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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2018: 0.27

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Volume 26, Issue 2


On Küneth’s correlation and its applications

Leonard Mdzinarishvili
Published Online: 2019-04-09 | DOI: https://doi.org/10.1515/gmj-2019-2021


Let 𝒦 be an abelian category that has enough injective objects, let T:𝒦A be any left exact covariant additive functor to an abelian category A and let T(i) be a right derived functor, u1, [S. Mardešić, Strong Shape and Homology, Springer Monogr. Math., Springer, Berlin, 2000]. If T(i)=0 for i2 and T(i)Cn=0 for all n, then there is an exact sequence


where C*={Cn} is a chain complex in the category 𝒦, Hn(C*) is the homology of the chain complex C*, TC* is a chain complex in the category A, and Hn(TC*) is the homology of the chain complex TC*. This exact sequence is the well known Künneth’s correlation. In the present paper Künneth’s correlation is generalized. Namely, the conditions are found under which the infinite exact sequence


holds, where T(2i+1)Hn+i+1=T(2i+1)Hn+i+1(C*), T(2i)Hn+i=T(2i)Hn+i(C*). The formula makes it possible to generalize Milnor’s formula for the cohomologies of an arbitrary complex, relatively to the Kolmogorov homology to the Alexandroff–Čech homology for a compact space, to a generative result of Massey for a local compact Hausdorff space X and a direct system {U} of open subsets U of X such that U¯ is a compact subset of X.

Keywords: Künneth’s correlation; inverse system of finite simplicial complex; Kolmogoroff; Čech; Massey homologies

MSC 2010: 55N10


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About the article

Received: 2018-09-15

Accepted: 2018-12-24

Published Online: 2019-04-09

Published in Print: 2019-06-01

Citation Information: Georgian Mathematical Journal, Volume 26, Issue 2, Pages 295–301, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2019-2021.

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