The Smith-Toda complex $V((p+1)/2)$ does not exist

Abstract

Using a generalized homotopy fixed point spectral sequence due to Hopkins and Miller, we show that the Smith-Toda complex $V((p+1)/2)$ does not exist for $p$ a prime greater than 5. This extends earlier results of Toda and Ravenel for the primes 2, 3, and 5. It is also shown that if $V((p-1)/2)$ exists, it is not a ring spectrum.