TITLE: heller functor of finite order
QUESTION [5 upvotes]: Hi,
let A be a finite dimensional selfinjective algebra.
Assume mod A is periodic.That is for every finite dimensional module with no projective summand we have: $ \Omega^{n} (M) =M $ for some n ,where $ \Omega $ is the Hellerfunctor(giving the kernel of a projective cover of M ).
When is it true that $ \Omega $ is of finite order,that is $ \Omega^{n} $ is isomorphic as a functor to the identity functor of the stable category of modules ?(Assume there is a common biggest period for all module if necassary)
Thanks for help

REPLY [5 votes]: Another example: If $A$ is a bialgebra over a field $k$ then $\Omega^n(k) = k$ implies $\Omega^n = id$. 
The proof is rather simple: Choose a projective resolution $P \to k$ such that $\Omega^i(k)$ is the kernel of $P_{i-1} \to P_{i-2}$.  Since $M$ is $k$-projective, 
$$P \otimes_k M\to k \otimes_k M \cong M$$ is a projective resolution of $M$ where $A$ acts diagonally on the tensor product (that's the place were the bialgebra structure is needed). If $f: M \to N$ is $A$-linear, then we obtain a commutative diagramm with exact rows 
$$\begin{array}{cccccccc}
0 & \to & M & \cong & k \otimes M & \to & P_{n-1} \otimes M & \to P_{n-2} \otimes M & \to \cdots  \newline
  &     & \scriptstyle f\; \displaystyle \downarrow &  & \scriptstyle id \otimes f\;\displaystyle\downarrow & & \scriptstyle id \otimes f\; \displaystyle\downarrow   & \scriptstyle id \otimes f\; \displaystyle \downarrow\newline  
0 & \to & N & \cong & k \otimes N & \to & P_{n-1} \otimes N & \to P_{n-2} \otimes N & \to \cdots  \newline
\end{array}$$
Since up to a projective summand $\Omega^n(M)$ is just the kernel in a projective resolution, we have $$M = \Omega^n(M) \oplus(\text{projective}) \text{ and }f = \Omega^n(f) \oplus p,$$
i.e. $\Omega^n(M) = M$ and $\Omega^n(f) = f$ in the stable category. Thus $\Omega^n = id$. 
Note: If $k$ is a commutative ring and $A$ projective as $k$-module then the same proof shows $\Omega^n = id$ on the stable category of finitely generated, $k$-projective $A$-modules.