Generic local rings on a spectrum between Golod and Gorenstein

Abstract

Artinian quotients R of the local ring Q=k[[x,y,z]] are classified by multiplicative structures on A=Tor⁎Q(R,k); in particular, R is Gorenstein if and only if A is a Poincaré duality algebra while R is Golod if and only if all products in A⩾1 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 Poincaré duality algebra P and a graded vector space V, on which P⩾1 acts trivially. That is, A is a trivial extension, A=P⋉V, and the extremes A=(k⊕Σk)⋉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.