Sharpness and semistar operations in Prüfer-like domains

Abstract

Let be a semistar operation on a domain D, the finite-type semistar operation associated to , and D a Prüfer -multiplication domain (PMD). For the special case of a Prüfer domain (where is equal to the identity semistar operation), we show that a nonzero prime P of D is sharp, that is, that , where the intersection is taken over the maximal ideals M of D that do not contain P, if and only if two closely related spectral semistar operations on D differ. We then give an appropriate definition of -sharpness for an arbitrary PMD D and show that a nonzero prime P of D is -sharp if and only if its extension to the -Nagata ring of D is sharp. Calling a PMD -sharp (-doublesharp) if each maximal (prime) -ideal of D is sharp, we also prove that such a D is -doublesharp if and only if each -linked overring of D is -sharp.