Periodic objects in Frobenius categories Let $A$ be a finite dimensional Gorenstein algebra and $C$ the stable module category of $A$.

Question: Does there always exist an indecomposable periodic object $X$ in C, that is an object with $\Omega^n(X) \cong X$ for some $n$?

edit: Simplified the conditions a bit from Frobenius categories to Gorenstein algebras.
 A: Assuming that the question is about finitely generated modules, I think that the following gives a finite dimensional Frobenius algebra $A$ that is a counterexample. In fact, for any non-projective finitely generated $A$-module $X$, the $\Omega$-orbit of $X$ contains modules of unbounded dimension.
Let $A$ be a $5$-dimensional local Frobenius algebra of radical length $3$. For example, let $k$ be a field, and let $A=k[x,y,z]/(xy,xz,yz,x^2-y^2, x^2-z^2)$, so that $A$ has a basis $\{1,x,y,z,x^2=y^2=z^2\}$.
Since $A$ is Frobenius, the syzygy of a non-projective indecomposable module is indecomposable.
Since $A$ also has radical length $3$, every non-projective indecomposable module has radical length at most $2$.
Let $X$ be such a module, and define its type to be $\pmatrix{a\\b}$, where $b=\dim\left(\text{soc}(X)\right)$, and $a+b=\dim(X)$. Unless $X$ is the simple module, $\text{soc}(X)=\text{rad}(X)$.
Unless $X$ is simple, its projective cover is free of rank $a$, and so $\Omega X$ has type $\pmatrix{3a-b\\a}$, which is $Mv$, where $M=\pmatrix{3&-1\\1&0}$ and $v$ is the type of $X$.
So unless some syzygy of $X$ is simple, the type of $\Omega^nX$ is $M^nv$. But $M$ has the two distinct eigenvalues $\frac{3\pm\sqrt{5}}{2}$, so $v=M^nv$ has no nonzero solution.
It remains to check that the simple module $S$ is not periodic. The type of $\Omega^nS$ is $M^n\pmatrix{1\\0}$ for $n>0$ (unless some earlier syzygy is simple) and it is not hard to show that this is not equal to $\pmatrix{0\\1}$ for any $n$, since the entries of $M$ grow too fast. Probably there is a slicker argument, but one can compute
$$M^n=\pmatrix{\frac{a^{n+1}-b^{n+1}}{a-b}&\frac{b^n-a^n}{a-b}\\
\frac{a^n-b^n}{a-b}&\frac{b^{n-1}-a^{n-1}}{a-b}},$$
where $a=\frac{3+\sqrt{5}}{2}$ and $b=\frac{3-\sqrt{5}}{2}$.
