Homological dimensions of the extension algebras of monomial algebras

The main objective of this paper is to study the dimension trees and further the homological dimensions of the extension algebras - dual and trivially twisted extensions - with a unified combinatorial approach using the two combinatorial algorithms - Topdown and Bottomup. We first present a more complete and clearer picture of a dimension tree, with which we are then able, on the one hand, to sharpen some results obtained before and furthermore reveal a few more hidden subtle homological phenomenons of or connections between the involved algebras; on the other hand, to provide two more efficient combinatorial algorithms for computing dimension trees, and consequently the homological dimensions as an application. We believe that the more refined complete structural information on dimension trees will be useful to study other homological properties of this class of extension algebras.