Coherent cohomology groups/positive characteristic/subgroups/description

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

$V=H^i(Y,𝒮)$.

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 $\phi :Y^{\prime }\to Y$.

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

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

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

$0⟶𝒮⟶{𝒮}^{\prime }⟶{𝒪}_Y⟶0$

and hence projective bundles

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

and the corresponding torsor

$T=\mathbb{P}({{𝒮}^{\prime }}^{*})-\mathbb{P}({𝒮}^{*})$.

Are there subspaces related to properties of $T$?