Duality and Tameness

An ideal $I$ in a Noetherian ring is called \textit{normal} if $I^n$ is integrally closed for all $n \geq 1$. Zariski proved that in two-dimensional regular local rings, every integrally closed ideal is normal. However, in dimension three and higher, this is no longer true in general, including monomial ideals in polynomial rings. In this paper, we study the normality of integrally closed monomial ideals in the polynomial ring $k[x,y,z]$ over a field $k$. We prove that every such ideal with at most seven minimal monomial generators is normal, thereby giving a sharp bound for normality in this setting. The proof is based on a detailed case-by-case analysis, combined with valuation-theoretic and combinatorial methods via Newton polyhedra.

math.AC↗

arXiv2026

On the Hilbert Series and Multiplicity of Generalized Binomial Edge Ideals

In this article, we compute the Hilbert series of the generalized binomial edge ideal associated with the join of two graphs. Consequently, we compute the Hilbert series of complete $r$-partite graphs. Further, we show that if a vertex satisfies a certain degree condition, then some Hilbert coefficients remain unchanged upon its removal. This provides a reduction technique for computing them. As an application, we compute the dimension and multiplicity of several classes of graphs.

math.AC↗

arXiv2026

On some problems concerning the growth of Betti numbers

We investigate several problems concerning the growth of Betti numbers over commutative noetherian rings. First, we establish uniform positive lower and finite upper bounds for the ratios of consecutive Betti numbers of finitely generated modules of infinite projective dimension over an arbitrary local ring, giving an affirmative answer to a question of Avramov. We also prove the upper semicontinuity of the curvature of residue fields of localizations over excellent rings. Next, using zeros of Hilbert series, we obtain criteria for exponential growth of Betti and Bass numbers of graded modules. As applications, we establish exponential growth of the Betti numbers of canonical modules for non-Gorenstein positively graded Cohen-Macaulay domains. Furthermore, we show that the first Betti number of the canonical module of a non-Gorenstein generically Gorenstein Cohen-Macaulay local ring is strictly larger than the zeroth. In particular, both questions of Jorgensen and Leuschke have affirmative answers for Cohen-Macaulay quotients of Gorenstein local rings by monomial ideals in regular sequences.

math.AC↗