Localization and Flatness in Quantale Theory

The study of flat ring morphisms is an important theme in commutative algebra. The purpose of this article is to develop an abstract theory of flatness in the framework of coherent quantales. The first question we must address is the definition of a notion of “flat quantale morphism” as an abstraction of flat ring morphisms. For this, we start from a characterization of the flat ring morphism in terms of the ideal residuation theory. The flat coherent quantale morphism is studied in relation to the localization of coherent quantales. The quantale generalizations of some classical theorems from the flat ring morphisms theory are proved. The Going-down and Going-up properties are then studied in connection with localization theory and flat quantale morphisms. As an application, characterizations of zero-dimensional coherent quantales are obtained, formulated in terms of Going-down, Going-up, and localization. We also prove two characterization theorems for the coherent quantales of dimension at most one. The results of the paper can be applied both in the theory of commutative rings and to other algebraic structures: F -rings, semirings, bounded distributive lattices, commutative monoids, etc.