Complexity one modules that are not periodic

Question

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 symmetric 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 $i \geq 1$.

Answer 1

Over a finite field, there are only finitely many modules with dimension less than $d$ for fixed $d$, so for some $n$ $M\cong\Omega^n$ for some $n$ if the algebra is self-injective with complexity one.

If every element of the field is a root of unity, then the field is a union of finite fields, and so the algebra and any finite dimensional module $M$ are defined over some finite field, and so the argument above shows that $M$ is periodic.