Symmetric modules over the infinite polynomial ring I: nilpotent quotients

TL;DR

This paper investigates the structure of modules over certain nilpotent quotients of an infinite symmetric polynomial ring, revealing their Grothendieck groups, Krull--Gabriel dimension, and derived category generators.

Contribution

It provides a detailed analysis of the category of modules over the quotients by specific symmetric prime ideals, advancing understanding of their algebraic properties.

Findings

Grothendieck group of the module category determined

Krull--Gabriel dimension shown to be s

Generators for the derived category obtained

Abstract

Cohen proved that the infinite variable polynomial ring $R=k[x_1,x_2,\ldots]$ is noetherian with respect to the action of the infinite symmetric group $\mathfrak{S}$. The first two authors began a program to understand the $\mathfrak{S}$-equivariant algebra of $R$ in detail. In previous work, they classified the $\mathfrak{S}$-prime ideals of $R$. An important example of an $\mathfrak{S}$-prime is the ideal $\mathfrak{h}_s$ generated by $(s+1)$st powers of the variables. In this paper, we study the category of $R/\mathfrak{h}_s$-modules. We obtain a number of results, and mention just three here: (a) we determine the Grothendieck group of the category; (b) we show that the Krull--Gabriel dimension is $s$; and (c) we obtain generators for the derived category. This paper will play a key role in subsequent work where we study general modules.