Local–Global Properties for Semistar Operations

Abstract

We study the “local” behavior of several relevant properties concerning semistar operations, like finite type, stable, spectral, e.a.b. and a.b. We deal with the “global” problem of building a new semistar operation on a given integral domain, by “gluing” a given homogeneous family of semistar operations defined on a set of localizations. We apply these results for studying the local–global behavior of the semistar Nagata ring and the semistar Kronecker function ring. We prove that an integral domain D is a Prüfer ⋆-multiplication domain if and only if all its localizations D _P are Prüfer ⋆ _P -multiplication domains.

Key Words:

• Star and semistar operation
• Prüfer domain
• Nagata ring
• Kronecker function ring

Mathematics Subject Classification:

• 13A15
• 13F05
• 13G05

Acknowledgments

During the preparation of this work the first named author was partially supported by a research grant MIUR 2001/2002 (Cofin 2000-MM 01192794). The second and third authors were partially supported by DGES BMF2001–2823 and FQM-266 (Junta de Andalucía Research Group).

Notes

^#Communicated by M. Ferrero.