Weibel’s vanishing conjecture for negative K-theory

Weibel’s conjecture asserts that for a noetherian scheme $X$ of finite dimension $d$, negative $K$-groups vanish below degree $-d$. Kerz, Strunk, and Tamme proved this standard formulation in full generality.

Known results

• Kerz–Strunk–Tamme (2018): $K_i(X)=0$ for $i<-d$; also homotopy invariance in the boundary range.
• Kelly–Saito–Tamme (2024): a derived-scheme generalization.
• Twisted negative $K$-theory satisfies an analogous bound for Azumaya algebras.

August 2026 non-noetherian counterexamples

Shane Kelly’s preprint claims explicit non-noetherian rings violating the analogous dimension bound, including $K_{-d}(R)\cong\mathbb Z$ for rings with prescribed dimensions and $K_{-d}(R)\ne0$ in dimension $1$. These examples do not contradict the standard noetherian theorem.

Current status (as of August 2026): The standard noetherian conjecture is settled; the broader non-noetherian analogue has claimed but unrefereed counterexamples.