A spectrum whose \(BP_ *\)-homology is \((BP_ *\slash{}I_ 5)[t_ 1]\) (Q1191889)

Denote by \(I_ 5\) the invariant prime ideal \((p,v_ 1,\dots,v_ 4)\) in the coefficient ring \(BP_ *\) of Brown-Peterson homology. Consider a prime number \(p>7\). The author constructs a spectrum \(W_ 1(4)\) with \(BP_ *(W_ 1(4))=(BP_ */I_ 5)[t_ 1]\), where \(| t_ 1|=2p-2\). It is obtained as the cofiber of a map which is shown to exist by an Adams spectral sequence argument. The existence of \(W_ 1(4)\) is necessary for the existence of the Smith-Toda spectrum \(V(4)\).