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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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1072-947X
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Volume 26, Issue 2

Issues

On Küneth’s correlation and its applications

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

Abstract

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

0T(1)Hn+1(C*)Hn(TC*)THn(C*)0,

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

T(2i+1)Hn+i+1T(3)Hn+2T(1)Hn+1Hn(TC*)THn(C*)T(2)Hn+1T(4)Hn+2T(2i)Hn+i

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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