Some explicit counter-examples to Weibel's conjecture

Technical Summary: Explicit Counter-examples to Weibel's Conjecture

Problem Statement
The paper addresses a question posed by Weibel in 1980 regarding the vanishing of negative K-theory for commutative Noetherian rings. Specifically, Question 1 asks: If $R$ is a commutative Noetherian ring of Krull dimension $d$, is $K_*(R) = 0$ for $* < -d$?

While the Kerz–Strunk–Tamme theorem (2018) provided a definitive affirmative answer for Noetherian rings, the status of this question for non-Noetherian rings remained open. Previous results suggested that if the Krull dimension is replaced by the "valuative dimension" (the supremum of Krull dimensions over all proper birational modifications), the vanishing holds. However, the paper notes that for non-Noetherian rings, the Krull dimension can be strictly less than the valuative dimension. The central problem addressed here is whether the Noetherian condition is essential for the vanishing of negative K-theory in terms of Krull dimension.

Methodology
The author constructs explicit counter-examples using a geometric strategy analogous to the topological construction of a sphere $S^d$ as a quotient of a disk $D^d$ by its boundary $\partial D^d$. The construction relies on three main components:

1. Milnor Squares: The rings $R$ are constructed as pullbacks (Milnor squares) of the form $R = R_\infty \times_{R_\infty/I_\infty} k$, where $k$ is a field. This setup creates a "quotient" scheme where the "boundary" is collapsed to a point.
2. Filtered Limits of Regular Schemes: The ring $R_\infty$ is defined as a filtered colimit (direct limit) of semilocalizations of regular schemes ($X_n$ or toric varieties $X_\Sigma$). Since these constituent rings are regular, they possess no negative K-theory. The non-vanishing of $K_*(R)$ arises entirely from the interaction between $R_\infty$ and its quotient $R_\infty/I_\infty$.
3. Iterative Blow-ups and Tropical Geometry:
   • For $d=2$: The construction begins with a smooth surface containing a simple normal crossing (snc) divisor arranged in a triangle. The author performs an infinite sequence of blow-ups at the intersection points of the divisor branches. This process iteratively replaces corners with smaller faces, creating a limit structure resembling a polyhedron with infinitely many faces.
   • For $d \ge 2$: The construction utilizes toric varieties. The author considers a filtered system of smooth projective toric varieties obtained by subdividing the fan. The "boundary" corresponds to the complement of the dense torus orbit. The intersection patterns of the boundary components are tracked using tropical varieties, which allow the author to ensure that the limit of the boundary configurations behaves combinatorially like a $(d-1)$-dimensional polyhedron.

Key Results

• Theorem 2 (Main Result): For all $d \ge 2$, there exists a ring $R$ such that $\text{Spec } R$ has exactly two points, $\text{Krull dim } R = 1$, and $K_{-d}(R) \cong \mathbb{Z}$.
• Proposition 4 ($d=2$ Case): A specific construction is provided where $R$ is formed from a sequence of blow-ups of a surface $X_0$ along the intersections of an snc divisor. The resulting ring $R$ has Krull dimension 1 and $K_{-2}(R) \cong \mathbb{Z}$.
• Proposition 6 ($d \ge 2$ Case): A generalization using toric varieties is presented. By taking the filtered limit of semilocalizations of toric varieties $X_\Sigma$ and their boundaries, the author constructs a ring $R$ with Krull dimension 1 and non-vanishing $K_{-d}(R)$.

Technical Mechanism for Non-Vanishing
The non-vanishing of $K_{-d}(R)$ is derived from the long exact sequence of K-theory associated with the Milnor square. Since $R_\infty$ and $k$ are regular, their negative K-groups vanish. The sequence reduces to an isomorphism:
$$K_{-d}(R) \cong K_{1-d}(R_\infty/I_\infty)$$
The term $K_{1-d}(R_\infty/I_\infty)$ is identified with the cdh-cohomology group $H^{d-1}_{cdh}(\text{Spec } R_\infty/I_\infty, \mathbb{Z})$. Due to the combinatorial structure of the boundary (constructed to resemble a $(d-1)$-sphere or polyhedron), this cohomology group is isomorphic to the singular cohomology of the corresponding polyhedron, which yields $\mathbb{Z}$ in degree $d-1$.

Significance and Claims
The paper claims to present the first known counter-examples to the non-Noetherian version of Weibel's question. Specifically, it demonstrates that the condition "Krull dimension $d$" is insufficient to guarantee the vanishing of $K_*(R)$ for $* < -d$ when the ring is not Noetherian.

The author notes that while the use of Milnor squares and finite polygons to generate negative K-theory classes is well-known (implicit in Bass's conductor formula), the specific construction using infinite polygons (for $d=2$) and toric varieties with tropical tracking (for $d \ge 2$) constitutes the novel contribution. The paper explicitly states that these examples show the necessity of the Noetherian hypothesis (or the replacement of Krull dimension with valuative dimension) for the vanishing theorem to hold.

Acknowledgement of Tools
The author transparently acknowledges the use of computer assistance in the development of the examples, particularly in identifying the infinite polygon construction in existing literature (Lazard) and in verifying technical details regarding toric intersections via tropical varieties. However, the core conceptual framework and the specific proofs for the $d \ge 2$ toric construction are attributed to the author.