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

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.