Glatte projektive Varietät/Endlicher Körper/Torsor/Kohomologische Dimension und endliche Trivialisierbarkeit/Problem/en

Let ${}Y$ denote a smooth projective variety of dimension ${}d$ over a finite field ${}K$. Let ${}{\mathcal {S}}$ be a vector bundle on ${}Y$ and ${}c\in H^{1}(Y,{\mathcal {S}})$ be a cohomolgy class with corresponding torsor

$T\longrightarrow Y.$

Does ${}T$ has cohomological dimension ${}d$ if and only if there exists a finite morphism

$\varphi \colon Y'\longrightarrow Y$

(where ${}Y'$ denotes another (smooth?) projective variety of dimension ${}d$) such that ${}\varphi ^{*}(c)=0$?