A singular analogue of Gersten's conjecture and applications to K-theoretic adeles

TL;DR

This paper extends Gersten's conjecture to singular, one-dimensional local rings, proving it under certain conditions, and explores applications to K-theoretic adeles and sheafified K-theory resolutions.

Contribution

It introduces a singular analogue of Gersten's conjecture for certain local rings and verifies it using cyclic homology methods, also relating K-theory resolutions to classical questions.

Findings

Verified the conjecture for reduced rings containing Q.

Established connections between K-theory adeles and classical localization.

Constructed a new sheafified K-theory resolution under the conjecture.

Abstract

The first part of this paper introduces an analogue, for one-dimensional, singular, complete local rings, of Gersten's injectivity conjecture for discrete valuation rings. Our main theorem is the verification of this conjecture when the ring is reduced and contains Q, using methods from cyclic/Hochschild homology and Artin-Rees type results due to A. Krishna. The second part of the paper describes the relationship between adele type resolutions of K-theory on a one-dimensional scheme and more classical questions in K-theory such as localisation and descent. In particular, we construct a new resolution of sheafified K-theory, conditionally upon the conjecture.