×

Some examples of lemmas of the first perturbation in cyclic homology. (Quelques exemples de lemmes de première perturbation en homologie cyclique.) (French) Zbl 0815.18008

We cite from the introduction: “To calculate the homology of a chain complex \((N,b)\), we usually have a smaller complex \((M,d)\) which is a deformation retract of the former one which allows us to limit ourself to the computation of \(H_ * (M,d)\). In such a situation, suppose \(t : N_ * \to N_{*-1}\) is a perturbation of \(b\) (i.e. \((b + t)^ 2 = 0)\). We can obtain (under suitable hypothesis) a perturbation \(\partial = \sum_{i \geq 1} \partial_ i\) of \(d\) on \(M\) such that \(H_ * (M,d + \partial) \simeq H_ * (N,b + t)\). Unfortunately, the complexity of \(\partial\) remains an insurmountable obstacle for the calculation of \(H_ * (M,d + \partial)\). We put forward examples for which the first perturbation \(\partial_ 1\) suffices to claim \(H_ * (M,d + \partial_ 1) \simeq H_ * (N,b + t)\).”
In the examples the cyclic and Hochschild homology of enveloping algebras and of truncated polynomial algebras are calculated as outlined above.

MSC:

18G60 Other (co)homology theories (MSC2010)
19D55 \(K\)-theory and homology; cyclic homology and cohomology
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
18G35 Chain complexes (category-theoretic aspects), dg categories
17B55 Homological methods in Lie (super)algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Borel A., Ann. Sci 7 pp 235– (1974)
[2] Brown R., Math. Reviews 93e : 55018 7 (1974)
[3] Brylinski J.L., J. Differential Geometry 28 pp 93– (1988)
[4] Cartan H., Homological algebra (1956)
[5] DOI: 10.1007/BF00538879 · Zbl 0743.13008 · doi:10.1007/BF00538879
[6] Connes A., Publ Math. Inst, Hautes Etudes Set 62 pp 257– (1986)
[7] Geller S., J. reine angew. Math 393 pp 39– (1989)
[8] DOI: 10.2307/1971283 · Zbl 0627.18004 · doi:10.2307/1971283
[9] Gugenheim V.K.A.M., Illinois J, Math 16 pp 398– (1972)
[10] Karoubi K., Asterisque 16 (1987)
[11] DOI: 10.1016/0021-8693(87)90086-X · Zbl 0617.16015 · doi:10.1016/0021-8693(87)90086-X
[12] DOI: 10.1007/BF01389366 · Zbl 0653.17007 · doi:10.1007/BF01389366
[13] DOI: 10.1515/crll.1990.408.159 · Zbl 0691.18002 · doi:10.1515/crll.1990.408.159
[14] DOI: 10.1007/BF00966115 · Zbl 0781.13008 · doi:10.1007/BF00966115
[15] Loday J.L., Springer Verlag 6 (1992)
[16] DOI: 10.1007/BF02566367 · Zbl 0565.17006 · doi:10.1007/BF02566367
[17] Serre J.R., Corps locaux (1968)
[18] DOI: 10.1090/S0002-9947-1961-0130900-1 · doi:10.1090/S0002-9947-1961-0130900-1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.