Ring homomorphisms and local rings with quasi-decomposable maximal ideal

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms.