# Excellent rings in transchromatic homotopy theory

## TL;DR

This paper verifies that key rings in transchromatic homotopy theory are Noetherian excellent normal domains, enabling the use of standard commutative algebra techniques for their analysis.

## Contribution

It demonstrates that several fundamental rings in transchromatic homotopy theory are Noetherian excellent normal domains, providing a foundation for algebraic methods in the field.

## Key findings

- Coefficients of iterated localizations of Morava E-theory are normal domains.
- Coefficients in the transchromatic character map form a normal domain.
- Key rings are established as Noetherian excellent normal domains.

## Abstract

The purpose of this note is to verify that several basic rings appearing in transchromatic homotopy theory are Noetherian excellent normal domains and thus amenable to standard techniques from commutative algebra. In particular, we show that the coefficients of iterated localizations of Morava $E$-theory at the Morava $K$-theories are normal domains and also that the coefficients in the transchromatic character map for a fixed group form a normal domain.