Higher colimits, derived functors and homology

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Higher colimits, derived functors and homology. / Ivanov, S. O.; Mikhailov, R. V.; Sosnilo, V. A.

In: Sbornik Mathematics, Vol. 210, No. 9, 09.2019, p. 1222-1258.

Research output: Contribution to journal › Article › peer-review

Author

Ivanov, S. O. ; Mikhailov, R. V. ; Sosnilo, V. A. / Higher colimits, derived functors and homology. In: Sbornik Mathematics. 2019 ; Vol. 210, No. 9. pp. 1222-1258.

BibTeX

@article{4584a3474eef4037b199287fa3e4817c,

title = "Higher colimits, derived functors and homology",

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{\textquoteright} 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.",

keywords = "Cyclic homology, Derived functors, Higher colimits, K-theory",

author = "Ivanov, {S. O.} and Mikhailov, {R. V.} and Sosnilo, {V. A.}",

year = "2019",

month = sep,

doi = "10.1070/SM9152",

language = "English",

volume = "210",

pages = "1222--1258",

journal = "Sbornik Mathematics",

issn = "1064-5616",

publisher = "Turpion Ltd.",

number = "9",

}

RIS

TY - JOUR

T1 - Higher colimits, derived functors and homology

AU - Ivanov, S. O.

AU - Mikhailov, R. V.

AU - Sosnilo, V. A.

PY - 2019/9

Y1 - 2019/9

N2 - 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.

AB - 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.

KW - Cyclic homology

KW - Derived functors

KW - Higher colimits

KW - K-theory

UR - http://www.scopus.com/inward/record.url?scp=85087453356&partnerID=8YFLogxK

U2 - 10.1070/SM9152

DO - 10.1070/SM9152

M3 - Article

AN - SCOPUS:85087453356

VL - 210

SP - 1222

EP - 1258

JO - Sbornik Mathematics

JF - Sbornik Mathematics

SN - 1064-5616

IS - 9

ER -

ID: 62108185