# On stratification for spaces with Noetherian mod $p$ cohomology

## TL;DR

This paper investigates conditions under which the module category over the cochain spectrum of a space with Noetherian mod p cohomology is stratified, linking algebraic and homotopical properties for various classes of spaces.

## Contribution

It establishes stratification criteria for module spectra over cochains on spaces, including classifying spaces of topological groups and H-spaces, extending previous results.

## Key findings

- Stratification holds for classifying spaces of many topological groups.
- Stratification is equivalent to a generalized Chouinard's theorem.
- Connects stratification to the generalized telescope conjecture.

## Abstract

Let $X$ be a topological space with Noetherian mod $p$ cohomology and let $C^*(X;\mathbb{F}_p)$ be the commutative ring spectrum of $\mathbb{F}_p$-valued cochains on $X$. The goal of this paper is to exhibit conditions under which the category of module spectra on $C^*(X;\mathbb{F}_p)$ is stratified in the sense of Benson, Iyengar, Krause, providing a classification of all its localizing subcategories. We establish stratification in this sense for classifying spaces of a large class of topological groups including Kac--Moody groups as well as whenever $X$ admits an $H$-space structure. More generally, using Lannes' theory we prove that stratification for $X$ is equivalent to a condition that generalizes Chouinard's theorem for finite groups. In particular, this relates the generalized telescope conjecture in this setting to a question in unstable homotopy theory.