Hochschild homology, global dimension, and truncated oriented cycles

TL;DR

This paper proves that bounded quiver and monomial algebras with truncated oriented cycles have infinite Hochschild homology and global dimensions, extending known results and confirming that finite global dimension algebras lack such cycles.

Contribution

It generalizes the relationship between truncated oriented cycles and infinite dimensions to broader classes of algebras, beyond local cases.

Findings

Bounded quiver algebras with 2-truncated oriented cycles have infinite Hochschild homology and global dimensions.

Monomial algebras with truncated oriented cycles also have infinite Hochschild homology and global dimensions.

Finite global dimension algebras cannot contain 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.