Projectively Wakamatsu Tilting Modules over One-Point Extensions

Abstract

Let $\hat{I}“=\hat{I}›[M]$ be the one-point extension of an algebra $\hat{I}›$ by a $\hat{I}›$-module $M$. We establish a method to lift projectively Wakamatsu tilting (PWT) modules from $\mathrm{mod}\hat{I}›$ to $\mathrm{mod}\hat{I}“$ 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 $\hat{I}“$-modules in terms of those over $\hat{I}›$. In particular, we establish a bijection \[ \mathrm{PWT}(Γ) \longleftrightarrow \mathrm{PWT}(Λ) \coprod \mathrm{RPWT}(Λ, S_i). \] which yields the counting formula about $|\mathrm{PWT}(\hat{I}“)|$.

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