Abstract
In this paper, we show that if an algebra KQ/I with an ideal I of KQ contained in \({R^{m}_{Q}}\) for an integer m ≥ 2 has an m-truncated cycle, then this algebra has infinitely many nonzero Hochschild homology groups, where R Q denotes the arrow ideal. Consequently, such an algebra of finite global dimension has no m-truncated cycles and satisfies an m-truncated cycles version of the “no loops conjecture".
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References
G. Ames, L. Cagliero, and P. Tirao, Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras, J. Algebra 322 (2009), 1466–1497.
Bardzell M. J.: The alternating syzygy behavior of monomial algebras. J. Algebra 188, 69–89 (1997)
R.-O. Buchweitz et al., Finite Hochschild cohomology without finite global dimension, Math. Res. Lett. 12 (2005), 805–816.
P.A. Bergh, Y. Han, and D. Madsen, Hochschild homology and truncated cycles, Proc. Am. Math. Soc. 140 (2012), 1133–1139.
Cibils C.: Cohomology of incidence algebras and simplicial complexes.. J. Pure Appl. Algebra 56, 221–232 (1989)
Cibils C.: Cyclic and Hochschild homology of 2-nilpotent algebras. K-theory. 4, 131–141 (1990)
Y. Han, Hochschild (co)homology dimension, J. Lond. Math. Soc. 73 (2006), 657–668.
D. Happel, Hochschild cohomology of finite-dimensional algebras, in Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39éme Année (Paris, 1987/1988), Lecture Notes in Mathematics 1404, Springer, Berlin, 1989, 108–126.
Igusa K.: Notes on the no loops conjecture. J. Pure Appl. Algebra 69, 161–176 (1990)
Igusa K., Liu S., Paquette C.: A proof of the strong no loop conjecture. Adv. Math 228, 2731–2742 (2011)
Itagaki T., Sanada K.: The dimension formula of the cyclic homology of truncated quiver algebras over a field of positive characteristic. J. Algebra 404, 200–221 (2014)
Lenzing H.: Nilpotence Elemente in Ringen von endlicher globaler Dimension. Math. Z. 108, 313–324 (1969)
J-L. Loday, Cyclic Homology, Springer-Verlag, Berlin (1992).
E. Sköldberg, The Hochschild homology of truncated and quadratic monomial algebras, J. Lond. Math. Soc. (2) 59 (1999), 76–86.
E. Sköldberg Cyclic homology of quadratic monomial algebras, J. Pure Appl. Algebra 156 (2001), 345–356.
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Itagaki, T., Sanada, K. Notes on the Hochschild homology dimension and truncated cycles. Arch. Math. 103, 219–228 (2014). https://doi.org/10.1007/s00013-014-0683-8
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DOI: https://doi.org/10.1007/s00013-014-0683-8