2021/05/27 by Lars Winther Christensen, Christensen, Lars Winther, Oana Veliche +1 · 1 citation
Mathematics · #13D02 #13D07 #13E10 #13P20 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Primary 13C05. Secondary 13A02
paper · pdf · doi:10.48550/arxiv.2105.13167
openalex publication_date 2021/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Artinian quotients R of the local ring Q = k[[x,y,z]] are classified by multiplicative structures on A = TorQ^*(R,k); in particular, R is Gorenstein if and only if A is a Poincare duality algebra while R is Golod if and only if all products in A>0 are trivial. There is empirical evidence that generic quotient rings with small socle ranks fall on a spectrum between Golod and Gorenstein in a very precise sense: The algebra A breaks up as a direct sum of a Poincare duality algebra P and a graded vector space V, on which P>0 acts trivially. That is, A is a trivial extension, A = P \ltimes V, and the extremes A = (k ⊕ Σk) \ltimes V and A = P correspond to R being Golod and Gorenstein, respectively. We prove that this observed behavior is, indeed, the generic behavior for graded quotients R of socle rank 2, and we show that the rank of P is controlled by the difference between the order and the degree of the socle polynomial of R.