Mdzinarishvili, L. D. The functor Ext and Kolmogorov homologies. (Russian) Zbl 0538.55004 Tr. Mat. Inst. Steklova 154, 180-185 (1983). The main result of this article is the following Theorem. Let \((\bar X,\bar A)=\{(X_ i,A_ i)\}\) be a countable inverse system of compact pairs \((X_ i,A_ i)\) and \((X,A)=\lim_{\leftarrow} (\bar X,\bar A),\quad \{G_ i,g_ i\}=\bar G\) a countable inverse system of Abelian groups, where \(g_ i\) is an epimorphism, and \(G=\lim_{\leftarrow} \bar G.\) Then for the Kolmogorov homology there exists an exact sequence \[ 0\to \lim^{(1)} H^ k_{q+1}(X_ i,A_ i,G_ i)\to H^ k_ q(X,A,G)\to \lim H^ k_ q(X_ i,A_ i,G_ i)\to 0. \] Results about generalized direct and inverse limits are also obtained. Moreover, the relations between homology dimensions related to different systems of coefficients are found. Reviewer: V.Sharko Cited in 1 Review MSC: 55N35 Other homology theories in algebraic topology 18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) 55M10 Dimension theory in algebraic topology Keywords:Kolmogorov homology theory; inverse system of compact pairs; lim-lim(1)- sequence; homology dimensions PDFBibTeX XMLCite \textit{L. D. Mdzinarishvili}, Tr. Mat. Inst. Steklova 154, 180--185 (1983; Zbl 0538.55004)