Golodness and residue-field Betti numbers of Γ_m(2,n)

Determine, for every 4≤n<m and S=Γ_m(2,n) with R=K[S], whether R is Golod; if it is not Golod, determine the smallest index i for which the Golod defect number 𝒟_i(R) is positive, determine whether the Poincaré series 𝒫_K^R(z) is rational, and determine the Betti numbers of the residue field K over R.

Background

The paper completely analyzes Golodness and the residue-field Poincaré series for the one-gap family Γ_m(n), proving that K[Γ_m(n)] is Golod exactly when n=2 and determining the first Golod defect for n≥3. It also computes the defining-ideal Betti numbers for the two-gap family Γ_m(2,n).

The corresponding homological behavior of K[Γ_m(2,n)] is not resolved in the paper. The open question asks for three related invariants: Golodness and the first failure of Serre’s upper bound, rationality of the residue-field Poincaré series, and the complete sequence of residue-field Betti numbers.

References

Although we compute the Betti numbers of the defining ideals of $K[\Gamma_m(2,n)]$, their Golodness remains to be analysed.

\begin{question} Let $4\leq n<m$, $S=\Gamma_m(2,n)$, and $R=K[S]$. \begin{enumerate}[leftmargin=*, label=(\alph*)] \item When is $R$ Golod? If not, what is the smallest $i$ such that $\mathcal{D}_i(R)\>0$? \item Is the Poincar e series $\mathcal{P}_KR(z)$ a rational function? \item What are the Betti numbers of $K$ over $R$? \end{enumerate} \end{question}

— The Betti numbers and Golodness of numerical semigroup rings of Sally type  (2609.26752 - Batavia et al., 22 Sep 2026) in Section 6, “Further Questions and Conjectures”