Some homological conjectures for quasi-stratified algebras. (English) Zbl 1118.16016
Let \(A\) be a finite dimensional algebra and \(I\) an idempotent ideal in \(A\), with \(I\) being projective as a left \(A\)-module. Then the cohomological properties of \(A\) are closely related to those of the quotient algebra \(A/I\). This fact has been used extensively in studying classes of algebras such as quasi-hereditary or standardly stratified algebras.
In the article under review, larger classes of algebras are defined by relaxing the conditions on \(I\), not requiring idempotent any more, but still keeping left projectivity. The largest idempotent ideal \(J\) contained in \(I\) plays a major role, too. Using inductive arguments, the authors then prove the Cartan determinant conjecture, the no loop conjecture and the strong no loop conjecture for the classes of algebras they have defined.
In the article under review, larger classes of algebras are defined by relaxing the conditions on \(I\), not requiring idempotent any more, but still keeping left projectivity. The largest idempotent ideal \(J\) contained in \(I\) plays a major role, too. Using inductive arguments, the authors then prove the Cartan determinant conjecture, the no loop conjecture and the strong no loop conjecture for the classes of algebras they have defined.
Reviewer: Steffen König (Köln)
MSC:
| 16G10 | Representations of associative Artinian rings |
| 16E10 | Homological dimension in associative algebras |
| 18G20 | Homological dimension (category-theoretic aspects) |
Keywords:
idempotent ideals; quasi-stratified algebras; Cartan determinant conjecture; no loop conjecture; quasi-hereditary algebras; standardly stratified algebrasReferences:
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