Email this article Higher colimits, derived functors and homology
- Authors: Ivanov S.O.1, Mikhailov R.V.1,2, Sosnilo V.A.1
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Affiliations:
- Laboratory of Modern Algebra and Applications, St. Petersburg State University
- St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
- Issue: Vol 210, No 9 (2019)
- Pages: 19-58
- Section: Articles
- URL: https://ogarev-online.ru/0368-8666/article/view/133282
- DOI: https://doi.org/10.4213/sm9152
- ID: 133282
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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.
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About the authors
Sergei Olegovich Ivanov
Laboratory of Modern Algebra and Applications, St. Petersburg State University
Email: ivanov.s.o.1986@gmail.com
Candidate of physico-mathematical sciences, no status
Roman Valerevich Mikhailov
Laboratory of Modern Algebra and Applications, St. Petersburg State University; St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of SciencesDoctor of physico-mathematical sciences, Head Scientist Researcher
Vladimir Aleksandrovich Sosnilo
Laboratory of Modern Algebra and Applications, St. Petersburg State University
References
- А. Картан, С. Эйленберг, Гомологическая алгебра, ИЛ, М., 1960, 510 с.
- П. Габриель, М. Цисман, Категории частных и теория гомотопий, Мир, М., 1971, 295 с.
- S. M. Gersten, “$K_3$ of a ring is $H_3$ of the Steinberg group”, Proc. Amer. Math. Soc, 37:2 (1973), 366–368
- S. M. Gersten, “K-theory of free rings”, Comm. Algebra, 1 (1974), 39–64
- G. Donadze, N. Inassaridze, M. Ladra, “Cyclic homology via derived functors”, Homology Homotopy Appl., 12:2 (2010), 321–334
- S. O. Ivanov, R. Mikhailov, “A higher limit approach to homology theories”, J. Pure Appl. Algebra, 219:6 (2015), 1915–1939
- F. Keune, “The relativization of $K_2$”, J. Algebra, 54:1 (1978), 159–177
- J.-L. Loday, Cyclic homology, Grundlehren Math. Wiss., 301, Springer-Verlag, Berlin, 1992, xviii+454 pp.
- J. Lurie, Higher topos theory, Ann. of Math. Stud., 170, Princeton Univ. Press, Princeton, NJ, 2009, xviii+925 pp.
- S. Mac Lane, Categories for the working mathematician, Grad. Texts in Math., 5, Springer-Verlag, New York–Berlin, 1971, ix+262 pp.
- B. A. Magurn, An algebraic introduction to $K$-theory, Encyclopedia Math. Appl., 87, Cambridge Univ. Press, Cambridge, 2002, xiv+676 pp.
- R. Mikhailov, I. B. S. Passi, “Generalized dimension subgroups and derived functors”, J. Pure Appl. Algebra, 220:6 (2016), 2143–2163
- R. Mikhailov, I. B. S. Passi, “Dimension quotients, Fox subgroups and limits of functors”, Forum Math., 31:2 (2019), 385–401
- D. G. Quillen, Homotopical algebra, Lecture Notes in Math., 43, Springer-Verlag, Berlin–New York, 1967, iv+156 pp.
- D. Quillen, “Higher algebraic K-theory. I”, Algebraic K-theory (Battelle Memorial Inst., Seattle, WA, 1972), v. I, Springer Lect. Notes Math., 341, Higher K-theories, Springer, Berlin, 1973, 85–147
- D. Quillen, “Cyclic cohomology and algebra extensions”, K-Theory, 3:3 (1989), 205–246
- C. A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math., 38, Cambridge Univ. Press, Cambridge, 1994, xiv+450 pp.
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