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A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer 
${{C}_{S}}(H)$ (and some of its prime factor algebras) of the Cartan element 
$H$ in the universal enveloping algebra 
$S$ of the Schrödinger (Lie) algebra. The simple ${{C}_{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $S$ (over the centre). It is proved that some (prime) factor algebras of $S$ and ${{C}_{S}}(H)$ are tensor homological
$/$Krull minimal.
A formula is provided to explicitly describe global dimensions of all kinds of tree type finite dimensional 
$k$-algebras for $k$ an algebraic closed field. In particular, it is pointed out that if the underlying tree type quiver has 
$n$ vertices, then the maximum global dimension is 
$n\,-\,1$.
