We define a projective link between maximal ideals, with respect to which an idealizer preserves being of finite global dimension. Let
D
D
be a local Dedekind domain with the quotient ring
K
K
. We show that for
2
≤
n
≤
5
2 \leq n \leq 5
, every tiled
D
D
-order of finite global dimension in
(
K
)
n
{(K)_n}
is obtained by iterating idealizers w.r.t. projective links from a hereditary order. For
n
≥
6
n \geq 6
, we give a tiled
D
D
-order in
(
K
)
n
{(K)_n}
without this property, which is also a counterexample to Tarsy’s conjecture, saying that the maximum finite global dimension of such an order is
n
−
1
n - 1
.