Truncated path algebras are homologically transparent

Abstract

It is shown that path algebras modulo relations of the form $\hat{I}›=KQ/I$, where $Q$ is a quiver, $K$ a coefficient field, and $I\subseteq KQ$ the ideal generated by all paths of a given length, can be readily analyzed homologically, while displaying a wealth of phenomena. In particular, the syzygies of their modules, and hence their finitistic dimensions, allow for smooth descriptions in terms of $Q$ and the Loewy length of $\hat{I}›$. The same is true for the distributions of projective dimensions attained on the irreducible components of the standard parametrizing varieties for the modules of fixed $K$- dimension.