A set of facts that I fnd puzzling is the behaviour of Krull dimension (in the sense of Gabriel and Rentschler, that is, for non-necessarily commutative rings) of Weyl algebras.

One has $\mathcal K(A_n(k))=n$ when $k$ is a (commutative!) field of characteristic zero, and this is very sensible. If $k$ is instead of positive characteristic, we have $\mathcal K(A_n(k))=2n$, which is the other sensible value... Now, if $k$ is a field of any characteristic and $D_n=\operatorname{Frac}A_n(k)$ is the $n$th Weyl field, then $\mathcal K(A_n(D))=2n$; this is already strange. More generally, $\mathcal K(A_n(D_m))=\min\{2n,n+m\}$ over a field of characteristic zero.

There is a paper by Goodearl, Hodges and Lenagan which is filled with information about this (and parallel onformation about global dimensions).