Structure of ind-pro completions of Noetherian rings

TL;DR

This paper investigates the structural properties of ind-pro completions of Noetherian rings, including dimension and inheritance of properties, with applications to algebra over fields.

Contribution

It provides new results on the structure and properties of ind-pro completions of Noetherian rings, especially along prime ideal flags.

Findings

Computed Krull dimension of ind-pro completions.

Established criteria for semilocality in finite type algebras.

Showed inheritance of properties like normality and regularity.

Abstract

We prove some results on the structure of ind-pro completions of Noetherian rings along flags of prime ideals. In particular, we compute the Krull dimension and deduce the criterion on semilocality in the case of essentially of finite type algebras over a field. We also show that ind-pro completion inherits properties of the base ring such as normality, regularity, local equidimensionality, etc.