On an example of Nagarajan

TL;DR

This paper discusses examples of formal power series rings with group actions where the invariant rings are not noetherian, extending Nagarajan's original example to various characteristics including zero.

Contribution

It extends Nagarajan's example to all positive prime characteristics and to characteristic zero, demonstrating non-noetherian invariant rings in broader contexts.

Findings

Existence of non-noetherian invariant rings in characteristic zero

Extension of Nagarajan's example to all positive prime characteristics

Sharpness of the examples in multiple ways

Abstract

K. R. Nagarajan constructed an example of a formal power series ring of dimension two, over a field of characteristic two, with the action of a cyclic group of order two, such that the ring of invariants is not noetherian. We point out how Nagarajan's example readily extends to each positive prime characteristic, and also to a characteristic zero example: There exists a formal power series ring of dimension two, over a field of characteristic zero, with an action of the infinite cyclic group, such that the ring of invariants is not noetherian. Both the positive characteristic and the characteristic zero examples are sharp in multiple ways.