Repetitive resolutions over classical orders and finite dimensional algebras

Abstract

Repetitiveness in projective and injective resolutions and its influence on homological dimensions are studied. Some variations on the theme of repetitiveness are introduced, and it is shown that the corresponding invariants lead to very good -- and quite accessible -- upper bounds on various finitistic dimensions in terms of individual modules. These invariants are the `repetition index' and the `syzygy type' of a module $M$ over an artinian ring $\Lambda$. The repetition index measures the degree of repetitiveness among non-projective direct summands of the syzygies of $M$, while the syzygy type of $M$ measures the number of indecomposable modules among direct summands of the syzygies of $M$. It is proved that if $T$ is a right $\Lambda$-module which contains an isomorphic copy of $\Lambda/J(\Lambda)$, then the left big finitistic dimension of $\Lambda$ is bounded above by the repetition index of $T$, which in turn is bounded above by the syzygy type of $T$. The finite dimensional $K$-algebras $\Lambda = {\cal O}/\pi{\cal O}$, where $\cal O$ is a classical order over a discrete valuation ring $D$ with uniformizing parameter $\pi$ and residue class field $K$, are investigated. It is proved that, if $\text{gl.dim.}\, {\cal O} =d<\infty$, then the global repetition index of $\Lambda$ is $d-1$ and all finitely generated $\Lambda$-modules have finite syzygy type. Various examples illustrating the results are presented.