Direct summands of infinite-dimensional polynomial rings

TL;DR

This paper investigates properties of pure subrings of infinite-dimensional polynomial rings generated by monomials, establishing key algebraic properties such as height-grade equality, weak Bourbaki unmixedness, and Cohen-Macaulayness in non-Noetherian contexts.

Contribution

It proves height equals grade for monomial-generated pure subrings and demonstrates Cohen-Macaulay properties for invariant rings under reductive group actions, providing new examples.

Findings

Height equals grade for all ideals in the subring.

The subring satisfies the weak Bourbaki unmixed property.

Invariant rings are Cohen-Macaulay, including non-Noetherian cases.

Abstract

Let k be a field and R a pure subring of the infinite-dimensional polynomial ring k[X1;...]. If R is generated by monomials, then we show that the equality of height and grade holds for all ideals of R. Also, we show R satisfies the weak Bourbaki unmixed property. As an application, we give the Cohen-Macaulay property of the invariant ring of the action of a linearly reductive group acting by k-automorphism on k[X1;...]. This provides several examples of non-Noetherian Cohen-Macaulay rings (e.g. Veronese, determinantal and Grassmanian rings).