Approximate cohomology

Let k be a field, G be an abelian group and r∈ N. Let L be an infinite dimensional k-vector space. For any m∈ End _k(L) we denote by r(m) ∈ [ 0 , ∞] the rank of m. We define by R(G, r, k) ∈ [ 0 , ∞] the minimal R such that for any map A: G→ End _k(L) with r(A(g^′+ g^{′ ′}) - A(g^′) - A(g^{′ ′})) ≤ r, g^′, g^{′ ′}∈ G there exists a homomorphism χ: G→ End _k(L) such that r(A(g) - χ(g)) ≤ R(G, r, k) for all g∈ 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 Approximate Cohomology groups HFk(V,M) [which is a purely algebraic analogue of the notion of ϵ-representation (Kazhdan in Isr. J. Math. 43:315–323, 1982)] and interperate our result as a computation of the group HF1(V,M) for some V-modules M.