Abstract
It has been shown by Cuntz and Quillen that in characteristic 0 the kernel of the square of the “noncommutative Laplacian” on the Hochschild and cyclic complexes contains the relevant homology information. In this note, we show that the same property holds for the plain kernel of this Laplacian, as in differential geometry. Using the same ideas, we define a variant of Hochschild homology and cyclic homology and show that we recover the classical definitions in characteristic 0.
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References
A. Connes, “Noncommutative differential geometry,” Publ. Math. IHES, 62, 257–360 (1985).
J. Cuntz and D. Quillen, “Operators on noncommutative differential forms and cyclic homology,” in: Geometry, Topology and Physics, International Press, Cambridge (1995), pp. 77–111.
M. Karoubi, Homologie cyclique et K-théorie, Soc. Math. de France (1987), Astérisque, Vol. 149.
C. Kassel, “Cyclic homology, comodules and mixed complexes,” J. Algebra, 107, 195–216 (1987).
J.-L. Loday and D. Quillen, “Cyclic homology and the Lie algebra homology of matrices,” Comment. Math. Helv., 59, 565–591 (1984).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 6, pp. 165–170, 2016.
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Karoubi, M. The Harmonic Decomposition in Cyclic Homology. J Math Sci 248, 776–779 (2020). https://doi.org/10.1007/s10958-020-04911-0
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DOI: https://doi.org/10.1007/s10958-020-04911-0