Finitistic dimension conjecture and radical-power extensions

Abstract

The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field has finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this paper, we investigate those extensions of Artin algebras in which some radical-power of smaller algebras is a nonzero one-sided ideal of bigger algebras. Our result can be formulated for an arbitrary ideal as follows: Let B⊆A be an extension of Artin algebras and I an ideal of B such that the full subcategory of B/I-modules is B-syzygy-finite. (1) If the extension is right-bounded (for example, Gpd(AB)<∞), IArad(B)⊆B and findim(A)<∞, then findim(B)<∞. (2) If Irad(B) is a left ideal of A and A is torsionless-finite, then findim(B)<∞. Particularly, if I is specified to a power of the radical of B, then our result not only generalizes some of results in the literature (see Corollary 1.2), but also provides new ways to detect algebras of finite finitistic dimensions.