Structure of ind-pro completions of Noetherian rings
Dmitry Badulin
TL;DR
This work studies ind-pro completions $C_\Delta R$ of Noetherian rings along flags of prime ideals, deriving a general dimension formula and flatness properties, and identifying when these completions are semilocal. It first treats the polynomial case to establish a semilocality criterion based on flag saturation, then extends to the general case of rings essentially of finite type over a field via Noether normalization. The main contributions include a precise dimension formula $\dim C_\Delta R = \mathrm{ht}\ \mathfrak{p}_0 + \mathrm{ht}(\mathfrak{p}_n/\mathfrak{p}_0) - n$, the semilocality criterion tied to saturation, and preservation of structural properties such as excellence and local equidimensionality under appropriate hypotheses. These results connect ind-pro completions to formal fibers and adelic local factors, deepening understanding of completion-localization interactions in Noetherian algebraic geometry.
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.