Notes on the Hochschild homology dimension and truncated cycles

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".