Smith-Toda complexes V(n) at every height when the prime may vary

Theorem 1.1: for every $n\ge0$ there are a prime $p$ and a finite $p$-local spectrum $X$ with $BP_*X\cong BP_*/(p,v_1,\ldots,v_n)$ as comodules with the canonical coaction and generator in degree zero. The companion constructs $V(4)$ at $p=1009$, answering the existence question Culver-Zhang recorded as open at every prime. Partial for the realization problem: no explicit or effective prime for $n\ge5$, no statement that $V(n)$ exists for all large $p$, no multiplicative structure, and nothing on the gap between Nave's nonexistence range and the constructed cases.

Theorem 1.1: for every $n\ge0$ there are a prime $p$ and a finite $p$-local spectrum $X$ with $BP_*X\cong BP_*/(p,v_1,\ldots,v_n)$ as comodules with the canonical coaction and generator in degree zero. The companion constructs $V(4)$ at $p=1009$, answering the existence question Culver-Zhang recorded as open at every prime. Partial for the realization problem: no explicit or effective prime for $n\ge5$, no statement that $V(n)$ exists for all large $p$, no multiplicative structure, and nothing on the gap between Nave's nonexistence range and the constructed cases.