On Artin algebras arising from Morita contexts

Abstract

We study Morita rings $Λ_{(ϕ,ψ)}=\bigl({smallmatrix} A &_AN_B_BM_A & B {smallmatrix}\bigr)$ in the context of Artin algebras from various perspectives. First we study covariant finite, contravariant finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms $\varphi$ and $\psi$ are zero. Further we give bounds for the global dimension of a Morita ring ${\Lambda }_{(0,0)}$, regarded as an Artin algebra, in terms of the global dimensions of $A$ and $B$ in the case when both $\varphi$ and $\psi$ are zero. We illustrate our bounds with some examples. Finally we investigate when a Morita ring is a Gorenstein Artin algebra and then we determine all the Gorenstein-projective modules over the Morita ring with $A=N=M=B=\Lambda$, where $\Lambda$ is an Artin algebra.

29 pages, revised version