Appoximate Cohomology

Let $k$ be a field, $G$ be an abelian group and $r\in \mathbb N$. Let $L$ be an infinite dimensional $k$-vector space. For any $m\in End_k(L)$ we denote by $r(m)\in [0,\infty ]$ the rank of $m$. We define by $R(G,r,k)\in [0,\infty]$ the minimal $R$ such that for any map $A:G \to End_k(L)$ with $r(A(g'+g'')-A(g')-A(g''))\leq r$, $g',g''\in G$ there exists a homomorphism $\chi :G\to End_k(L)$ such that $r(A(g)-\chi (g))\leq R(G, r, k)$ for all $g\in G$. We show the finiteness of $R(G,r,k)$ for the case when $k$ is a finite field, $G=V$ is a $k$-vector space $V$ of countable dimension. We actually prove a generalization of this result. In addition we introduce a notion of {\it Approximate Cohomology} groups $H^k_{\mathcal F} (V,M)$ (which is a purely algebraic analogue of the notion of $\epsilon$-representation (\cite{ep})) and interperate our result as a computation of the group $H^1_{\mathcal F} (V,M)$ for some $V$-modules $M$.