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 Mat. Sb., 2019, Volume 210, Number 9, Pages 19–58 (Mi msb9152)

Higher colimits, derived functors and homology

S. O. Ivanova, R. V. Mikhailovab, V. A. Sosniloa

a Laboratory of Modern Algebra and Applications, St. Petersburg State University, St. Petersburg, Russia
b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, St. Petersburg, Russia

Abstract: We develop a theory of higher colimits over categories of free presentations. We show that different homology functors such as Hochschild and cyclic homology of algebras over a field of characteristic zero, simplicial derived functors, and group homology can be obtained as higher colimits of simply defined functors. Connes' exact sequence linking Hochschild and cyclic homology was obtained using this approach as a corollary of a simple short exact sequence. As an application of the developed theory, we show that the third reduced $K$-functor can be defined as the colimit of the second reduced $K$-functor applied to the fibre square of a free presentation of an algebra. We also prove a Hopf-type formula for odd-dimensional cyclic homology of an algebra over a field of characteristic zero.
Bibliography: 17 titles.

Keywords: higher colimits, derived functors, $K$-theory, cyclic homology.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 14.W03.31.0030 This research was carried out with the support of a grant of the Government of the Russian Federation for state support of scientific research conducted under the auspices of leading scientists (ag. no. 14.W03.31.0030).

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DOI: https://doi.org/10.4213/sm9152

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English version:
Sbornik: Mathematics, 2019, 210:9, 1222–1258

Bibliographic databases:

UDC: 512.664+512.666
MSC: 16E40, 18A30, 18E25
Received: 25.07.2018 and 08.04.2019

Citation: S. O. Ivanov, R. V. Mikhailov, V. A. Sosnilo, “Higher colimits, derived functors and homology”, Mat. Sb., 210:9 (2019), 19–58; Sb. Math., 210:9 (2019), 1222–1258

Citation in format AMSBIB
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