${\bf p}=(p_1,\dots,p_n)$ and let $$k[{\bf T},{\bf X}]=k[T_0,\dots,T_d,X_1,\dots,X_n]$$ $$\label{relations}
h_i:=X_i^{p_i}-\ell_i({\bf T)}$$ Now consider the $k$-algebra $$R=R({\bf L},{\bf p}):=k[{\bf T},{\bf X}]/(h_i\;|\;1\leq i\leq n).$$ As in the previous section, let $${\mathbb{L}}={\mathbb{L}}({\bf p}) = {\langle {\vec x}_1,\cdots,{\vec x}_n,{\vec c}\rangle}/({p_i{\vec x}_i-{\vec c}}\;|\;
1\leq i\leq n)$$ We give $R$ an ${\mathbb{L}}$-grading by defining ${\rm deg}\;X_i={\vec x}_i$ and ${\rm deg}\;T_i={\vec c}$.

We will soon encounter several different graded rings so it useful to establish the following notation:

Let $G$ be an abelian group and $A=\bigoplus_{g\in G}A_g$ be a right noetherian $G$-graded ring which is finitely generated over $k$ and $\dim_kA_g<\infty$ for all $g\in G$. We denote by ${\operatorname{\mathsf{mod}}}^GA$ the category of finitely generated $G$-graded $A$-modules, and by ${\operatorname{\mathsf{mod}}}^G_0A$ the full subcategory of ${\operatorname{\mathsf{mod}}}^GA$ of finite dimensional modules. We let $$\operatorname{\mathsf{qgr}}A:={\operatorname{\mathsf{mod}}}^G A/{\operatorname{\mathsf{mod}}}^G_0A.$$

We apply this definition to the setting of GL projective spaces:

For the ${\mathbb{L}}$-graded $k$-algebra $R=R({\bf L}, {\bf p})$, we call $$\operatorname{\mathsf{coh}}{\mathbb{X}}=\operatorname{\mathsf{coh}}{\mathbb{X}}({\bf L}, {\bf p}) :=\operatorname{\mathsf{qgr}}R({\bf L}, {\bf p})$$ the category of *coherent sheaves on GL projective space ${{\mathbb{X}}}$ of type* $({\bf L},{\bf p})$.

In the rest of this section, we will prove the following connection between GL projective spaces and GL orders.

\[graded vs order\] Let ${\mathbb{X}}={\mathbb{X}}({\bf L}, {\bf p})$ be a GL projective space 