Jaffard-Ohm correspondence and Hochster duality

摘要

We study categorical aspects of the Jaffard-Ohm correspondence between abelian l-groups and Bezout domains and show that this correspondence is close to a localization. For this purpose, we establish a general extension theorem for valuations with value group that is an abelian l-group. As an application, we prove Anderson's conjecture which refines the Jaffard-Ohm. correspondence. We then extend the correspondence to sheaves on spectral spaces and show that the spectrum of a Bezout domain and the spectrum of its corresponding abelian 1-group provide a concrete example for Hochster's duality of spectral spaces.