[Paper Review] Obtaining intermediate rings of a local profinite Galois extension without localization

Daniel G. Davis|arXiv (Cornell University)|Jun 16, 2010

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This paper demonstrates that intermediate $K(n)$-local spectra in profinite Galois extensions—such as $E_n^{hH}$, $L_{K(n)}(S^0)$, and $L_{K(n)}(X)$—can be obtained directly as homotopy fixed points of a single discrete $G_n$-spectrum $F_n$ without requiring a tower of spectra or subsequent $K(n)$-localization. The key result is that $E_n^{hU} \simeq (F_n)^{hU}$ for any open subgroup $U \leq G_n$, eliminating the need for localization after fixed points.

ABSTRACT

Let E_n be the Lubin-Tate spectrum and let G_n be the nth extended Morava stabilizer group. Then there is a discrete G_n-spectrum F_n, with L_{K(n)}(F_n) \simeq E_n, that has the property that (F_n)^{hU} \simeq E_n^{hU}, for every open subgroup U of G_n. In particular, (F_n)^{hG_n} \simeq L_{K(n)}(S^0). More generally, for any closed subgroup H of G_n, there is a discrete H-spectrum Z_{n, H}, such that (Z_{n, H})^{hH} \simeq E_n^{hH}. These conclusions are obtained from results about consistent k-local profinite G-Galois extensions E of finite vcd, where L_k(-) is L_M(L_T(-)), with M a finite spectrum and T smashing. For example, we show that L_k(E^{hH}) \simeq E^{hH}, for every open subgroup H of G.

Motivation & Objective

• To show that $K(n)$-local spectra like $E_n^{hH}$ can be obtained without $K(n)$-localizing after taking homotopy fixed points.
• To eliminate the need for a tower of discrete $G_n$-spectra in constructing $K(n)$-local homotopy fixed points.
• To establish that a single discrete $G_n$-spectrum $F_n$ suffices to recover $L_{K(n)}(S^0)$ and $L_{K(n)}(X)$ via homotopy fixed points.
• To generalize this result to consistent profaithful $k$-local profinite $G$-Galois extensions of finite virtual cohomological dimension (vcd).

Proposed method

• Construct $F_n$ as the colimit of fibrant replacements of $E_n^{dhN}$ over open normal subgroups $N \triangleleft_o G_n$, yielding a discrete $G_n$-spectrum.
• Use the equivalence $E_n^{hH} \simeq \operatorname{holim}_i (F_n \wedge M_i)^{hH}$, where $\{M_i\}$ is a tower of generalized Moore spectra.
• Show that $L_k(E^{hH}) \simeq \left(\operatorname{colim}_{V \triangleleft_o H} L_k((E_{fG})^V)\right)^{hH}$ for consistent $k$-local profinite $G$-Galois extensions.
• Prove that for profaithful extensions, $L_k(E^{hH}) \simeq \left(\operatorname{colim}_{V \triangleleft_o H} L_k((E_{fG})^V)\right)^{hH}$, with the colimit independent of $H$'s structure.
• Establish a weak equivalence $\operatorname{colim}_{V \triangleleft_o H} L_k((E_{fG})^V) \xrightarrow{\simeq} \operatorname{colim}_{V \triangleleft_o H} L_k((E_{fH})^V)$ in $\Sigma\mathrm{Sp}_H$, enabling direct computation.
• Leverage fibrant replacement functors and $K(n)$-localization functors to relate $E_n^{hH}$ to fixed points of $F_n$.

Experimental results

Research questions

• RQ1 Can $E_n^{hH}$ be obtained as the homotopy fixed points of a single discrete $G_n$-spectrum without $K(n)$-localization?
• RQ2 Is the use of a tower of spectra necessary to construct $K(n)$-local homotopy fixed points?
• RQ3 Can the construction of $L_{K(n)}(S^0)$ be achieved via homotopy fixed points of $F_n$ alone?
• RQ4 Does the profaithful $k$-local Galois extension structure allow a uniform construction of $L_k(E^{hH})$ across subgroups $H$?
• RQ5 Can the dependence of the fixed-point spectrum on $H$ be removed in favor of a universal discrete $H$-spectrum?

Key findings

• For any open subgroup $U \leq G_n$, $E_n^{hU} \simeq (F_n)^{hU}$, showing that $K(n)$-local spectra arise directly from fixed points of a single discrete $G_n$-spectrum.
• The spectrum $F_n$ is a discrete $G_n$-spectrum satisfying $(F_n)^{hG_n} \simeq L_{K(n)}(S^0)$, recovering the Hopkins-Miller spectrum without $K(n)$-localization.
• For any closed subgroup $H \leq G_n$, there exists a discrete $H$-spectrum $Z_{n,H}$ such that $(Z_{n,H})^{hH} \simeq E_n^{hH}$, providing a uniform construction for each $H$.
• The equivalence $L_k(E^{hH}) \simeq \left(\operatorname{colim}_{V \triangleleft_o H} L_k((E_{fG})^V)\right)^{hH}$ holds for consistent profaithful $k$-local profinite $G$-Galois extensions of finite vcd.
• The colimit $\operatorname{colim}_{V \triangleleft_o H} L_k((E_{fG})^V)$ is a discrete $H$-spectrum whose $H$-homotopy fixed points yield $L_k(E^{hH})$ without further localization.
• The construction is generalized to $k$-local profinite Galois extensions where $L_k(-) = L_M(L_T(-))$, with $M$ finite and $T$ smashing, and the result holds under profaithfulness.