We observe that there exists an associative finite dimensional $\mathbb{ C}$-algebra $A$ of finite global dimension, such that the bounded derived category $D^b(A)$ of finite dimensional $A$-modules admits an admissible subcategory $\mathcal{ P}$ with vanishing Grothendieck group $K_0(\mathcal{ P})$. In other words, $\mathcal{ P} \subseteq D^b(A)$ is a phantom. Using tilting theory, this follows directly from a very recent example of a phantom for a smooth rational surface due to Krah. By work of Aihara & Iyama, this also leads to new examples of presilting objects that cannot be completed to silting objects.
Categories: math.RT math.AG