FINITISTIC DIMENSIONS AND PIECEWISE HEREDITARY PROPERTY OF SKEW GROUP ALGEBRAS

Let Λ be a finite-dimensional algebra and G be a finite group whose elements act on Λ as algebra automorphisms. Assume that Λ has a complete set E of primitive orthogonal idempotents, closed under the action of a Sylow p-subgroup S ≤ G. If the action of S on E is free, we show that the skew group algebra Λ G and Λ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra ΛS is a direct summand of the ΛS-bimodule Λ. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce a criterion for Λ G to be piecewise hereditary.

FINITISTIC DIMENSIONS OF TRIVIALLY TWISTED EXTENSIONS OF MONOMIAL ALGEBRAS

We use the algorithms Bottomup and Topdown to study the finitistic dimensions of a class of extension algebras: the trivially twisted extensions of monomial algebras. A hybrid algorithm for determination of the finitistic dimension of the aforementioned extension is presented, and in particular, conditions for construction of such extensions that have relatively larger finitistic dimension are investigated.

A Graph Approach to a Splitting Theorem

By associating a graph to the ring under discussion, we propose a graph approach to extend a well-known ring splitting theorem due to Zaks, and we then describe the global and finitistic dimensions of the ring in the language of the graph.