Big polynomial rings with imperfect coefficient fields

TL;DR

This paper extends previous results by demonstrating that the inverse limit of standard-graded polynomial rings remains a polynomial ring even over arbitrary coefficient fields, and also establishes similar results for ultraproducts of polynomial rings.

Contribution

It generalizes the known polynomial ring structure of inverse limits from perfect to arbitrary coefficient fields and introduces analogous results for ultraproducts.

Findings

Inverse limit of polynomial rings over arbitrary fields is a polynomial ring.

Ultraproducts of polynomial rings also form polynomial rings.

Results hold in uncountably many variables.

Abstract

We previously showed that the inverse limit of standard-graded polynomial rings with perfect coefficient field is a polynomial ring, in an uncountable number of variables. In this paper, we show that the same result holds with arbitrary coefficient field. We also prove an analogous result for ultraproducts of polynomial rings.