h\in H\}=\operatorname{\mathsf{proj}}^GA$.

By a standard argument in Morita theory, one can show that the functor $$F:=\bigoplus_{h\in H}{\operatorname{Hom}\nolimits}_A^G(P,-(h)):\operatorname{\mathsf{Mod}}^GA\to\operatorname{\mathsf{Mod}}^HA^{[H]}$$ is an equivalence.

Let $A$ be a ${\mathbb{Z}}$-graded ring. If we choose $\left\{ 0,1 \right\}$ as the representatives of the two cosets of ${\mathbb{Z}}/2{\mathbb{Z}}$ then $$A^{[2{\mathbb{Z}}]}=\bigoplus_{i\in 2{\mathbb{Z}}}(A^{[2{\mathbb{Z}}]})_i\ \ \ \mbox{where}\ \ \ (A^{[2{\mathbb{Z}}]})_{i}=
\begin{bmatrix}A_{i}&{A_{i-1}}\\A_{i+1}&A_{i}\end{bmatrix}.$$ A ${\mathbb{Z}}$-graded $A$-module $M=\bigoplus_{i\in{\mathbb{Z}}}M_i$ corresponds to the $2{\mathbb{Z}}$-graded $A^{[2{\mathbb{Z}}]}$-module $
     \bigoplus_{i\in 2{\mathbb{Z}}}\begin{bmatrix}
          M_{i}\\M_{i+1}
     \end{bmatrix}
     $.

Geigle-Lenzing projective spaces.
---------------------------------

We now introduce Geigle-Lenzing (GL) projective spaces, or more precisely, the category of coherent sheaves on them. The technique of studying a category resembling a category of sheaves without ever explicitly mentioning a topological space is especially prominent in noncommutative algebraic geometry. We use the same notation as in [@HIMO] where much more information can be found. Our goal is to show that the category of coherent sheaves on GL projective spaces is equivalent to the module category of a GL order on ${\mathbb{P}}^d$.

To define GL weighted ${\mathbb{P}}^d$, as before we choose $n$ hyperplanes ${\bf L}=(L_1,\dots,
L_n)$ in ${\mathbb{P}}^d_{T_0:\dots : T_d}$ satisfying Assumption \[general position\]. We may assume the hyperplanes are given as zeros of the linear polynomials $$\ell_i({\bf T})=\sum_{j=0}^{d}\lambda_{i,j}T_j.$$ Also fix an $n$-tuple of positive integers (the *weights*) 