Ultraproducts of Noetherian Local Rings: Properties and Applications

Cuny New York City College Of Technology, Brooklyn NY

Investigators

Abstract

In past research, the proposer has used ultraproducts in commutative algebra as a means to transfer results from one characteristic to another and to prove the existence of uniform bounds. Under the support of the NSF project "Transfer, Definability and Constructibility", this approach cumulated in the introduction of an alternative tight closure operation in characteristic zero. The key idea is that by taking ultraproducts of the positive characteristic Frobenii, one gets an ultra-Frobenius in characteristic zero acting on a canonically defined overring. By this technique, complications from reduction to positive characteristic have been avoided and the elegant arguments from positive characteristic have been recovered. Through his novel approach, the proposer was able to improve results from classical tight closure theory, to successfully classify rational singularities of finite type over a field of characteristic zero, and to solve an open problem on Kawamata-Viehweg vanishing for quotients of Fano varieties. The author purports that these results can be extended to arbitrary Noetherian local rings containing a field, if one better understands the structures that are in the center of this transfer: ultraproducts of Noetherian local rings. He proposes a comprehensive study of their local properties, using variants of classical notions from commutative algebra, cast in a more general framework of local rings of finite embedding dimension. Many applications and benefits are expected to follow from his study, some of the more important of which are: a parallel development of tight closure in positive and in zero characteristic leading to a unified and simplified treatment, making the theory more accessible and appealing to a general mathematical audience; sharper results on uniform bounds; the study of rational singularities in a more general setup; and, asymptotical solutions to the Homological Conjectures in mixed characteristic. The synergy between concepts from model theory, commutative algebra, homological algebra, ring theory, topology and algebraic geometry, is what makes this project particularly interesting, and the exchange of ideas between these mathematical disciplines will contribute substantially to the advancement of the subject. This is one of the first studies in its kind that brings these notions together via a single concept, to wit, the ultraproduct of Noetherian local rings. In response to the proposer's previous success in using ultraproducts, other researchers in algebra and ring theory have already turned their attention to such ultraproducts. It is the goal of the author to engage in a productive interaction with these researchers, in the hope that this eventually will lead to the organization of an international conference on the use of ultraproducts in ring theory.