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Quotient rings of 𝐻𝔽₂∧ℍ𝔽₂
Transactions of the American Mathematical Society ( IF 1.3 ) Pub Date : 2021-09-29 , DOI: 10.1090/tran/8512
Agnès Beaudry , Michael Hill , Tyler Lawson , XiaoLin Danny Shi , Mingcong Zeng

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.


中文翻译:

𝐻𝔽₂∧ℍ𝔽₂的商环

摘要:我们研究交换环谱$H\mathbb F_2\wedge H\mathbb F_2$ 上的模,其系数群是Milnor 生成器集合的对偶Steenrod 代数的商。我们证明这些商中很少有接受代数结构的,但那些可以简单构造的商可以简单地构造:杀死关联代数类别中的生成器 $\xi _k$ 自由地杀死更高的生成器 $\xi _{k+n}$ . 使用关于对偶 Steenrod 代数中的共轭运算的新信息,我们还考虑了 Milnor 生成器族及其共轭的商。这使我们能够产生一族关联的 $H\mathbb F_2\wedge H\mathbb F_2$-代数,其系数环是有限维的,并表现出意想不到的对偶特征。
更新日期:2021-11-09
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