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 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 $T$ ?