Quotient rings of $H\mathbb{F}_2 \wedge H\mathbb{F}_2$

TL;DR

This paper investigates modules over the spectrum $H\mathbb{F}_2 \wedge H\mathbb{F}_2$, identifying which quotients admit algebra structures, and uses these to compute homotopy groups with new duality insights.

Contribution

It characterizes algebra structures on quotients of the dual Steenrod algebra and constructs new associative algebras with unique duality properties for homotopy computations.

Findings

Few quotients admit algebra structures.

Killing a generator kills higher generators predictably.

Constructed finite-dimensional algebras with unexpected duality features.

Abstract

We study modules over the commutative ring spectrum $H\mathbb F_2\wedge H\mathbb F_2$, whose coefficient groups are quotients of the dual Steenrod algebra by collections of the Milnor generators. We show that very few of these quotients admit algebra structures, but those that do can be constructed simply: killing a generator $\xi_k$ in the category of associative algebras freely kills the higher generators $\xi_{k+n}$. Using new information about the conjugation operation in the dual Steenrod algebra, we also consider quotients by families of Milnor generators and their conjugates. This allows us to produce a family of associative $H\mathbb F_2\wedge H\mathbb F_2$-algebras whose coefficient rings are finite-dimensional and exhibit unexpected duality features. We then use these algebras to give detailed computations of the homotopy groups of several modules over this ring spectrum.