论文标题
紧凑型$ p $ - 亚种分析组和色度同型双重性的双重化领域
Dualizing spheres for compact $p$-adic analytic groups and duality in chromatic homotopy
论文作者
论文摘要
本文的主要目的是研究$ k（n）$ - 本地类别中的杀手型二元性。 $ k（n）$ - 本地类别中的主要参与者之一是lubin-tate Spectrum $ e_n $，其同质群体在隐性特征$ p $中对正式的高度$ n $的正式组定律进行了分类。众所周知，$ e_n $是自动划分的。但是，这并不能完全考虑到Morava稳定器组$ \ MATHBB {G} _N $的行动，甚至是其正式组的自动形态的亚组。在本文中，我们发现$ \ mathbb {g} _n $ - equivariant dual $ e_n $实际上是$ e_n $ twist twest twest twest，twist twest twest twest twist twest twist twest twest，$ \ mathbb {g} _n $。该领域是组$ \ mathbb {g} _n $的双重化模块，我们为任何紧凑型$ p $ -p $ -Adic Analytic Group $ \ MATHCAL {G} $构造和研究了这样的对象$ i _ {\ Mathcal {G}} $。如果我们将$ \ MATHCAL {G} $在$ i _ {\ Mathcal {G}} $上的动作限制为某些类型的小型子组，我们将识别$ i _ {\ Mathcal {G}} $，具有特定代表范围的Sphere s s s lie ealgebra of lie eLgebra of $ \ nathcal of $ \ mathcal {g} $ {g} $。这是通过$ p $ complete球谱的分类，并在特征类别上的小学Abelian $ p $ - 组采取了动作，然后对所讨论的特征类进行了特定的比较。正如我们在本文的后面部分所证明的那样，该设置使该理论非常易于计算，确定了$ k（n）$ - 本地Spanier-whitehead duals $ e_n^{hh} $，用于选择$ p $和$ n $的选择和$ n $和有限的子群体$ h $ h $ h $ h $ h $ h $ h $ h $ h $ h $ \ mathbb {g mathbb {
The primary goal of this paper is to study Spanier-Whitehead duality in the $K(n)$-local category. One of the key players in the $K(n)$-local category is the Lubin-Tate spectrum $E_n$, whose homotopy groups classify deformations of a formal group law of height $n$, in the implicit characteristic $p$. It is known that $E_n$ is self-dual up to a shift; however, that does not fully take into account the action of the Morava stabilizer group $\mathbb{G}_n$, or even its subgroup of automorphisms of the formal group in question. In this paper we find that the $\mathbb{G}_n$-equivariant dual of $E_n$ is in fact $E_n$ twisted by a sphere with a non-trivial (when $n>1$) action by $\mathbb{G}_n$. This sphere is a dualizing module for the group $\mathbb{G}_n$, and we construct and study such an object $I_{\mathcal{G}}$ for any compact $p$-adic analytic group $\mathcal{G}$. If we restrict the action of $\mathcal{G}$ on $I_{\mathcal{G}}$ to certain type of small subgroups, we identify $I_{\mathcal{G}}$ with a specific representation sphere coming from the Lie algebra of $\mathcal{G}$. This is done by a classification of $p$-complete sphere spectra with an action by an elementary abelian $p$-group in terms of characteristic classes, and then a specific comparison of the characteristic classes in question. The setup makes the theory quite accessible for computations, as we demonstrate in the later sections of this paper, determining the $K(n)$-local Spanier-Whitehead duals of $E_n^{hH}$ for select choices of $p$ and $n$ and finite subgroups $H$ of $\mathbb{G}_n$.
