Finite Smith–Toda Complexes at Varying Primes

For every nonnegative integer n, we construct a Smith–Toda complex $V(n)$ at some prime p depending on n. It is a finite p-local spectrum whose Brown–Peterson homology is ${\mathrm{BP}}_{\ast }/(p,v_1,\dots ,v_n)$ with its canonical comodule structure and generator in degree zero. Each listed generator is killed to its first power.