Local Artinian rings and the Fröberg relation (Q1360202)

Let \(R\) be a commutative Artinian local ring, with maximal ideal \({\mathfrak m}\), and residue field \(k\). Define the Poincaré series \(P_R(t)= \sum_{i\geq 0}\dim(\text{Tor}_i^R(k,k))t^i\) and the Hilbert series \(H_R(t)= \sum_{i\geq 0}\dim ({\mathfrak m}^i/{\mathfrak m}^{i+1})t^i\). \(R\) is called a Fröberg ring if \(P_R(t)= H_R(-t)^{-1}\). The main result is that if \({\mathfrak m}^3=0\) and \({\mathfrak m}\cdot\text{ann }x={\mathfrak m}^2\) for all \(x\in{\mathfrak m}/{\mathfrak m}^2\), then \(R\) is a Fröberg ring. Some partial converses are given. This is motivated by a preliminary result showing that certain Witt rings are Fröberg rings.