Projectively Wakamatsu Tilting Modules over One-Point Extensions

Let $\mathrm{\Gamma }=\mathrm{\Lambda }[M]$ be the one-point extension of an algebra $\mathrm{\Lambda }$ by a $\mathrm{\Lambda }$-module $M$. We establish a method to lift projectively Wakamatsu tilting (PWT) modules from $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Lambda }$ to $\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\Gamma }$ by adding the new projective module, and prove that this lifting process perfectly preserves mutation relations under certain homological conditions. Furthermore, for source point extensions of representation-finite algebras, we obtain a complete classification of PWT $\mathrm{\Gamma }$-modules in terms of those over $\mathrm{\Lambda }$. In particular, we establish a bijection $$\mathrm{P}\mathrm{W}\mathrm{T}(\mathrm{\Gamma })⟷\mathrm{P}\mathrm{W}\mathrm{T}(\mathrm{\Lambda })\coprod \mathrm{R}\mathrm{P}\mathrm{W}\mathrm{T}(\mathrm{\Lambda },S_i)\mathrm{.}$$ which yields the counting formula about $\mathrm{\mid }\mathrm{P}\mathrm{W}\mathrm{T}(\mathrm{\Gamma })\mathrm{\mid }$.