The Picard group of topological modular forms via descent theory

By A. Mathew, Vesna Stojanoska. 2014

82 citations recorded by the source corpus.

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 $\Sigma TMF $ (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 $Tmf$.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/39ecb9bff56d87ccd05875017edad880fc4541ae

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In Geometry and Topology

1. Lennart Meier
2. A. Mathew
3. M. Hill
4. B. Bhatt
5. Peter Scholze
6. N. Axel Naumann
7. J. Noel
8. Vesna Stojanoska

In Mathematical Physics

1. A. Mathew
2. Lennart Meier
3. M. Hill
4. B. Bhatt
5. Peter Scholze
6. N. Axel Naumann
7. J. Noel
8. Vesna Stojanoska

Citations in the corpus

• The Galois group of a stable homotopy theory (2014, background): “For some applications of these ideas to computations, see the paper [52] (for descriptions of thick subcategories) and [28,62,34] (for calculations of certain Picard groups).”
• The C2–spectrum Tmf1(3) and its invertible modules (2015, methodology): “The Picard group was first introduced into stable homotopy theory by Hopkins; recent work of Mathew and Stojanoska [MS14] then significantly extended our toolbox for its computation.”
• A simple universal property of Thom ring spectra (2014, unknown): “…-monoidal ∞ -category inherits an O -monoidal structure has been used implicitly before (see for example, [ABG11, Definition 8.5 and Remark 8.6] or [MS16, Proposition 2.2.3]), but as far the authors know, no explicit description has appeared in the literature, which is why we feel justified in…”