Alexandroff topology of algebras over an integral domain

יום ד', 04/12/2019 - 10:30

Let S be an integral domain with field of fractions F, and let A be an F-algebra.  An S-subalgebra R of A is called S-nice if R lies over S and the localization of R with respect to S\{0} is A.  Let X be the set of all S-nice subalgebras of A.  We define a notion of open sets on X which makes this set a T_0-Alexandroff space.  This enables us to study the algebraic structure of X from a topological point of view.  We prove that an irreducible subset of X has a supremum with respect to the specialization order.  We present equivalent conditions for an open set of X to be irreducible and characterize the irreducible components of X.  We also characterize quasi-compactness of subsets of a T_0-Alexandroff topological space.