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arXiv · 1004.0748

Hochschild homology, global dimension, and truncated oriented cycles

Yang Han
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Abstract

It is shown that a bounded quiver algebra having a 2-truncated oriented cycle is of infinite Hochschild homology dimension and global dimension, which generalizes a result of Solotar and Vigué-Poirrier to nonlocal ungraded algebras having a 2-truncated oriented cycle of arbitrary length. Therefore, a bounded quiver algebra of finite global dimension has no 2-truncated oriented cycles. Note that the well-known "no loops conjecture", which has been proved to be true already, says that a bounded quiver algebra of finite global dimension has no loops, i.e., truncated oriented cycles of length 1. Moreover, it is shown that a monomial algebra having a truncated oriented cycle is of infinite Hochschild homology dimension and global dimension. Consequently, a monomial algebra of finite global dimension has no truncated oriented cycles.

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Yang Han. 2010-04-06. Hochschild homology, global dimension, and truncated oriented cycles. https://arxiv.org/abs/1004.0748

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