capitalised upon. In this paper, we aim to show that the language of orders gives a quite effective tool to study weighted projective lines and their generalizations.

Recently in [@HIMO], *Geigle-Lenzing* (*GL*) *projective spaces* were introduced as a higher dimensional generalization of Geigle-Lenzing weighted projective lines, and their representation theory was studied. In this paper we will introduce a certain class of orders on ${\mathbb{P}}^d$ which we call *Geigle-Lenzing* (*GL*) *orders* on ${\mathbb{P}}^d$ and prove that they actually give the category of coherent sheaves on GL projective spaces:

Let ${\mathbb{X}}$ be a GL projective space and $\Lambda$ be a GL order of the same type. There exists an equivalence $$\operatorname{\mathsf{coh}}{\mathbb{X}}\simeq {\operatorname{\mathsf{mod}}}\Lambda.$$

After Beilinson’s work [@Be], various projective varieties are known to be derived equivalent to non-commutative algebras: for example, Hirzebruch surfaces [@Ki], rational surfaces [@HP], homogeneous spaces [@Kap; @Kan; @BLV] and so on. The notion of tilting bundles is crucial to construct derived equivalences. In representation theory, tilting bundles on Geigle-Lenzing weighted projective lines [@GL] play an important role since they give Ringel’s canonical algebras [@R] as their endomorphism algebras. One of the basic results in [@HIMO] (see also [@Ba; @IU]) is the existence of tilting bundles on GL projective 