On the vanishing of Twisted negative K-theory and homotopy invariance

TL;DR

This work extends Weibel's conjectures and homotopy invariance from ordinary K-theory to twisted K-theory defined by Azumaya algebras, with a focus on Noetherian schemes and Prüfer-type rings. It establishes vanishing of twisted negative K-groups for $n$ exceeding the scheme's dimension and demonstrates twisted $K$-theory regularity under polynomial extensions, via Quillen's generalized projective bundle formula and Severi-Brauer fibrations. The paper also derives Morita-invariance-related structure through the Brauer group, analyzes twisted K-theory on weakly regular stably coherent rings, and provides an injection result for twisted versus untwisted K-theory over valuation rings. Collectively, these results deepen understanding of twisted K-theory invariants, their base-change behavior, and the interplay with Brauer groups and Severi-Brauer varieties, with implications for homotopy-invariance in broader algebraic settings.

Abstract

In this article, we revisit Weibel's conjecture for twisted $K$-theory. We also examine the vanishing of twisted negative $K$-groups for Prüfer domains. Furthermore, we observe that the homotopy invariance of twisted $K$-theory holds for (finite-dimensional) Prüfer domains.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• Remark 2.4
• Theorem 3.1
• proof