# Depth and detection for Noetherian unstable algebras

## TL;DR

This paper extends classical theorems on the depth of cohomology rings from finite groups to a broad class of algebraic and topological groups, including Lie groups, profinite groups, and Kac--Moody groups, using unstable algebra techniques.

## Contribution

It generalizes Duflot and Carlson's theorems on cohomology ring depth to Noetherian unstable algebras associated with various groups and modules, broadening their applicability.

## Key findings

- Proves depth theorems for cohomology rings of diverse groups.
- Extends classical results to unstable modules and broader group classes.
- Connects the results to p-local compact groups and modular invariant theory.

## Abstract

For a connected Noetherian unstable algebra $R$ over the mod $p$ Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of $R$, originally proved when $R$ is the mod $p$ cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when $R$ is the mod $p$ cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac--Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely generated unstable $R$-modules. Moreover, we explain the results in the case of the $p$-local compact groups of Broto, Levi, and Oliver, as well as in the modular invariant theory of finite groups.