Approximate cohomology

Abstract

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 $m\in {\text {End}}_k(L)$ 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 $A:G \rightarrow {\text {End}}_k(L)$ with $r(A(g'+g'')-A(g')-A(g''))\le r$, $g',g''\in G$ there exists a homomorphism $\chi :G\rightarrow {\text {End}}_k(L)$ such that $r(A(g)-\chi (g))\le R(G, r, k)$ 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 Approximate Cohomology groups $H^k_\mathcal F(V,M)$ [which is a purely algebraic analogue of the notion of $\epsilon$-representation (Kazhdan in Isr. J. Math. 43:315–323, 1982)] and interperate our result as a computation of the group $H^1_\mathcal F(V,M)$ for some V-modules M.