Cohomological Dimension in Pro-p Towers

We give a proof without use of perfectoid geometry of the following vanishing theorem of Scholze: for $X\subset \mathbb{P}^n$ a projective scheme of dimension $d$ over an algebraically closed characteristic $0$ field, and $X_r$ the inverse image of $X$ via the map that assigns $(x_0^{p^r}: \dots : x_n^{p^r})$ to the homogeneous coordinates $(x_0:\ldots :x_n)$, the induced map $H^i(X, {{\mathbb{F}}}_p)\to H^i(X_r, {{\mathbb{F}}}_p)$ on étale cohomology dies for $i>d$ and $r$ large. Our proof holds in characteristic $\ell \neq p$ as well.