automatically implies the vanishing of the corresponding $E_2$-terms of Chow-weight spectral sequences (and of the factors of the corresponding Chow-weight filtrations) for any (co)homology of these motives. Our methods of bounding weights can be applied to various "motivic" triangulated categories; this yields some new statements on (constructible) complexes of étale sheaves. We also relate the weights of $K$-motives with rational coefficients to that of Beilinson motives; the Chow-weight spectral sequences converging to their ${\mathbb{Q}}_l$-étale (co)homology yield Deligne-type weights for the latter. Somewhat surprisingly, we are able to prove in certain ("extreme") cases that the corresponding weight bounds coming from étale (co)homology are precise. We illustrate these statements by simple examples.

A few corrections made; references updated

References in corpus (2)

• Borel-Moore motivic homology and weight structure on mixed motives
• Intersecting the dimension filtration with the slice one for (relative) motivic categories

Cited by in corpus (7)

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• On morphisms killing weights, weight complexes, and Eilenberg-Maclane (co)homology of spectra
• Künneth formulas for motives and additivity of traces
• On constructing weight structures and extending them to idempotent extensions
• Intersecting the dimension filtration with the slice one for (relative) motivic categories
• K-Motives and Koszul Duality
• Dimensional homotopy t-structure in motivic homotopy theory