[Paper Review] Iterated homotopy fixed points for the Lubin-Tate spectrum, with an Appendix: An example of a discrete G-spectrum that is not hyperfibrant

Daniel G. Davis, Ben Wieland|arXiv (Cornell University)|Oct 29, 2006

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This paper establishes a construction of iterated homotopy fixed point spectra for the Lubin-Tate spectrum $E_n$ with respect to closed normal subgroups $H \trianglelefteq K \leq G_n$, where $G_n$ is the extended Morava stabilizer group. It proves that $(E_n^{hH})^{hK/H} \simeq E_n^{hK}$, extending Devinatz and Hopkins' result to non-finite quotients $K/H$, using Bousfield localization and continuous $G$-spectrum theory.

ABSTRACT

When G is a profinite group and H and K are closed subgroups, with H normal in K, it is not known, in general, how to form the iterated homotopy fixed point spectrum (Z^{hH})^{hK/H}, where Z is a continuous G-spectrum and all group actions are to be continuous. However, we show that, if G=G_n, the extended Morava stabilizer group, and Z=L_{K(n)}(E_n \wedge X), where L_{K(n)} is Bousfield localization with respect to Morava K-theory, E_n is the Lubin-Tate spectrum, and X is any spectrum with trivial G_n-action, then the iterated homotopy fixed point spectrum can always be constructed. Also, we show that (E_n^{hH})^{hK/H} is just E_n^{hK}, extending a result of Devinatz and Hopkins.

Motivation & Objective

• To resolve the problem of constructing iterated homotopy fixed point spectra $(Z^{hH})^{hK/H}$ when $H$ is not open in $K$, particularly for profinite $G$-spectra.
• To extend the known equivalence $(E_n^{hH})^{hK/H} \simeq E_n^{hK}$ beyond the case where $K/H$ is finite.
• To provide a general framework for forming iterated homotopy fixed points in chromatic homotopy theory using $K(n)$-localization.
• To demonstrate that $\widehat{L}(E_n^{dhH} \wedge X)$ is a continuous $K/H$-spectrum, enabling the construction of iterated homotopy fixed points.
• To construct a discrete $G$-spectrum that is not hyperfibrant, answering a question about the limitations of the hyperfibrant condition in equivariant stable homotopy theory.

Proposed method

• Uses Bousfield localization $\widehat{L} = L_{K(n)}$ to construct $\widehat{L}(E_n^{dhH} \wedge X)$ as a continuous $K/H$-spectrum with diagonal $K/H$-action.
• Applies the theory of continuous $G$-spectra from [2] to define homotopy fixed points as total right derived functors.
• Leverages the fact that $E_n^{hK} \simeq E_n^{dhK}$ for all closed $K \leq G_n$, established in [4], to relate the constructed iterated fixed points to known $K$-fixed points.
• Constructs a $\mathbb{Z}_q$-equivariant isomorphism $\pi_0(X^{h\mathbb{Z}/p}) \cong \prod_{n \geq 0} \mathbb{Z}/p[\mathbb{Z}/q^n]$ to analyze the module structure.
• Uses a $G$-equivariant injection $i: \mathbb{Z}_q \to \prod_{n \geq 0} \mathbb{Z}/p[\mathbb{Z}/q^n]$ to show that the product is not a discrete $\mathbb{Z}_q$-module, proving non-hyperfibrancy.
• Applies contradiction via continuity of the action: if the product were discrete, $\mathbb{Z}_q$ would be a discrete $\mathbb{Z}_q$-set, which it is not.

Experimental results

Research questions

• RQ1 Can iterated homotopy fixed points $(Z^{hH})^{hK/H}$ be constructed when $H$ is not open in $K$ for a continuous $G$-spectrum $Z$?
• RQ2 Does the equivalence $(E_n^{hH})^{hK/H} \simeq E_n^{hK}$ hold when $K/H$ is profinite, not finite?
• RQ3 Is there a discrete $G$-spectrum that is not hyperfibrant, and if so, can it be explicitly constructed?
• RQ4 Can the $K/H$-action on $E_n^{dhH}$ be lifted to a continuous $K/H$-spectrum structure on $\widehat{L}(E_n^{dhH} \wedge X)$?
• RQ5 Is the product $\prod_{n \geq 0} \mathbb{Z}/p[\mathbb{Z}/q^n]$ a discrete $\mathbb{Z}_q$-module, and what does this imply for the hyperfibrant condition?

Key findings

• The spectrum $\widehat{L}(E_n^{dhH} \wedge X)$ is a continuous $K/H$-spectrum with diagonal $K/H$-action, enabling the construction of iterated homotopy fixed points.
• The iterated homotopy fixed point spectrum $(E_n^{hH})^{hK/H}$ is equivalent to $E_n^{hK}$, even when $K/H$ is profinite.
• The $\mathbb{Z}_q$-action on $\pi_0(X^{h\mathbb{Z}/p}) \cong \prod_{n \geq 0} \mathbb{Z}/p[\mathbb{Z}/q^n]$ is not discrete, which implies the spectrum $X$ is not hyperfibrant.
• The construction of $\widehat{L}(E_n^{dhH} \wedge X)$ as a continuous $K/H$-spectrum allows the application of the theory of [2] to define $(\widehat{L}(E_n^{dhH} \wedge X))^{hK/H}$.
• The product $\prod_{n \geq 0} \mathbb{Z}/p[\mathbb{Z}/q^n]$ is not a discrete $\mathbb{Z}_q$-module, as shown by contradiction via continuity of the action.
• An explicit example of a discrete $G$-spectrum that is not hyperfibrant is constructed, resolving a question in equivariant stable homotopy theory.