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• W2406464145 abstract "We study certain triangulated categories of $K$-motives $DK(-)$ over a wide class of base schemes, and define certain 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) $mathcal{K}_*(-)$. 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 $f_*(1_Y)$ ($1_Y$ 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 etale 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 $mathbb{Q}_l$-etale (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 etale (co)homology are precise. We illustrate these statements by simple examples." @default.
• W2406464145 created "2016-06-24" @default.
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• W2406464145 date "2016-05-26" @default.
• W2406464145 modified "2023-09-27" @default.
• W2406464145 title "On relative $K$-motives, weights for them, and negative $K$-groups" @default.
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