Gross-Hopkins对偶与Gorenstein条件

格罗斯-霍普金斯对偶

Gross-Hopkins duality

本文给出了Hopkins和Gross关于K（n）-局部稳定同伦范畴中Brown-Comenetz对偶与Spanier-Whitehead对偶的一个结果的一个新的简单证明.

Gross-Hopkins Duals of Higher Real K-theory Spectra

We determine the Gross-Hopkins duals of certain higher real $K$-theory spectra. More specifically, let $p$ be an odd prime, and consider the Morava $E$-theory spectrum of height $n=p-1$. It is known, in the expert circles, that for certain finite subgroups $G$ of the Morava stabilizer group, the homotopy fixed point spectra $E_n^{hG}$ are Gross-Hopkins self-dual up to a shift. In this paper, we determine the shift for those finite subgroups $G$ which contain $p$-torsion. This generalizes previous results for $n=2$ and $p=3$.

The Lubin-Tate stack and Gross-Hopkins duality

Morava $E$-theory $E$ is an $E_\infty$-ring with an action of the Morava stabilizer group $Γ$. We study the derived stack $\operatorname{Spf} E/Γ$. Descent-theoretic techniques allow us to deduce a theorem of Hopkins-Mahowald-Sadofsky on the $K(n)$-local Picard group, as well as a recent result of Barthel-Beaudry-Stojanoska on the Anderson duals of higher real $K$-theories.

通过Gross-Hopkins对偶性研究奇异Picard群与色消失现象

Exotic Picard groups and chromatic vanishing via the Gross-Hopkins duality

本文利用Gross-Hopkins对偶性研究奇异K(h)-局部Picard群与色消失猜想，证明高度3素数5下奇异元素不可被V(1)检测、同调消失猜想在I_{h-1}模下成立且κₕ为零，为κ₃消失证明奠基。

Picard groups and duality for Real Morava $E$-theories

We show, at the prime 2, that the Picard group of invertible modules over $E_n^{hC_2}$ is cyclic. Here, $E_n$ is the height $n$ Lubin--Tate spectrum and its $C_2$-action is induced from the formal inverse of its associated formal group law. We further show that $E_n^{hC_2}$ is Gross--Hopkins self-dual and determine the exact shift. Our results generalize the well-known results when $n = 1$.

代数与拓扑中的对偶

Duality in algebra and topology

本文从交换代数中汲取一些经典思想，主要是涉及对偶的思想，并将它们应用于代数拓扑学中。为了实现这一点，我们解释的性质，普通交换环在这样一种方式，他们可以扩展到更一般的环，出现在同伦理论。在我们研究的环中，有空间上的上链的微分分次环，闭路空间上的链的微分分次环，以及各种环谱，例如，有限谱的Spanier-Whitehead图或球谱的色局部化。也许本文最重要的贡献是概念框架，它使我们能够查看所有以下对偶：庞加莱对偶流形，Gorenstein对偶交换环，本森-卡尔森对偶有限群的上同调环，庞加莱对偶群，格罗斯-霍普金斯对偶色稳定同伦理论，作为一个单一的现象的例子。除了建立这个框架，虽然，我们证明了一些新的结果，无论是在代数和拓扑，并给出了一些旧的结果新的证明。