论文标题
异国情调的Picard群体和色彩消失通过大牛皮二元性
Exotic Picard groups and chromatic vanishing via the Gross-Hopkins duality
论文作者
论文摘要
在本文中，我们研究了异国情调的$ k（h）$ - 本地PICARD组$κ_H$当$ 2p-1 = h^2 $和同源色的消失猜想时，当$ p-1 $不划分$ h $时。主要思想是使用毛大奖金二元性将两个问题与色素同型理论中的某些希腊字母元素计算相关联。然后，Miller-Ravenel-Wilson的经典结果暗示着一个高度$ 3 $和Prime $ 5 $的外来元素，并未被Type-2 $ $ 2 $复杂$ v（1）$检测到。对于同源消失的猜想，我们证明它将其保存为不变的prime理想$ i_ {h-1} $。我们进一步表明，这种消失的猜想的特殊案例暗示了异国情调的Picard Group $κ_H$在高度$ 3 $和Prime $ 5 $时为零。这两种结果都可以被认为是证明$κ_3$以$ 5 $的消失的第一步。
In this paper, we study the exotic $K(h)$-local Picard groups $κ_h$ when $2p-1=h^2$ and the homological Chromatic Vanishing Conjecture when $p-1$ does not divide $h$. The main idea is to use the Gross-Hopkins duality to relate both questions to certain Greek letter element computations in chromatic homotopy theory. Classical results of Miller-Ravenel-Wilson then imply that an exotic element at height $3$ and prime $5$ is not detected by the type-$2$ complex $V(1)$. For the homological Vanishing Conjecture, we prove it holds modulo the invariant prime ideal $I_{h-1}$. We further show that this special case of the Vanishing Conjecture implies the exotic Picard group $κ_h$ is zero at height $3$ and prime $5$. Both results can be thought of as a first step towards proving the vanishing of $κ_3$ at prime $5$.
