关于\(\mathbb{E}_\infty\)空间同调上的层级结构与循环幂运算
Level structures and cyclic power operations on the homology of $\mathbb{E}_\infty$ spaces
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中文总结 AI 辅助
本文基于代数几何，借助Morava \(E\)-理论的幂运算理论，探讨周期复配边的循环幂运算，描述了\(2\)周期偶数\(\mathbb{E}_\infty\)环\(E\)同调上幂运算的计算方法，并给出\(p = 2\)时的具体示例计算。
中文摘要 AI 辅助
在本文档中，我们讨论如何依据Ando、Hopkins和Strickland所发展的Morava \(E\)-理论的幂运算理论，通过代数几何来理解周期复配边的循环幂运算。利用此观点，我们描述了如何计算\(E\)同调上的幂运算，其中\(E\)是\(2\)周期的偶数\(\mathbb{E}_\infty\)环。然后我们给出了\(p = 2\)时的一些具体示例计算；特别地，我们计算了加法运算\(\Delta\)对\(E = KU, E_2\)时不可分解元\(\widehat{Q}(E_0(BU))\)的作用。
英文摘要
In this document, we discuss how cyclic power operations for periodic complex bordism can be understood through algebraic geometry, following from the theory of power operations for Morava $E$-theory developed by Ando, Hopkins, and Strickland. Using this perspective, we describe how one computes power operations on the $E$-homology of $BU$ and $BU\times \mathbb{Z}$, where $E$ is a 2-periodic even $\mathbb{E}_\infty$ ring. Then we provide some explicit example computations at $p=2$; in particular, we compute the action of the additive operations $Δ$ on the indecomposables $\widehat{Q}(E_0(BU))$ for $E=KU, E_2$.
