Intersections of quasi-local domains

Source abstract

Let R = ⋂ V i R = \bigcap {{V_i}} be an intersection of quasi-local domains with a common quotient field K K . Our goal is to find conditions on the V i {V_i} ’s in order to get some or all of V i {V_i} ’s to be localizations of R R . We show for example that if V 1 {V_1} is a 1 1 -dimensional valuation domain and if V 1 ⊉ V 2 {V_1} \nsupseteq {V_2} , then both V 1 {V_1} and V 2 {V_2} are localizations of R = V 1 ∩ V 2 R = {V_1} \cap {V_2} .