Sharpness and semistar operations in Prüfer-like domains

Abstract

Let $\star$ be a semistar operation on a domain D, ${\star }_f$ the finite-type semistar operation associated to $\star$, and D a Prüfer $\star$-multiplication domain (P$\star$MD). For the special case of a Prüfer domain (where $\star$ is equal to the identity semistar operation), we show that a nonzero prime P of D is sharp, that is, that $D_P⊉\cap D_M$, 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 ${\star }_f$-sharpness for an arbitrary P$\star$MD D and show that a nonzero prime P of D is ${\star }_f$-sharp if and only if its extension to the $\star$-Nagata ring of D is sharp. Calling a P$\star$MD ${\star }_f$-sharp (${\star }_f$-doublesharp) if each maximal (prime) ${\star }_f$-ideal of D is sharp, we also prove that such a D is ${\star }_f$-doublesharp if and only if each $(\star ,t)$-linked overring of D is ${\star }_f$-sharp.

Keywords:

• Divisorial closure
• essential valuations
• Prüfer domain
• sharp ideal
• star operation

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15; 13F05; 13G05; 13F20; 13F30; 13B25; 13B02; 14A05

Acknowledgements

The authors thank the referee for his/her comments and suggestions, which contributed to improving the final version of this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

M. Fontana was partially supported by GNSAGA of Istituto Nazionale di Alta Matematica. E. Houston was supported by a grant from the Simons Foundation (#354565). M. H. Park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2018R1D1A1B07048928).