  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.

Commutative Algebra