Gorenstein Prüfer domains with non-Prüfer integral closure

Abstract

There exists a one-dimensional local Gorenstein Prüfer domain whose integral closure is not Prüfer. We prove that the recently constructed coherent counterexample to the normalization problem has this stronger Gorenstein property. The argument establishes permanence of divisoriality of finitely generated ideals under ultraproducts and quotients by divided primes. It also supplies uniform matrix factorizations: in an ultraproduct of plane-curve domains, every nonzero ideal generated by $r$ elements is a cycle of a two-periodic totally acyclic complex with free terms of rank $r$. The divided prime defining the one-dimensional example is flat, so these complexes descend through the quotient and localization. A specified two-generated ideal remains noninvertible in every integral overring. Over the normalization, its associated Laurent overring has nonzero first Tor with every injective cogenerator, although both rings are Gorenstein flat over the original domain. Examples distinguish this failure from a nonnormal Gorenstein curve with Prüfer normalization and show that the periodicity conclusion is stronger than the Gorenstein Prüfer property alone.