Arithmetical rank and cohomological dimension of generalized binomial edge ideals

TL;DR

This paper investigates algebraic properties of generalized binomial edge ideals associated with graphs, providing bounds on their cohomological dimension, conditions for cohomological complete intersections, and cases where arithmetical rank equals projective dimension.

Contribution

It introduces new bounds and conditions for generalized binomial edge ideals, linking their algebraic invariants to graph properties and extending previous results.

Findings

Lower bound for cohomological dimension of $J_m(G)$

Conditions for $J_m(G)$ to be cohomologically complete intersection

Equality of arithmetical rank and projective dimension for $J_2(G)$ in certain cases

Abstract

Let $G$ be a connected and simple graph on the vertex set $[n]$. To the graph $G$ one can associate the generalized binomial edge ideal $J_{m}(G)$ in the polynomial ring $R=K[x_{ij}: i \in [m], j \in [n]]$. We provide a lower bound for the cohomological dimension of $J_{m}(G)$. We also study when $J_{m}(G)$ is a cohomologically complete intersection. Finally, we show that the arithmetical rank of $J_{2}(G)$ equals the projective dimension of $R/J_{2}(G)$ in several cases.