论文标题
彩色傅立叶变换
The Chromatic Fourier Transform
论文作者
论文摘要
我们开发了一个高级半二元傅立叶变换的一般理论，其中包括有限的Abelian群体的经典离散傅立叶变换，高度$ n = 0 $，以及$ e_n $ - （CO）$π$ - finite Spectra的$ e_n $ - （CO）同源性，由霍普金斯和lurie建立的，高度为$ n \ ge 1 $。我们使用该理论将上述二元性推广到三个不同的方向。首先，我们将其从$ \ mathbb {z} $ - 模块光谱扩展到所有（适当有限的）光谱，并使用它来计算$ e_n $的差异光谱。其次，我们通过用$ t（n）$替换$ e_n $的$ e_n $将其提升到望远镜设置，从而在望远镜环境中推断出有关亲戚，艾伦贝格（Eilenberg）的各种成果，艾伦贝格（Eilenberg） - 摩尔公式和galois扩展。第三，我们将它们的结果分类为两个对称的单体$ \ infty $ - $ k（n）$ - 本地$ e_n $ modules的本地系统的类别，并将其与（半二维）红移现象相关联。
We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height $n=0$, as well as a certain duality for the $E_n$-(co)homology of $π$-finite spectra, established by Hopkins and Lurie, at heights $n\ge 1$. We use this theory to generalize said duality in three different directions. First, we extend it from $\mathbb{Z}$-module spectra to all (suitably finite) spectra and use it to compute the discrepancy spectrum of $E_n$. Second, we lift it to the telescopic setting by replacing $E_n$ with $T(n)$-local higher cyclotomic extensions, from which we deduce various results on affineness, Eilenberg--Moore formulas and Galois extensions in the telescopic setting. Third, we categorify their result into an equivalence of two symmetric monoidal $\infty$-categories of local systems of $K(n)$-local $E_n$-modules, and relate it to (semiadditive) redshift phenomena.
