Appoximate 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 $m\in En{d}_k(L)$ 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 $A:G\to En{d}_k(L)$ with $r(A(g^{\prime }+g^{\prime \prime })-A(g^{\prime })-A(g^{\prime \prime }))\le r$, $g^{\prime },g^{\prime \prime }\in G$ 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 $g\in G$. 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 $H_{\mathcal{F}}^k(V,M)$ (which is a purely algebraic analogue of the notion of $\epsilon$-representation (\cite{ep})) and interperate our result as a computation of the group $H_{\mathcal{F}}^1(V,M)$ for some $V$-modules $M$.

References in corpus (1)

• An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm