Is the following true:


For any $n$ there exists $p\_0$ s.t. for any finite group $G$ of Lie type of rank $\leq n$ and characteristic $p\geq p\_0$ and any (irreducible) $\mathbb F\_p$ representation $V$ of $G$ of dimension $\leq n$  we have 
$$H^1(G,V)=0$$



Vanishing of first co-homology with coefficients modular representations of small dimension
I think the answer to your question is yes for example with $p\_0 = n+3$.


See the following paper:



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> "Small Representations Are Completely Reducible", Robert M. Guralnick, J. Algebra (1999).
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Theorem A in this paper says the following:



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> Let $k$ be a field of positive characteristic $p$. Let $G$ be a finite group containing no nontrivial normal $p$-subgroup. Let $V$ be a $kG$-module such that $G$ acts faithfully on $V$.
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> (a) If $\dim V \leq p − 2$, then $V$ is completely reducible;
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> (b) If $\dim V \leq p − 3$, then $H^1(G, V) = 0$.
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