2017/07/13 by Anjan K. Gupta, Gupta, Anjan, Shekhar, Shrikant · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1707.04056
openalex publication_date 2017/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We say that a local ring R is good, in the sense of Roos, if all finitely generated R-modules have rational Poincaré 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 \mathfrakm 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) \mathfrakm4 = 0 and \mathfrakm2 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.