Homological Dimensions of Extensions of Artinian Rings by Right Projective Bimodules

ABSTRACT

Let Λ be an Artinian ring and let B be a right projective Λ-Λ-bimodule. Recall that a ring Γ is an extension of Λ by B if we have a short exact sequence of Abelian groups , where f is a homomorphism of rings. Recall that a Λ-module M is called liftable to Γ if there exists a Γ-module X such that X/BX≅ M and Tor ^Γ _n (Λ,X) = 0 for all n > 0. The finitistic dimension of Γ is the supremum of the projective dimensions of finitely generated Γ-modules of finite projective dimension. We study Γ-modules of finite projective dimension and we give upper bounds on the finitistic dimension of Γ in terms of the finitistic dimension of liftable Λ-modules.

ACKNOWLEDGMENT

The results of this article were obtained as a part of my Ph.D. thesis. I would like to thank my supervisor Prof. Sverre O. Smal{øt} for his guidance. I would also like to thank the referee for several helpful remarks.