A Study of Good and Bad Artinian Gorenstein local Rings

We say that a local ring $R$ is good, in the sense of Roos, if all finitely generated $R$-modules have rational Poincar\'e series that share a common denominator; otherwise, $R$ is said to be bad. An important class of good rings is the class of generalized Golod rings. In this paper, we show that connected sums of Artinian Gorenstein generalized Golod rings are good. We provide a criterion for decomposing Artinian Gorenstein local rings as connected sums. As a key application, we prove that a Gorenstein local ring $R$ with maximal ideal $\mathfrak{m}$ is good under either of the following conditions: (1) the multiplicity of $R$ is at most $12$ and its $h$-vector is different from $(1, 5, 5, 1)$, (2) $\mathfrak{m}^4$ = 0 and $\mathfrak{m}^2$ is generated by at most four elements. The above result records partial progress towards resolving a question posed by L.~Avramov. We also present examples of bad Artinian Gorenstein local rings of any multiplicity greater than or equal to $18$. In all these cases, the results establishing that the rings are good are obtained by showing that the rings are generalized Golod rings.