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For a finite acyclic quiver $Q$ and the corresponding preprojective algebra$\Pi$, we study the factor algebra $\Pi_w$ associated with a element $w$ in theCoxeter group introduced by Buan-Iyama-Reiten-Scott. The algebra $\Pi_w$ has anatural $\mathbb{Z}$-grading, and we prove that$\underline{\mathsf{Sub}}^{\mathbb{Z}}\Pi_w$ has a tilting object $M$.Moreover, we show that the endomorphism algebra of $M$ is isomorphic to thestable Auslander algebra of a certain torsion free class of $\mathsf{mod}\,kQ$.

Tilting theory of preprojective algebras and $c$-sortable elements