On the finitistic dimension conjecture I: related to representation-finite algebras

Abstract

We use the class of representation-finite algebras to investigate the finitistic dimension conjecture. In this way we obtain a large class of algebras for which the finitistic dimension conjecture holds. The main results in this paper are: (1) Let A be an artin algebra and let I j ,1⩽ j⩽ n be a family of ideals in A with I 1 I 2⋯ I n =0, such that proj.dim( A I j )<∞ and proj.dim( I j ) A =0 for all j⩾3. If A/ I 1 and A/ I 2 are representation-finite and if A/ I j has finite finitistic dimension for j⩾3, then the finitistic dimension of A is finite. In particular, the finitistic dimension conjecture is true for algebras obtained from representation-finite algebras by forming dual extensions, trivially twisted extensions, Hochschild extensions, matrix algebras and tensor products with algebras of radical-square-zero. (2) Let A, B and C be three artin algebras with the same identity such that (i) C⊆ B⊆ A, and (ii) the Jacobson radical of C is a left ideal of B and the Jacobson radical of B is a left ideal of A. If A is representation-finite, then C has finite finitistic dimension. We also provide a way to construct algebras satisfying all conditions in (2), and this leads to a new reformulation of the finitistic dimension conjecture.