Hochschild homology, global dimension, and truncated oriented cycles
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 Vigu\'{e}-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.
Keywords
Cite
@article{arxiv.1004.0748,
title = {Hochschild homology, global dimension, and truncated oriented cycles},
author = {Yang Han},
journal= {arXiv preprint arXiv:1004.0748},
year = {2010}
}