Generic local rings on a spectrum between Golod and Gorenstein

TL;DR

This paper investigates the structure of generic local rings between Golod and Gorenstein types, revealing a spectrum characterized by algebraic decompositions and proving this behavior for socle rank 2 quotients.

Contribution

It establishes that generic graded quotients with socle rank 2 exhibit a predictable algebraic structure, extending empirical observations to a rigorous proof.

Findings

Algebra A decomposes as a trivial extension of a Poincare duality algebra and a graded vector space.

The behavior is proven to be generic for socle rank 2 quotients.

The rank of P is linked to the socle polynomial's order and degree.

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 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 \oplus \Sigma k) \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.