Let $A$ be a finite dimensional algebra with a module $M$ who has minimal injective coresolution $(I_i)$. Define the complexity $cx(M)$ as $cx(M):= \inf \{ t \geq 0 | \dim(I_i) \leq a i^{t-1}$ for all $i \geq n_0$ for some $n_0 \in \mathbb{N}$ and some $a \in \mathbb{R} \}$.
Question: Is there an algebra with indecomposable modules $M_i$ having arbitrary large but finite complexity?
I have no experience with complexity so I am not sure whether this questions belongs to MO.
