Matthew Morrow

PRO CDH-DESCENT FOR CYCLIC HOMOLOGY AND -THEORY

In this paper, we prove that cyclic homology, topological cyclic homology, and algebraic

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

K

An introduction to higher dimensional local fields and adeles

-theory satisfy a pro Mayer–Vietoris property with respect to abstract blow-up squares of varieties, in both zero and finite characteristic. This may be interpreted as the well-definedness of

-theory of one-dimensional rings via pro-excision

K

Integral

-theory with compact support.

p

The first part of this article introduces an analogue, for one-dimensional, singular, complete local rings, of Gerstens 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.