On the homotopy groups of E(n)-local spectra with unusual invariant ideals

TL;DR

This paper investigates the complex structure of the E(n)-based Adams E_2-term for certain spectra, revealing unexpected complexity at odd primes and providing explicit calculations at prime two for specific localized spectra.

Contribution

It analyzes the E(n)-based Adams E_2-term for spectra with unusual invariant ideals, highlighting complexities at odd primes and computing the E_infty-term at prime two.

Findings

E(n)-based Adams E_2-term complexity is unexpectedly high at odd primes.

Explicit determination of E_infty-term for pi_*(L_2T(1)/(v_1)) at prime two.

Computation of homotopy groups estimates for L_nT(m) spectra.

Abstract

Let E(n) and T(m) for nonnegative integers n and m denote the Johnson-Wilson and the Ravenel spectra, respectively. Given a spectrum whose E(n)_*-homology is E(n)_*(T(m))/(v_1,...,v_{n-1}), then each homotopy group of it estimates the order of each homotopy group of L_nT(m). We here study the E(n)-based Adams E_2-term of it and present that the determination of the E_2-term is unexpectedly complex for odd prime case. At the prime two, we determine the E_{infty}-term for pi_*(L_2T(1)/(v_1)), whose computation is easier than that of pi_*(L_2T(1)) as we expect.