and $\Lambda=\Lambda({\bf L}, {\bf p})$ be a GL order of type $({\bf L}, {\bf p})$. Then we have an equivalence $$\operatorname{\mathsf{coh}}{\mathbb{X}}\simeq {\operatorname{\mathsf{mod}}}\Lambda.$$

For the subgroup ${\mathbb{Z}}{\vec c}$ of ${\mathbb{L}}$, we have $${\mathbb{Z}}{\vec c}\simeq{\mathbb{Z}}\ \mbox{ and }\ {\mathbb{L}}/{\mathbb{Z}}{\vec c}\simeq\prod_{i=1}^n{\mathbb{Z}}/ p_i {\mathbb{Z}}.$$ A key role in the proof is played by a ${\mathbb{Z}}$-graded subring $$S:=\bigoplus_{\ell\in{\mathbb{Z}}}R_{\ell{\vec c}}$$ which is isomorphic to the polynomial ring $k[{\bf T}]$. By (\[relations\]), we have $X_i^{p_i}=\ell_i({\bf T})\in S$ for all $1\leq i\leq n$. Now let us consider the ${\mathbb{Z}}$-graded ring $R^{[{\mathbb{Z}}{\vec c}]}$.

\[prop:standardfrorm\] We have an isomorphism of ${\mathbb{Z}}$-graded $k$-algebras $$R^{[{\mathbb{Z}}{\vec c}]}\simeq {\rm T}_{p_1}(S,X_1^{p_1})\otimes_S\cdots\otimes_S {\rm T}_{p_n}(S,X_n^{p_n}),$$ where we regard ${\rm T}_{p_i}(S,X_i^{p_i}):={\rm T}_{p_i}(S,(X_i^{p_i}))$ as a ${\mathbb{Z}}$-graded $k$-algebra whose degree $\ell$-part consists of elements such that lower diagonal entries are in $S_\ell$ and upper diagonal entries are in $X_i^{p_i}S_{\ell-1}$.

Let $I:=\{\sum_{i=1}^na_i{\vec x}_i\mid 0\le a_i\le p_i-1\}$ be a complete set of representatives of ${\mathbb{L}}/{\mathbb{Z}}{\vec c}$. Let $S_i:=k[X_i^{p_i}]$ for $1\le i\le n$. For ${\vec x}=\sum_{i=1}^na_i{\vec x}_i$ and ${\vec y}=\sum_{i=1}^nb_i{\vec x}_i$ in $I$, we have $$\begin{aligned}
R_{{\vec x}-{\vec y}+{\mathbb{Z}}{\vec c}}=(\prod_{i=1}^nX_i^{a_i-b_i+\epsilon_ip_i})S=(X_1^{a_1-b_1+\epsilon_1p_1}S_1)\otimes_k\cdots\otimes_k(X_n^{a_n-b_n+\epsilon_np_n}S_n),\end{aligned}$$ where $\epsilon_i:=0$ if $a_i\ge b_i$ and $1$ otherwise. Thus we have isomorphisms $$\begin{aligned}
R^{[{\mathbb{Z}}{\vec c}]}&=&\bigotimes_{i=1}^n\left[
\begin{array}{ccccc}
S_i&X_i^{p_i-1}S_i&\cdots&X_i^2S_i&X_iS_i\\
X_iS_i&S_i&\cdots&X_i^3S_i&X_i^2S_i\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
X_i^{p_i-2}S_i&X_i^{p_i-3}S_i&\cdots&S_i&X_i^{p_i-1}S_i\\
X_i^{p_i-1}S_i&X_i^{p_i-2}S_i&\cdots&X_iS_i&S_i
\end{array}\right]\\
&\simeq&{\rm T}_{p_1}(S_1,X_1^{p_1})\otimes_k\cdots\otimes_k{\rm T}_{p_n}(S_n,X_n^{p_n})\\
&=&{\rm T}_{p_1}(S,X_1^{p_1})\otimes_S\cdots\otimes_S {\rm T}_{p_n}(S,X_n^{p_n})\end{aligned}$$ of ${\mathbb{Z}}$-graded $k$-algebras.

Let 