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 $m\in En{d}_k(L)$ we denote by $r(m)\in [0,\mathrm{\infty }]$ the rank of $m$. We define by $R(G,r,k)\in [0,\mathrm{\infty }]$ the minimal $R$ such that for any map $A:G\to En{d}_k(L)$ with $r(A(g^{\mathrm{\prime }}+g^{\mathrm{\prime }\mathrm{\prime }})-A(g^{\mathrm{\prime }})-A(g^{\mathrm{\prime }\mathrm{\prime }}))\le r$, $g^{\mathrm{\prime }},g^{\mathrm{\prime }\mathrm{\prime }}\in G$ there exists a homomorphism $\chi :G\to En{d}_k(L)$ such that $r(A(g)-\chi (g))\le R(G,r,k)$ for all $g\in G$\mathrm{.}Weshowthefinitenessof$R(G,r,k)$forthecasewhen$k$ 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 $H_{\mathcal{F}}^k(V,M)$ (which is a purely algebraic analogue of the notion of $ϵ$-representation (\cite{ep})) and interperate our result as a computation of the group $H_{\mathcal{F}}^1(V,M)$ for some $V$-modules $M$.