Spaces with Noetherian cohomology

TL;DR

This paper investigates whether the cohomology of p-compact groups with Noetherian twisted coefficients is itself Noetherian, extending classical results to a broader algebraic context over p-adic integers.

Contribution

It generalizes the Evens-Venkov Theorem to p-compact groups over p-adic integers, analyzing Noetherian properties of their cohomology with twisted coefficients.

Findings

Cohomology with Noetherian twisted coefficients is Noetherian over p-adic integers.

Comparison of Noetherianity over finite fields and p-adic integers.

Extension of classical theorems to p-compact groups in a new algebraic setting.

Abstract

Is the cohomology of the classifying space of a p-compact group, with Noetherian twisted coefficients, a Noetherian module? This note provides, over the ring of p-adic integers, such a generalization to p-compact groups of the Evens-Venkov Theorem. We consider the cohomology of a space with coefficients in a module, and we compare Noetherianity over the field with p elements, with Noetherianity over the p-adic integers, in the case when the fundamental group is a finite p-group.