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
AI总结：
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。
摘要：
本文研究了当2p-1=h²时的奇异K(h)-局部Picard群κₕ，以及当p-1不整除h时的同调色消失猜想。核心思想是利用Gross-Hopkins对偶性将这两个问题与色同伦理论中特定希腊字母元素的计算关联起来。Miller-Ravenel-Wilson的经典结果表明，高度3、素数5下的奇异元素无法被2型复形V(1)检测到。对于同调消失猜想，我们证明其在不变素理想I_{h-1}模下成立，并进一步指出该猜想的这一特殊情况可推导出高度3、素数5下的奇异Picard群κₕ为零。这两个结果可视为证明素数5下κ₃消失的第一步。
英文摘要：
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 $5$.
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