Resumen de 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∈Endk(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→Endk(L) with r(A(g′+g′′)−A(g′)−A(g′′))≤r , g′,g′′∈G there exists a homomorphism χ:G→Endk(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 HkF(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 H1F(V,M) for some V-modules M.
