Minimal Order Morphisms Original Research Article

An order is a one-dimensional, residually finite, Noetherian domain. A finite birational morphism ƒ: A → B between two orders can be decomposed by means of minimal morphisms: inert, decomposed, or ramified. We determine a least upper bound for the number of minimal morphisms appearing in a decomposition of ƒ. This number is reached in the canonical decomposition ƒ = ƒ1ring operatorƒ2ring operatorƒ3 where ƒ1: A″ → B (resp. ƒ2: A′ → A″, ƒ3:A → A′) is composed only of inert morphisms (resp. decomposed, ramified). The orders A′ and A″ are unique and are respectively the seminormalization and the quasinormalization of A in B. The canonical decomposition always exists. Though there is a decomposition with a minimal number of minimal order morphisms, we can only give a lower bound for this number. All this applies especially to the orders of algebraic number fields and an example is given.