Arithmetical rank and cohomological dimension of generalized binomial edge ideals

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 $R\mathrm{/}{J}_2(G)$ in several cases.