Coherent cohomology groups/positive characteristic/subgroups/description

Let $Y$ be a projective variety over a field of positive characteristic $K$, ${\mathcal {S}}$ a coherent sheaf on $Y$ and consider

$V=H^{i}(Y,{\mathcal {S}})$.

Do there exist any interesting $K$-subspaces of $V$?

• Classes which are annihilated by some Frobenius power.

• Classes which are annihilated under some finite extension $\varphi :Y'\rightarrow Y$.

• Classes $c$ with the property: there exists a $z\in \Gamma (Y,{\mathcal {O}}(k))$, $z\neq 0$, such that

$zF^{e*}(c)=0$ for all $e$.

• For $i=1$ and a vector bundles ${\mathcal {S}}$ one can also look for properties of the geometric torsor defined by a class $c$. The class defines an extension

$0\longrightarrow {\mathcal {S}}\longrightarrow {{\mathcal {S}}'}\longrightarrow {\mathcal {O}}_{Y}\longrightarrow 0$

and hence projective bundles

${\mathbb {P} }({\mathcal {S}}^{*})\subset {\mathbb {P} }({{\mathcal {S}}'}^{*})$

and the corresponding torsor

$T={\mathbb {P} }({{\mathcal {S}}'}^{*})-{\mathbb {P} }({\mathcal {S}}^{*})$.

Are there subspaces related to properties of $T$?