On the vanishing of negative homotopy K -theory

On very effective hermitian K-theory

“…We show that holim q→∞ f q ψ is a map between contractible motivic spectra, i.e., ψ is a map between slice complete spectra. For KGL/2 this follows by the argument prior to Lemma 2.6: By[11] we know f q KGL is q-connected in the sense of[20, Definition 3.16]. Thus holim q→∞ f q KGL ∼ = * , and likewise for f 0 KGL/2.…”

Vanishing theorems for the negative K-theory of stacks

“…Our goal now is to use the cdh-descent for homotopy K-theory to prove the vanishing theorems for negative K-theory. In order to apply cdh-descent, Kerz and Strunk [24] used the idea of killing classes in the negative K-theory of schemes using Gruson-Raynaud flatification [15]. In §7.1, we prove an analog of this result for stacks.…”

“…Before a complete proof of Weibel's conjecture for schemes appeared in [25], Kelly [23] used the alteration methods of de Jong and Gabber to show that the vanishing conjecture for negative K-theory holds in characteristic p > 0 if one is allowed to invert p. Later, Kerz and Strunk [24] gave a different proof of Kelly's theorem by proving Weibel's conjecture for negative homotopy K-theory, or KH-theory, a variant of Ktheory introduced by Weibel [46]. In their proof, Kerz and Strunk used the method of flatification by blow-up instead of alterations.…”

On the vanishing of relative negative K-theory

“…For the second assertion, consider the spectral sequence K q (S × ∆ p ) ⇒ KH p+q (S). By the first part S is K −n -regular for n ≥ d. Therefore, K −n (S) ∼ = KH −n (S) for n ≥ d. By Theorem 1 of [7], KH −n (S) = 0 for n > d. Hence the result.…”

“…By Theorem 1.2, KH −n (π) = 0 for n > d+1. We have KH −n (X) = 0 for n > d by Theorem 1 of [7]. Then the exact sequence (2.3) implies that KH −n (P t X ) = 0 for n > d. [11].…”