For the Thom spectra y(n) between S and HF2, the paper proves that K(m)_*(TP(y(n))) vanishes for 1≤m≤n, so TP raises chromatic height by one.
Stable homotopy and generalised homology
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A homological approach to chromatic complexity of algebraic K-theory
For the Thom spectra y(n) between S and HF2, the paper proves that K(m)_*(TP(y(n))) vanishes for 1≤m≤n, so TP raises chromatic height by one.