Quotient rings of 𝐻𝔽₂∧ℍ𝔽₂

Beaudry, Agnès; Hill, Michael; Lawson, Tyler; Shi, XiaoLin Danny; Zeng, Mingcong

doi:10.1090/tran/8512

Quotient rings of $H\mathbb {F}_2 \wedge H\mathbb {F}_2$
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by Agnès Beaudry, Michael A. Hill, Tyler Lawson, XiaoLin Danny Shi and Mingcong Zeng PDF

Trans. Amer. Math. Soc. 374 (2021), 8949-8988 Request permission

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.

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