A Singular Analogue of Gersten's Conjecture and Applications to K-theoretic Adèles

Abstract

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