Questions:

1. Is there a module of complexity one that is not periodic over a selfinjective algebra over a finite field?

2. Is there a module of complexity one that is not periodic over a selfinjective algebra over a finite field?

One may replace finite field by any field that consists only of roots of unity.

Here complexity one means that the terms $P_i$ of a minimal projective resolution of the module have bounded dimensions. Periodic means that $\Omega^i(M) \cong M$ for some $i \geq 1$.