Complexity and periodicity
Let $M$ be a finitely generated module over an Artin algebra. By considering the lengths of the modules in the minimal projective resolution of $M$, we obtain the Betti sequence of $M$. This sequence must be bounded if $M$ is eventually periodic, but the converse fails to hold in general. We give conditions under which it holds, using techniques from Hochschild cohomology. We also provide a result which under certain conditions guarantees the existence of periodic modules. Finally, we study the case when an element in the Hochschild cohomology ring “generates” the periodicity of a module.