A finite dimensional algebra with a phantom (a corollary of an example by J. Krah)

Abstract

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.