Jaffard-Ohm correspondence and Hochster duality

We study categorical aspects of the Jaffard-Ohm correspondence between abelian l-groups and Bézout 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 Bézout domain and the spectrum of its corresponding abelian l-group provide a concrete example for Hochster's duality of spectral spaces.