A splitting theorem for connected Morava K-theories (Q923423)

Let k(n) (n\(\geq 1)\) be the connective Morava K-theories for a prime p; k(n) is a BP-module spectrum with \(\pi_*(k(n))={\mathbb{Z}}_ p[v_ n]\), where \(v_ n\) has degree \(2(p^ n-1)\). For a connected locally finite spectrum X, the author proves that all \(k(n)^*\)-torsion of \(k(n)^*(X)\) is annihilated by \(v^ e_ n\) (e\(\geq 1)\) if and only if k(n)\(\wedge X\) is homotopy equivalent to a wedge of suspensions of spectra k(n) and \({}^ rk(n)\) (0\(\leq r\leq e-1)\), where \({}^ rk(n)\) is the r-th Postnikov factor of k(n). The main tool used is the Bockstein spectral sequence for \(k(n)^*(X)\).