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 Ij,1less-than-or-equals, slantjless-than-or-equals, slantn be a family of ideals in A with I1I2cdots, three dots, centeredIn=0, such that proj.dim(AIj)<∞ and proj.dim(Ij)A=0 for all jgreater-or-equal, slanted3. If A/I1 and A/I2 are representation-finite and if A/Ij has finite finitistic dimension for jgreater-or-equal, slanted3, 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) Csubset of or equal toBsubset of or equal toA, 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.