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

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 less than or equal to j less than or equal to n be a family of ideals in A with I1I2 (...) I-n = 0, such that proj.dim(I-A(j)) < infinity and proj.dim(I-j)(A) = 0 for all j greater than or equal to 3. If A/I-1 and A/I-2 are representation-finite and if A/I-j has finite finitistic dimension for j greater than or equal to 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 subset of or equal to B subset of or equal to 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. (C) 2004 Elsevier B.V. All rights reserved.