Some Homological Conjectures for Quasi-stratified Algebras

In this paper, we are mainly concerned with the Cartan determinant conjecture and the no loop conjecture. If A is an artin algebra of finite global dimension, the first conjecture claims that the Cartan determinant of A is equal to 1, while the second one states that every simple A-module admits only the trivial self-extension. Among numerous partial solutions to these conjectures such as those in [6, 14, 15, 22, 24], we observe particularly that both of them have been established for standardly stratified algebras; see [4, 21]. This class of algebras serves as a generalization of quasi-hereditary algebras introduced by Cline, Parshall and Scott; see, for example, [8]. The key idea for studying standardly stratified algebras is to relate the homological properties of an algebra A to those of A/I with I an idempotent projective ideal. We shall pursue further in this line by relaxing the condition that I be idempotent. This enables us to generalize many results found in [4, 11, 21, 24]. More importantly, it leads us to the introduction of two new classes of algebras, called quasi-stratified and ultimate-hereditary algebras, which include standardly stratified and quasihereditary algebras, respectively. We shall show that the finiteness of the global dimension of a quasi-stratified algebra is equivalent to the Cartan determinant equal to one, as well as to the algebra being ultimate-hereditary. Moreover, in this case, we prove that every simple module admits only the trivial selfextension.