On a conjecture of Vorst

We prove the following result. Let k be an infinite perfect field of positive characteristic and assume that strong resolution of singularities holds over k. Let R be a localization of a commutative d-dimensional k-algebra of finite type and suppose that R is K_{d+1}-regular. Then R is a regular ring.

💡 Research Summary

The paper addresses a long‑standing conjecture of Vorst, which posits that K‑regularity in a certain degree forces a ring to be regular. Specifically, the author proves that if k is an infinite perfect field of positive characteristic p, and if strong resolution of singularities holds over k, then any localization R of a finitely generated d‑dimensional k‑algebra that is K_{d+1}‑regular must be a regular ring.

The argument proceeds through several sophisticated layers of modern algebraic K‑theory and birational geometry. First, the notion of K_{n}‑regularity is recalled: a ring R is K_{n}‑regular if the natural maps K_i(R) → K_i(R