Maximal self-orthogonal modules and a new generalization of tilting modules

We introduce a generalization of tilting modules of finite projective dimension, projectively Wakamatsu tilting modules, which are self-orthogonal and Ext-progenerators in their Ext-perpendicular categories. Under a certain finiteness condition, we prove that the following modules coincide: projectively Wakamatsu tilting, Wakamatsu tilting, maximal self-orthogonal, and self-orthogonal modules with the same rank as the algebra. This provides another proof of the weak Gorensteinness of representation-finite algebras. To prove this, we introduce Bongartz completion of self-orthogonal modules and characterize its existence. Moreover, we study a binary relation on Wakamatsu tilting modules which extends the poset of tilting modules, and use it to prove that every self-orthogonal module over a representation-finite Iwanaga-Gorenstein algebra has finite projective dimension. Finally, we discuss several conjectures related to self-orthogonal modules and their connections to famous homological conjectures.