We investigate when the filtration induced by Beilinson’s spectral sequence splits non-canonically into a direct sum decomposition. We conclude that for any vector bundle \({{\mathcal {E}}}\) on a projective space over an algebraically closed field of characteristic \(p>0\) there exists \(r_{0}\) such that for \(r\ge r_{0}\) the Frobenius pushforward \({\textsf{F}^{r}_{*}}{{\mathcal {E}}}\) decomposes as a direct sum of line bundles and exterior powers of the cotangent bundle (we also give a variant for the "toric Frobenius map" valid in any characteristic). As an application we give a short proof of Klyachko’s theorem for vanishing of the cohomology of toric vector bundles on projective spaces.
