拓扑模形式的Picard群的下降理论
The Picard group of topological modular forms via descent theory
摘要：
本文首先阐述了研究$\mathbf{E}_{\infty}$ -环谱的Picard群的下降理论方法，从而引出了Picard谱的研究。然后，我们开发的工具，从而获得的皮卡德光谱的相关下降谱序列中的差异的有效和明确的确定。作为一个主要应用，我们计算了拓扑模形式周期谱$TMF$和非周期非连通$Tmf$的Picard群.我们发现，$\mathrm{Pic} (TMF)$是循环的顺序576，产生的悬浮$ΣTMF $（结果最初由于霍普金斯），而$\mathrm{Pic}(Tmf) = \mathbb{Z}\oplus \mathbb{Z}/24$。特别地，我们证明了存在一个可逆的$Tmf$ -模，它不等价于$Tmf$的一个悬置.
英文摘要：
This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of $\mathbf{E}_{\infty}$-ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms $TMF$ and the non-periodic and non-connective $Tmf$. We find that $\mathrm{Pic} (TMF)$ is cyclic of order 576, generated by the suspension $ΣTMF $ (a result originally due to Hopkins), while $\mathrm{Pic}(Tmf) = \mathbb{Z}\oplus \mathbb{Z}/24$. In particular, we show that there exists an invertible $Tmf$-module which is not equivalent to a suspension of $Tmf$.
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