Local--Global Properties for Semistar Operations

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 Pr{\"u}fer $\star$--multiplication domain if and only if all its localizations $D_{P}$ are Pr{\"u}fer $\star_P$--multiplication domains.