The Krull filtration of the category of unstable modules over the Steenrod algebra

TL;DR

This paper explores the algebraic properties of the Krull filtration in the category of unstable modules over the Steenrod algebra, highlighting its applications in topology and recent advances in understanding nonrealization conjectures.

Contribution

It provides a detailed analysis of the algebraic structure of the Krull filtration and connects it to topological applications and recent progress in the field.

Findings

Characterization of the Krull filtration in algebraic terms

Application of the filtration to topological nonrealization conjectures

Recent proofs of conjectures using the filtration

Abstract

In the early 1990's, Lionel Schwartz gave a lovely characterization of the Krull filtration of U, the category of unstable modules over the mod p Steenrod algebra. Soon after, this filtration was used by the author as an organizational tool in posing and studying some topological nonrealization conjectures. In recent years the Krull filtration of U has been similarly used by Castellana, Crespo, and Scherer in their study of H--spaces with finiteness conditions, and Gaudens and Schwartz have given a proof of some of my conjectures. In light of these topological applications, it seems timely to better expose the algebraic properties of the Krull filtration.