[Paper Review] The Lubin-Tate stack and Gross-Hopkins duality

Sanath K. Devalapurkar|arXiv (Cornell University)|Nov 13, 2017

Homotopy and Cohomology in Algebraic Topology12 references3 citations

TL;DR

This paper establishes a derived algebraic geometry framework for the Lubin-Tate stack $①\operatorname{Spf}E/\Gamma$ to re-derive and conceptually explain key results in $K(n)$-local stable homotopy theory, including the Hopkins-Mahowald-Sadofsky theorem on the $K(n)$-local Picard group and Barthel-Beaudry-Stojanoska's Anderson self-duality of higher real $K$-theories. The key contribution is a descent-theoretic proof of Gross-Hopkins duality via the monodromy action on line bundles over the Lubin-Tate stack.

ABSTRACT

Morava $E$-theory $E$ is an $E_\infty$-ring with an action of the Morava stabilizer group $Γ$. We study the derived stack $\operatorname{Spf} E/Γ$. Descent-theoretic techniques allow us to deduce a theorem of Hopkins-Mahowald-Sadofsky on the $K(n)$-local Picard group, as well as a recent result of Barthel-Beaudry-Stojanoska on the Anderson duals of higher real $K$-theories.

Motivation & Objective

• To re-derive the Hopkins-Mahowald-Sadofsky theorem on the $K(n)$-local Picard group using descent-theoretic techniques on the Lubin-Tate stack $\operatorname{Spf}E/\Gamma$.
• To provide a conceptual explanation for the Anderson self-duality of higher real $K$-theories, particularly $EO_{p-1}$, via derived duality and quasicoherent sheaves.
• To establish a criterion for identifying dualizing sheaves on even periodic derived Deligne-Mumford stacks, using homotopical and cohomological conditions on spectra.
• To prove Spanier-Whitehead self-duality of $E_{n(p-1)}^{hG}$ for $G=C_p$ at height $n(p-1)$, using spectral sequence computations and the HFPSS.

Proposed method

• Use of derived algebraic geometry to interpret $\operatorname{Spf}E/\Gamma$ as the Lubin-Tate stack, enabling descent-theoretic analysis of $K(n)$-local spectra.
• Application of the equivalence $\operatorname{QCoh}(\operatorname{Spf}E/\Gamma) \simeq L_{K(n)}\mathrm{Sp}$ to relate quasicoherent sheaves to $K(n)$-local categories.
• Construction of the monodromy action of $\Gamma$ on line bundles over $\operatorname{Spf}E/\Gamma$ to interpret the map $\operatorname{Pic}_n \to H^1_c(\Gamma; E_0^\times)$.
• Computation of the $E_2$-page of the homotopy fixed point spectral sequence (HFPSS) for $E_{n(p-1)}^{hC_p}$ using the Hill-Hopkins-Ravenel result on differentials.
• Use of Toda-style and $d_{2p^i-1}$-differentials to show that $\delta_n^N \gamma$ is a permanent cycle, implying $DE^{hC_p} \simeq \Sigma^k E^{hC_p}$.
• Leveraging the Leray-Hochschild-Serre spectral sequence to extend the result from $C_p$ to general finite subgroups $G \subset \Gamma$ with Sylow $p$-subgroup $C_p$.

Experimental results

Research questions

• RQ1 How can the Hopkins-Mahowald-Sadofsky theorem on the $K(n)$-local Picard group be re-derived using descent theory on the Lubin-Tate stack?
• RQ2 What is the geometric interpretation of the map $\operatorname{Pic}_n \to H^1_c(\Gamma; E_0^\times)$ in terms of monodromy on line bundles over $\operatorname{Spf}E/\Gamma$?
• RQ3 Why does $L_{K(n)}I_{\mathbf{Z}}EO_{p-1} \simeq \Sigma^{(p-1)^2 - 1}EO_{p-1}$ hold, and what is the underlying geometric reason for Anderson self-duality in higher real $K$-theory?
• RQ4 Can the Spanier-Whitehead self-duality of $E_{n(p-1)}^{hC_p}$ be established via spectral sequence methods and the HFPSS?
• RQ5 What conditions ensure that a spectrum $R$ is a dualizing sheaf for $\operatorname{Spec}S$, and how can this be detected via homotopical data?

Key findings

• The equivalence $\operatorname{QCoh}(\operatorname{Spf}E/\Gamma) \simeq L_{K(n)}\mathrm{Sp}$ is established, providing a derived geometric interpretation of $K(n)$-local stable homotopy theory.
• The Hopkins-Mahowald-Sadofsky theorem is re-derived as a descent-theoretic statement along the étale cover $\operatorname{Spf}E \to \operatorname{Spf}E/\Gamma$.
• The map $\operatorname{Pic}_n \to H^1_c(\Gamma; E_0^\times)$ is interpreted as the monodromy action of the line bundle corresponding to a $K(n)$-locally invertible spectrum over $\operatorname{Spf}E/\Gamma$.
• A criterion is proven: a $p$-complete spectrum $R$ is a dualizing sheaf for $\operatorname{Spec}S$ if and only if $\operatorname{Map}(H\mathbf{Z}/p, R) \simeq \Sigma^{-1}H\mathbf{Z}/p$.
• The Spanier-Whitehead self-duality of $E_{n(p-1)}^{hC_p}$ is proven by showing that $\delta_n^N \gamma$ is a permanent cycle in the HFPSS for $DE^{hC_p}$, implying $DE^{hC_p} \simeq \Sigma^k E^{hC_p}$.
• The result extends to all finite subgroups $G \subset \Gamma$ with Sylow $p$-subgroup $C_p$, via norm invariance in the HFPSS.