The Picard group of topological modular forms via descent theory

Mathew, Akhil¹ ; Stojanoska, Vesna²

¹ Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, United States
² Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green Steet, Urbana, IL 61801, United States

Geometry & topology, Tome 20 (2016) no. 6, pp. 3133-3217.

Résumé

This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of $E_{\infty }$–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 nonperiodic and nonconnective $Tmf$. We find that $Pic\left(TMF\right)$ is cyclic of order $576$, generated by the suspension $\Sigma TMF$ (a result originally due to Hopkins), while $Pic\left(Tmf\right)=\mathbb{Z}\oplus \mathbb{Z}∕24$. In particular, we show that there exists an invertible $Tmf$–module which is not equivalent to a suspension of $Tmf$.

Détail

DOI : 10.2140/gt.2016.20.3133

Classification : 14C22, 55N34, 55P43, 55S35, 55T99, 55P47
Keywords: Picard groups and spectra, descent, topological modular forms

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